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is minimal, it follows
w
p = (,lx)x. Proposition 5.5.6.9 ( 0 ) Let E be a complex Hzlbert space and C'subalgebra of L ( E ) actzng irreducibly on E . If
F
a
3 n K ( E ) # {O) > then K ( E ) c 3 .
Step 1
There is a finitedimensional vector subspace
F of E with (0) # F # E and
TF
E3
There is a u E (Re3) n K(E)\Cl
By Theorem 5.5.6.1 a
3
e , there is an orthonormal basis A of E and an
f E co(4) such that f is not constant and
Takc
tr
E f(A)\{O) anti p i t I
B
:=
f ( a ) , F := Ker (01  U)
5.5 Orthonormal Bases
25.7
By Theorem 5.5.6.1 a + b : F is finitedimensional and by Theorem 5.5.6.1 a + f & j (arid Proposition 5.5.1.7),
Step 2
3 2 E E\{O), (.1x)x E 3
Let F be a minimal vector subspace of E with the properties described in Step 1 . Given u E 3 ,define
and put
Since X F E F ,it follows that ? is a subalgebra of C ( F ) . By Proposition 5.3.2.25, 3. acts irreducibly on F . By Step 1, F is onedin~ensional.Hencc
for every x E F , llxll = 1 Stcp 3
Given x, y E E , if (.lx)x E
F ,then (.lx)y E F
We may assume that
By Proposition 5.3.2.20 b) , x is cyclic for in 3 such that
F .Hence there is a sequence (u,),~Lv
lim u,x = y
n+m
We get (.lz)y = lirn (.Ix)unx = lim n+m
Step 4
n+m
K(E) c F
By Stcp 2: 3, and Corollary 5.5.1.11,
IL, o
((.lx)x) E 3
254
5. Hilbert Spaces
Hence by Corollary 5.2.3.4, r(E) c 7.
m
Remark. A similar result does not hold in the real case as the following example shows: E.=IR 2
C o r o l l a r y 5.5.6.10
,
jr:=
~
a
,
9
Let E be a complex Hilbert space and J~ an order a 
complete C*subalgebra of s
acting irreducibly on E . If P r ~ \ { 0 }
has a
minimal element, then c 7. Let p be a minimal element of Pr~\{0}. By Corollary 4.3.2.12 a =, d, pjrp is onedimel~sional. So, by Proposition 5.3.2.20 b), there is an x c E with p
<I~)~.
Hence p 6 K~(E) and by Proposition 5.5.6.9,
r(E) P r o p o s i t i o n 5.5.6.11
7.
m
( 0 ) Let E be a complex C*algebra and (H, tp),
(K,~p) irreducible representations of E such that Ker~ = Kerr
p(E) N K~(H) # {0}, then the two representations are equivalent. Put F := K e r ~ = K e r r 1
G := ~ (LS(H)).
5.5 Orthonormal Bases
Let cp and and let
255
4be the factorizations through G/E' of ylG and $JIG,respectively, + G / F
q :G
be the quotient map ( F and G are closed ideals of E ). By Proposition 5.5.6.9, K(H)
c Y(E)>
so that
cp : G / F 4K ( H ) is an isometry. Take < , q E K , < # O , a n d ~ > O . T a k ex ~ G \ F . T h e r e i s a< o € K such that (*x)
# 0.
By Proposition 5.3.2.20 b), there are y, z E E such that II(*Y)(*x)~o 711 <
E
5,
E
I(*zX

< 2(llzll llyll + 1)
We get that  (@Y)(*x)
<
Since yxz E G , it follolvs that is cyclic for $(G). By Proposition 5.3.2.20 a), is an irreducible representation of G . Hence
4
is an irreducible representation of K ( H ) . By Theorem 5.5.1.24, this representation is the identical representation. Hence there is an isometry
5. Hilbert Spaces
256
for every
7)
E K ( H ) . We get
(*x)

0
u = (*qx)
0
u = u 0 (Lpqx) = u 0 (cpx)
for every x E G . Take
X E
E and ~ E G We . have x y ~ G , s that o
Take ( E H and g E H\{O) . Since g is cyclic for cp(G) (Proposition 5.3.2.20
b ) ) ,there is a filter
5
on G such that
I;?(cpy)o = t We get
Hence ( H , y) and ( K ,tb) are eqriivalent.
W
Corollary 5.5.6.12 ( 0 ) Let H be a cornplez Hilbert space, E a C * subalgebra of L ( H ) containing K ( H ) , and cp : E t E and zsometry of C* algebras. T h e n there is o unitary operator u o n H such that for each x 6 E ,
cpx = uxuI u is unique up to a factor of absolute value 1 .
If 1;. derlotes the identity map E + E , the11 ( H , cp) and ( H ,Q) are irreducible representatio~~s of E . By Proposition 5.5.6.11, the two representations are equivalent. The uniqueness follows from Proposition 5.4.1.7.
5.5 Orthonormal Bases
257
Corollary 5.5.6.13 ( 0 ) Let (EL),,, be a family of (complex) Hilbert spaces, E the C*d i r e c t s u m of the family (K(E,)),,, , and F a C * subalgebra of E such that: 1) for every
L
E
I , the representation
is irreducible, 2) i f
L,
X E I are distinct, then p, and px are not equzvalent representations.
Then R e E c F ( E = F ) . Take distinct L , A E I and a E K(E,) . By Proposition 4.2.4.18, it is sufficient to show that there is an x E F with
x,=a,
xx=O.
By l ) , 2), and Proposition 5.3.6.11, Ker 9,# Ker p x Since K ( E A )is simple, Ker px is a maximal closed ideal of F . Hence p,(Ker p A ) is a nonzero closed ideal of K(E,) (Theorem 4.2.6.6). Since K(E,) is simple,
Hence there is an x E F with 2,
= p,x = a , ,
xx
= (FAX = 0
5. Hilbert Spaces
258
5.5.7 Examples of Real C*Algebras Definition 5.5.7.1 ( 0 ) A n involutive Hilbert space is a Hilbert space H endowed with a conjugate linear involution
H
+H ,
(the involution of H )
such that
(flri) = 0 for all <,7 E H . For e v e y set T , @ ( T )will be considered a n involutive Hilbert space with respect to the canonical involution
If G and H are involutive Hilbert spaces and x E L(G,H ) then (by Definition 2.3.1 . l ) z:G+H, < e x ? .
Every involutive Hilbert space is an invol~~tive Banach space with respect to its canonical norm.
a) x E L(G,H )
(0)
Let F, G , H be znvolutive Hilbert spaces. 5 E L(G, H ) , )1Z11 = llxll, Z = x , Z* = x* .
Proposition 5.5.7.2
b ) L(G,H ) endowed with the znvolution C(G,H ) + C(G,H ) , x
HZ
is a n involutive Banach space.
c) x ~ L ( F , G ) , y ~ L ( G , H ) + y o z = y o 5 . d) The map
C ( H ) + C ( H ), x
M
z
zs a conjugate znvolution of the C*algebra C ( H )
e) Assume IK = C and let E be a n C'subalgebra of C ( H ) such that XEETEE. Then the map
i +x E~ , (x,y)ct(x+iy,~+ijj) is a n zsomorphism of C' algebras.
5.5 Orthonomal Bases
a ) Take ( < , q )E G x H . Then
 (2<11,)
= ( Z l v ) = (xZl7) = (Clx*?) =
Hence
The other relations are easy to see. b) follows from a ) . c) follows from Proposition 2.3.2.22 i). d ) follows from a ) arid c). e) follows from d) and Theorem 4.1.1.8 e) . Proposition 5.5.7.3
Let E be an involutive Banach algebra,
a conjugate involution, and x1 E EL such that 
xl(Z) = xl(x)
for every x E E . In the sequel we use the notation of Theorem 5.4.1.2. a ) Z E F for every x E F . Put
for every X E E I F and x E X . b) X, Y E E / F
+ (XIP) = (XlY) , IlX11 = llXll,
=X.
c) E / F endowed with the involution

is an involutive normed space. We extend this involution to E / F .

d) E I F is an involutive Hilbert space.
5. Hzlbert Spaces
260
a ) We have

f (F) = xl(Z'Z) = xl(s'z) = xl(x*x) = 0
b) For (x, y ) E
Xx Y,
c) Since the m a p is obviously a conjugate linear involutive the assertion follows from b). d) follows from b) and c) . e) Take Y' E E I F and y E 1 ' . Then
so that cpx = gJZ
Proposition 5.5.7.4 ( 0 ) Let H be an involutive complex (real) Hilbert space. Then H has an orthonormal baszs A such that
for every
<EA
Let 2L be the set of orthonormal sets A of H such that
<
E A . Order 2l by inclusion and take a maximal elemrnt A E I I for every (Zorn). Assume A is not an orthonormal basis of H . Then there is a [ E A' with ((((( = 1. Since
for every 71 E A , it follows
F
A'. Let 0 E IR such that (
Pllt
5.5 Orthonormal Bases
Then 7 E A" ,
and
Hence A U 17 E 2l and this contradicts the maximality of A . Remark. It follou~sfrom this proposition that every involutive complex Hilbert space is isomorphic to P2(T) for some set T . Proposition 5.5.7.5
(0)
Let H be a n involutive Hzlbert space and
a) E is a closed involutive vector subspace of L ( H ) b) If 3 is a n upward dzrected set of L ( H ) contained i n E and if x is its supremum i n L ( H ) , then x E E . c) If x is a self normal operator on H belonging to E , then f (x) E E for every bounded Bore1 fiinctzon j on ~ ( x. ) d) If IK = C and if x is a unitary operator o n H belonging to E , then for each n E K there zs a unitary operator y o n H belonging to E such that y" = x . Moreover, there zs an orthonormal baszs A of H such that xf = for every = A .
<
<
e) If IK = C and if x, y are unitary operators o n H belonging to E , then there is a unitary operator 2 o n H such that
262
5. Hzlbert Spaces
a) E is obviously a vector subspace of & ( H ) . By Proposition 5.5.7.2 a), it is closed. b) Take <,71 E H and let 5 be the upper section filter of F .By Theorem 5.3.3.14 c),
Hence x' = 2 and x E E . c) By Theorem 5.3.3.14, L ( H ) is Corder complete, so by Corollary 4.3.2.5 b), f ( x ) is welldefined. We denote by B the set of bounded Borel functions on a ( x ) and put a , : = {f E B I f(x)*
=fo}.
If P E IK[s,t ] , then by Proposition 5.5.7.2 a),c) (and Corollary 4.1.3.8) the map a(x)
+ IK,
a
P(a,Z)
belongs to Boo.By a) and the WeierstrassStone Theorem, C(a(x)) C & . If ( fn)nElh. is an increasing (decreasing) sequence in Bo with supremum (infimum) f in B , then by b), f E Bo . Hence Bo = B . d ) By Corollary 4.3.2.5 h) (and Theorem 5.3.3.14), there is a bounded Borel function f on a ( x ) such that f (x) is unitary and f (x)" = x . By c), f (x) E E . In particular, there is a unitary operator y on H belonging to E such that y2 = x . By Proposition 5.5.7.4, we may assume H = e2(T) for some set T . b). We Then (yet)ttr is an orthonormal basis of H (Proposition 5.5.5.9 d have
*
x P t = xYet = xy8et = y2y*et= yet for every t E T . e) By d ) , there are unitary operators xo, yo on H belonging to E such that
Put
Then z is a unitary operator on H and we have
5.5 Orthonormal Bases
Proposition 5.5.7.6 ( 0 ) Take operator on H := P2(T),and
IK
=
269
C . Let T be a set, x a unitary
Then the following are equivalent.
If these conditions are fulfilled and i f E is a C*subalgebra of L ( H ) such that u ( E ) C E , then
F
:= { a E
E ( a= u ( a ) )
is a real C* subalgebra of E such that the map
is an isomorphism of complex C*algebras. Given a E L ( H ) , we have (by Proposition 5.5.7.2 d)) u 2 ( a )= x
~

x =*xZae*xf .
I f a ) is fulfilled, then xf=fl,
?x'=fl,
so that u 2 = 1 . If u2 = 1 , then (xZ)a = a(xZ) for every a E L ( H ) . Hence XZ E C1 (Corollary 5.3.2.14). Putting a := Z' in the above equality, we get
Hence there is an cu E
R such that
264
5. Hilbert Spaces
Hence Z = ax'
from which it follows that cr E (1, + I ) . BY b),
Hence the last assertion follows from Proposition 2.3.1.43 a ) (and Corollary 4.1.1.21, Proposition 4.1.1.24).
Proposition 5.5.7.7 ( 0 ) Assume IK = C .Let T be a set and x, y unitary operators on H := e2(T) such that
in H such that if we put
a ) There is a family ([,),,I
C(,,O):=EL, for every
L
E
C(L,I)
:=x<,
I , then (
b) If T zs finite then Card T 2s even. c) There zs a unitary operator z on EI such that
a ) Take < , q E H . \Z1e have (zE1q) = (E~x'v)= (?\%V)= (Elq) = (x5il<), 
xx< = xz< = xx*[ = [. It follows that
Let 0 be the set of orthonormal sets A of H such that [lx5j for all [, 17 E A . Since f2 is inductively ordered by inclusion, it contains a maximal
5.5 Orthonormal Bases
265
element A (Zorn's Lemrna). Assume A u {xf I ( E A) is not an orthonormal basis of H . Then there is an
with 11q11 = 1 . By the above remark, A u { ~ )E 0 ,contradicting the maximality of A . Hence A U {xf 1 ( 6 A ) is an orthonormal basis of H . By the above remark, is a family in H such that (Cx),,,,~,,,) is an orthonormal basis of H . b) follows from a). c) Let (F,),,, arid ( v ~ ) ~ be ~ ,families , in H with the property desc:ribed in a ) with respect to x and y , respectively. We may assume that I = L (Corollary 5.5.2.2). Let z be the unitary operator on H , defined by
for every
L
E
I (Propositions 5.5.1.22 and 5.5.5.9 c
+ d ) . Take
L
E I . Then
Hence
Proposition 5.5.7.8 ( 0 ) Assume IK = C . Let T be a set and s,y unitary operators o n H := t 2 ( T )such that z = z* ,
ij = y*
(resp. Z = x* ,
Let E be a C*subalgebra of L ( H ) such that zEz' = E for any unitary operator z o n H and such that a~E==+ii€E.
= ye)
5. Hilbert Spaces
266
Put
E, := { a E E I xtix* = a ) , EY := { a E E I yay* = a ) T h e n there is a unitary operator z o n H such that
and the m a p
E,
+E y ,
a Hzaz*
is a n isomorphism of real C *algebras (Proposztion 5.5.7.6) By Proposition 5.5.7.5 e ) (by Proposition 5.5.7.7 c ) ) , there is a unitary
operator z on H such that z x = yz. Take a E E . Then 
yzaz* y* = yzhz'y*
=
zxiix'z* ,
so that ( a E E,)
(a = xhx*) e(zaz' = yzaz'y')
++
(zaz* E E,)
Since the map E+E,
azaz*
is an isomorphism o f complex C*algebras (Proposition 4.1.1.24), it follows from the above that the map
is an isomorphism o f real C'algebras. Corollary 5.5.7.9 ( 7 ) Let T be a set and HK := e2(T,IK),(i.e. the Hilbert space H over IK ). Let x be a unitary operator on Hc such that Z = x* and put
Then the real C*algebras E and E n K(Hc) (Proposition 5.5.7.6) are isomorphic to L ( H R ) and K ( H R ) , respectively.
5.5 Orthonormal Bases
By Proposition 5.5.7.8, E (resp. E n K ( H c ) ) is isomorphic t o
(resp. { a E K ( H c ) 1 a = 3) ) .
{a E L(Hc) I a = Z )
If we identify H R canonically with a subset of H c , then for a E C ( H c ) a=Za(HR)c
HR.
Hence E (resp. E n K ( H c ) ) is isomorphic t o L ( H I R ) (resp. K ( H R ) ). Proposition 5.5.7.10 define
(0)
Let
IK = C and let T be a set. Given t
t1 := e(t,o),
. ET ,
t" := e (t.1) .
For s, t E T and cp E IR , define
{
5:t
:= e'?(.ltl)sl
+ e'?
Y$
:= e'?(.ltl)s"

(.lt")sl'
e'?(.lt")sl
Further, let x be the unztary operator on H such that
for all t E T (Propositzons 5.5.1.22 and 5.5.5.9 c of L ( H ) such that
+ d),
E a C*subalgebra
and put
F ( E ) := F := { a E E I XZX*= a ) . a ) F is a real C*subalgebra of E such that the map ;t
E,
(a,b)cta+ib
is an isomorphism of complex algebras. If E is simple then F is .simple and purely real.
5. Halbert Spaces
268
b) For all q, r, s, t E T and p, I) E IR,
{
x:,
= cos
yx;,
y$ = cos
y:t
(a*xLV =
7
+ sin y x l t + sin y yf, ,
(yrt)' = YZ
xfrx!t = 6,,x::',
= 6,,x,,
Y:,Y$
xfry$ = 6rs~;'"
7
v+*
yrrx!t = brsy$+*,
1
c) For every t E T with x i E E , x& is a minimal element of P r F\{O) .
d ) For every t E T with xf, E E the map
M + X:~FX\, cr+pi+ y j +6k ++
ax: + p x : , +yy:, +6yp,
is an isomorphism of real C' algebras (Example 4.1.1.31). e)
If there is a t
E T with x: E E
, then F is a purely real C*algebra
f ) Assume there is a t E T with x i E E . If S is a set and
G := K ( e 2 ( S ) )(resp. G
:= K ( e Z ( S )nmS) )
then F zs not zsomorphzc to G . a ) We have
By Proposition 5.5.7.6, F is a real C'subalgebra of E and the map
is an isomorphism of complex C*algebras. By Proposition 4.3.5.3 a is simple then F is simple and purely real. b) LVe have
x:, = (cos
X P
+ i sin 2
+ b , if
( ) s t+ <:or  i sin  (.lt")s" 2
"7
(
7
"7
=
E
269
5.5 Orthononnal Bases
"'P = cos ((.lt')sl
2
"P (z(.lt1)s'  i(.ltl')s") + (.I t")srl) + sin 2
"9 6 = (cos 1 + i sin
=
2
"'P
= cos ((. ltl)s" 
(. lt")sl)
2
"P (z(.l tl)s" + i(.l ttl)s') = + sin 2 "P + sin y,, 2
= cos y,,
2
I
By Proposition 5.3.2.13 b),
i
(xyt)*= e'? (.ls1)t' + e'? (.Isl')trl = xkv , (y,:)'
I IS")^'
= e'':

(.ls1)tt1= ylP, .
By Proposition 5.3.2.13 e), x ~ ~ r=: (el?(.r')q'
+ e'%(.rrl)q")
(e'?(.lt')sl
(sllrl)(.~tl)ql+ e'V ( s " j T ~ ~ ) ( . ~ t= ~ ~brs)z:*q "
= el
ygy; = (e'?(rl)q"  e'? (.lr")ql) = e'
(*u) 2
(.ltl)r"
 1 

(
2
e  ' 7 (.ltll)sl) =
"(*u)
(sl\rl)(. 1 tl')ql
~ " = 2(el?

,
(sl~~l)(~~t")q"el~(sll~r'l)(~~tl)ql=brsz~~;w,
z ~ v ,= $ (er%(.r~)q'+ r  ' ~ ( . / T l l ) q " )(elB(.lt/)sll 
+ e'Y(.ltl')s")

+ el+ *,"I P+*)
~(!8+Ui)
= ef7 (s'Irr) (. Itl)q"  ef&
(s"ITb) (. Itl')ql = brS *::y

(.I<)v = (.I?)q, 0
x* = (.(xf)x7j
(Proposition 5.3.2.13 c),d)). It follows that
,
(.jtl),Tl + e  ~ 7 ( . 1 t ~ ~ )=s ~ ~ )
For <,?1 E H ,
x0
i17(.jt1~)s1) =
(s"~rt1) (. 1t1)q1'= brsy$'
IT^)^')
(.lrt)qu  e'?

.
5. Hilbert Spaces
2 70
c) By b), x: E Pr F . Take p E Pr F\{0} with p 5 x i . Then p = px;tp = (.lptt)pt' + (.lpt")pt" (Corollary 4.2.7.6 a
+ f, Proposition
5.3.2.13 c),d)) and
for all s E T\{t) (Corollary 4.2.7.6 a a , lj, y, 6 E C such that
+ c). By Proposition
5.3.2.4, there are
+ Pt" pt" = yt' + 6t". pt' = at'
Thus p = (./at1+ Dttl)(at' + pt")
+ (.I$'+ bt")(yt1+ at") =
+(pa + Ty)(.lt")tl + (/Dl2 + 1612)(.lt")t". Since p E F , p
+ ly)2)(.)t1r)t'' (a3+ y8)(.lt")t1(@ + 67)(.t')t1' + (IPI2+ 16I2)(.1t')t'
= xpx* = (1a12

(Proposition 5.3.2.13 c),d)),so that
+
Ia12 IyI2 = llj12
+ 1612, 5 P + 76 = (@ + 67)
and therefore
+
p = (laI2 IYl'')z;
+ (ap + 76)(.lt1)t"

alj
+ 76(.ltr1)t'.
Furthermore, P = p* = (la['
+ Iyl2)x: + CYP+ y6(.lt")tt
(Proposition 5.3.2.13 b)), so that
sp+7a=o

( a p + 76)(.lt1)t"
5.5 Orthonomal Bases
and therefore P = (laI2+ Ir12)xPC,
1=l
l ~ l l= (laI2+lr12)11x~tll= 1012+ lrI2,
p = xPt . d ) Put
a:=xft,
b:=x:,,
c : = ~ , ' , , d:=ypt.
By b), a E P r F ,
b2 = c2 = d2 = a
Take y E F . By Proposition 5.3.2.13 c),e), there are a : P, y , 6 E C ,such that
aya = a a + Pb + yc
+ 6d
Since aya E F , by b),
aya=xayax* = ~ i a + p b + y c + 6 d . Since a , b, c, d are linearly independent, a , P, y , 6 E R . Hcncc aFa is the vector subspace of F generated by a, b, c, d and therefore the map
is an isomorphism of involutive algebras. e ) Assume thc contrary and put
2 72
5. Hilbert Spaces
for every y E E , i.e. u2 = 1 . By Proposition 2.3.1.43 d), there is a linear map v : E + E , such that
and
v ( y z ) = ( v y ) z = y ( v z ) , vy' = (vy)* for all y, z E E . Then for y E F ,
vy=vuy=uvy,
vyE F .
Hence v ( F ) c F . For every y E F ,
v (x:t~x:t)= x:t(vy)x::
7
so that
v (xftF X : ~ )= x:~F X ;
.
By c), we may replace X ~ ~ F by X :El. ~ Then
vi=
= (vi)*, vj=vj* = (vj)*.
Hence there are a , 13 E R with
vi = a l ,
v j = pl
which is a contradiction. f) Let v : F t G be an isomorphism of real C*a1gebras.B~c), sft is a minimal element of Pr F\{O} . Then ~ ( x : is ~ a) minimal element of PrG\{O) . By Proposition 5.5.6.8, there is a E e 2 ( S ) such that ~ ( x ! = ~ )(.I<)<. Then V ( X ~ ~ FisXisomorphic ~,) to (.l<)
<
Proposition 5.5.7.11
(0)
We assume IK =C. Let T be a set. Put
H : = ~ ~ ( T )H, R : = H ~ R ~ .
5.5 Orthonormal Bases
Let E be a C*subalgebra of C ( H ) containing K ( H ) such that zEz* = E for every unitary operator z on H and aEE==+iiEE Further put
and let F be a real C*subalgebra of E for which there is an isomorphisnl of complex C *algebras v : F + E such that
for every a E F . a)
If u is a conjugate involution on E , then there is a unitary operator x on H such that
for every a E E and
u is unique up to a factor of absolute value 1
b)
There zs a unitary operator x on H such that
F = { a E E I xiix* = a } , x* = Z
or x * = Z,
and v ( ( a ,b)) = a + ib 0
for every ( a ,b) E F . x is unique up to a factor of absolute value 1 .
c ) a ( H I R )C HR for every a E G . For a E G, define

a : HR
Put

+ H R ,
<
a<.
G:={Z~~EG}.
5. Hilbert Spaces
d)
2. zs a C'subalgebra
of L(HR) containing K(HR) such that zGz* = 2.
for every unitary operator z on HR . e) If E = L ( H ) (resp. E = K ( H ) ) then
G = L(HR)
(resp.
G = K(HR)).
f) If Z = x* then there is a unitary operator z on H such that z G * E
for every a E F and the map

F
a ~ z a z *
is an isomorphzsm of real C'algebras. g) If x* = Z then there is a unitary operator z on H such that
z F z W= F ( E ) (Proposition 5.5.7.10) and the map
F
+F ( E ) ,
a+ zaz'i
is an isomorphism of real C*algebras. a ) The map E+E,
aiZ
is an isomorphism of complex C*algebras. By Corollary 5.5.6.12, there is a unitary operator x on H , unique up to a factor of absolute value 1 , such that
for every a E E . By Proposition 5.5.7.6 b =. a,
b) By Proposition 2.3.1.43 b), there is a conjugate involutive u on E such that
By a), thcrc is a unitary operator x on H , unique up to a factor of absolute value 1, such that
5.5 Orthonoma1 Bases
and
~ = x *or z =  x * . By Proposition 5.5.7.6, the map
is an isomorphism of complex C*algebras. It follows
v ( ( a ,b ) ) = a + ib
F
for all ( a ,b) E . C) For HR
<
so that a< E
HR
arid
a(HR) C HR .
d) For b E L ( H R ) , define
Take b
C ( H R ) . Then
IEL(H).
0
0
b=b,
0
b=b.
Hence if E E , then b E G . It is easy to see that G is a Casubalgebra of L ( H m ) . Since E K ( H ) whenever b E K ( H R ) , we deduce that K ( H m ) C G .
i
NOWtake b E
g . Then
i E G . Since
is a unitary operator on H ,
Since
we get that 0
A
zbz* E G , zbz* E
e,
zez* = G
5. Hilbert Spaces
2 76
e) is easy to see. f) By Proposition 5.5.7.8, there is a unitary operator z on H such that
and the map
is an isomorphism of real C*algebras. For a E F ,
ZG* E G and the map
FG.

azaz*
is an isomorphism of real C'algebras. g) follows from Proposition 5.5.7.8
Corollary 5.5.7.12
(0)
Assume that IK = C . Let S, T be sets and let := e2(T), respectively, such that
x,y be unitary operators o n e 2 ( S ) and H
Further, let E be a C*subalgebra of L ( H ) contaznzng K ( H ) such that
for every unitary operator z o n H and such that
Put
F := { a E E I yay' = a ) ,
G
:= { b E
K ( e 2 ( s ) 1) X ~ X *= b} .
If either S or T zs nonempty, then the real C *algebras F and G (Proposition 5.5.7.6) and F and K ( e 2 ( S ) ) are not isomorphic. By Proposition 5.5.7.11 d),c),f),g), F is isomorphic to F ( E ) of Example 5.5.7.10 and G is isomorphic to K(Y2(S)n R").The assertion follows from Proposition 5.5.7.10 f ) .
5.5 Orthonormal Bases
277
Corollary 5.5.7.13 ( 0 ) W e adopt the hypotheses and notation of Proposition 5.5.7.11. Let CE be the class of real C *algebras A such that the complex 0
C*algebras A and E are isomorphic.
a ) If n := C a r d T is an odd natural number, then every element of & is isomorphic to R,,,, . b) If C a r d T is not an odd natural number, then & decomposes into precisely two zsomorphism classes: those of the first class are isomorphic to G while the elements of the second are isomorphic to F ( E ) (Proposition 5.5.7.10). If in addition E = L ( H ) (resp. E = K ( H ) ) then 5 = L(HN) (resp. = K(Hm)).
c
a & b . Let .F be the set of real C' subalgebras B of E such that the map
is an isomorphisni of complex C * algebras. By Proposition 2.3.1.43 c), every element of CE is isomorphic to an element of 3 . Tako B 3 . By Proposition 5.5.7.11 b), there is a unitary operator x on H such that
and

If x* = 3 ,then by Proposition 5.5.7.11 f), B is isomorphic to G . If x* =  3 , then by Proposition 5.5.7.7 b), CardT is not an odd natural number and by Proposition 5.5.7.11 g), B is isomorphic to F ( E ) . Hence if n := C a r d T is an odd natural numbor, then by Proposition 5.5.7.1 1 e), each element of & is isomorphic: to b,, By Corollary 5.5.7.12, G and F ( E ) are not isomorphic. Hence if Card T is not an odd natural number, then CE decomposcs in precisely two isomorphism classes: the elements of first class are isomorphic to G while the elements of the second are isomorphic to F ( E ) . The last assertion of b) was proved in Proposition 5.5.7.11 e). Corollary 5.5.7.14 Let E be a complex C'algebra for which there is a faithful irreducible representation (H, p) such that:
2 78
5. Halbert Spaces
Let further & be the class of real C *  algebras for which E is isomorphic to their complexifications. Put
F

for all F, G E & . Then
G :e F and G are isomorphic

is a n equzvalence relation o n & and Card &/ E { 1 , 2 ) .
Since the representation ( H ,cp) is faithful, we may identify E with cp(E) . By 1 ) and Proposition 5.5.6.9, K ( H ) c v ( E ) and the assertion follows from Corollary 5.5.7.13 a),b) .
Corollary 5.5.7.15
(0)
W e assume lK = C . Let T be a nonem.pty set,
n E Pi,and let & be the class of real C ' a l g e b r a s E such that E is isomorphic to K ( H ) " . W e set E

F
:
E and F are isomorphic
for all E , F E & , E ( p , q ) := K ( H ) Px K ( H I R ) qx F ( K ( H ) ) "  ~ ~ ( ~2 +~9
5 n),
for all p,q E N U { 0 ) , and
If Card T is an odd natural number (is not a n odd natural number), then
5.5 Orthonormal Bases
and the map
is bijective. I n particular 1 Carci (el) =  ( n 2
(card
+ 1 + cor2 "" 2 )
(el) = 41 ( ( n + 2)'
 sin2
2
By Corollary 5.5.7.13 a), if C a r d T is an odd natural number, K ( H R ) is the only real C*algebra that has complexification isomorphic to K ( H ) . By Corollary 5.5.7.13 b), if C a r d T is not an odd natural number then there are exactly two nonisomorphic real C*algebras, namely K(HlR) and F(IC(H)), the complexification of which are isomorphic: with K ( H ) . Now we use Proposition 4.3.5.8. If Card T is an odd natural number then
and
1 Card (El) =  ( n 2
+ 1 + cos2 )nn2
If C a r d T is not an odd natural number then
and Card (t/)= 1
nl (n k)n + 51 x (n X + 1 + cos2 )= 
k=O
1 =  ((n
4
+ 2 ) ~ sin2 "" 2 ) 
,
Example 5.5.7.16 W e use the notation and conventions of Proposition 5.5.7.10. Let C denote the set of all
(f,9,(P, $) E
myXTx IR:XT x IRTXTx R T x T
5. Hilbert Spaces
280
such that the family
is summable and for each ( f ,g, cp, q!~)E C let z( f , g, cp, 111) denote the sum of the above famzly. a)
If ( f , g , c p , $ ) E C , q , r , s , t ~ T and , O E R ,then
+
2 ( j 2 ( s .t ) s i n 2 ( l ( 0 y ( s , t ) )+ g2(s.t ) sin2 (0 2 2 X
Hence z ( f , g, cp, I,!J)= 0 implies that f b)
=g =0
+ d ( s .t ) ) )xi.
=
.
Take f , g, h, k E KtTXT such that the family
( f ( s ,t)x;t + g ( s , t)x:t + h ( s , t)y,Ot+ k ( s ,t ) ~ : t ) , , ~ ~ ~ zs summable. b l ) If the svm is 0 then: f=g=h=k=O. bS) The sum zs selfadjoint iff
h ( s ,t ) =  h ( t , s ) ,
k ( s ,t ) =  k ( t , s )
for all s , t E T . c ) If E = K ( H ) , then: cl)
F is the closed real vector subspace of E generate by {xrt I
8,t
E T ,cp E
IR) u {Y:
I s, t E T ,
E
w
and F =
{ ~ ( fg,, cp, $1 I ( f , s,(P,q ~E) z).
5.5 Orthononnal Bases
c2) F is simple and purely real.
( f , g, cp, $1
~ 3 )If

E C then z ( f ,g, cp, $) is selfadjoint i f f
f ( s ,t ) # 0 * cp(s,t ) + cp(t,5 )
( s , t ) = f ( t ,s ) ,
f
281
g ( s , t ) = g ( t , s ) , g(s, t ) # 0
0
* cp(s,t )  ~ ( ts ), .= 2
(mod 4 ) , (mod 4 )
for all s, t E T . d ) If T is finite, then Dim F = (2Card T)'
Cxi
and
, Dim Re F
=
2(Card T)'  Card T
is a unit of L ( H ) .
LET
a) Put z := z ( f , g , cp,+). B y Proposition 5.5.7.10 b),
8+v(s,t) @+tl(s,t) ~ : , ~ z ~ t )( ,~ =t)xqr 7 f + g ( ~t)yqT ,
=f
y:,z?/;,
8lp(s,t)
( s ,t)xpt
x:~zYP' = f ( s ,t)yqr8v(s,t)


g ( s , t )y, B ;Q(", L )
g ( s ,t ) ~ : ; ~ ( ~ ' ~ )
= f ( 3 , t)y,rQ+ds,O  g ( s , t)x,;B+Q("")
Y:.?~X;~
8 .
Y:~zYP,
x ~ . +~ ~ x =~ ~ =
0
2f ( s , t ) sin 5 ( 0
Y : ~ Z " : ~  X:~ZY:,
+ cp(s,t ) ) x &+ 2g(s,t ) sin :(0 + $ ( s , t))$, =
= 2g(s, t ) sin :(0
2 ( f 2 ( s :t ) sin2 5 ( 0 = f ( s ,t ) sin
+ $ ( s , t))x:, + 2f ( s ,t ) sin :(O + p(s, t ) ) y i ,
+ p ( s , t ) )+ g2(s,t ) sin2 :(0 + $ ( s , t ) ) ) ~ : =,
;(e + 4%~ ) ) ( X : , ~ Z X : , + Y : ~ Z Y ~ ) , ) 
 g ( s ,t ) sin ;(0 +*(st t ) ) ( ~ : ~ zx ~,8,~2Y:,). :~
Assume that z = 0 . Taking R such that K
sin  ( R 2 we see that
+ p ( s , t ) )# 0 ,
sin ?(R 2
+ + ( s ,t ) ) # 0 ,
282
5. Hilbert Spaces
f ( s ,t ) = g(s, 1 ) = 0 . Since s and t are arbitrary, f = g = 0 . b l ) For s, t E I R , define
Hence there are cp, $ E IRTxT such that
i
f = Fcos;cp
h = Gcos%$
g = Fsin%cp,
k=Gsin;$.
By Proposition 5.5.7.10 b ) , ~ ( st ), ~ ~ =' ~ F ("r ,t ) (cos m
2
x
:
+ sin 2
= h ( s , t)ysot+ k ( s ,t ) ) ~ :. ,
Hence ( F ,G , cp, $) E C and z ( F , G , cp,$)
=0.
By a ) ,
F=G=O. Hence
b2) By Proposition 5.5.7.10 b ) ,
C ( f ( s ,t)x9t + g ( s , t)x:, + h ( s ,t1y:t + k ( s ,t)y:t)* = = C ( f ( s ,t)xp,  g(s, t)x:,  h ( s , t ) ~ : , k ( s , t ) ~ : , )= s,tEl'
=
C ( f ( t 7s)xPt  g ( t , s)xjt  h ( t ,~ ) y : t k ( s 7t1y.L) .
s,lET
By b l ) ,the sum is sclfadjoint i f f
5.5 Orthononnal Bases
for all s , t E T . cl) Let G be the closed real vector subspace of E generated by {xrt ~ s , ~ E T , ~ ~ E R ) u { Y Z I ~ , ~ E T , ~ ~ E I R )
Then { z ( f , 9,cp, $1
I (f,g , P. *)
E
El C G C F
(Proposition 5.5.7.10 b)). Take z E F . By the last assertion of Theorem 5.5.5.1,
Hence by Propositiori 5.3.2.13 c),d),
Thus ( z t l J s l )= (ztl'ls")
,
(ztrlls") = (ztllsl)
(zt1(sI1)=  ( ~ t ~ ~ l s, ~ ()z t r l J s l )= (ztllsl')
for all s, t E T , so that
Given s , t E T , define ( f , g , cp, $) E
x 1~;~''
*TxT
IRTXT'
by
284
5. Hilbert Spaces
f ( s , t ) e ' : ~ ( := ~ ~(ztfls') ~) g ( s , t)e1T*(3J):= ( z t fIsff). Then
( f , g, cp, @) E C and
Hence
c2) By Corollary 5.5.1.13 b ) , E is simple so that the assertion follows from Proposition 5.5.7.10 a). c3) By Proposition 5.5.7.10 b),

g ( t , r) (cos % ( t , s ) d 2
K
+ sin +(t. 2
s ) ~ . : , ).
By b l ) and Proposition 5.5.7.10 b), z( f , g , cp, @) is selfadjoint iff for all s , t E T ,
f ( s , t ) cos ;cp(s, t ) = f ( t , s ) cos ; p ( t , s )
f ( s ,t ) sin :cp(s, t ) = f ( t , s ) sin ;cp(t, s ) g ( s , t ) cos ;dJ(s, t ) =  g ( t , s ) cos ;dJ(t, S ) g ( s , t ) sin :@(s,t ) =  g ( t , s ) sin ;dJ(t, 3) ,
+
f ( s , t ) 2 = ( f ( s ,t ) cos :cp(s, t ) ) 2 ( f ( s ,t ) sin ; p ( s , t ) ) 2=
+
= ( f ( t ,S ) cos ; v ( t , s ) ) ~ ( f ( t , S ) sin :cp(t. s))' = f ( t , s ) ~ ,
g ( s , t)"
( g ( s ,t ) cos ;$(s, t ) ) 2+ ( g ( s ,t ) sin ; $ ( s , t ) ) 2=
= g ( t , s ) cos :@(t,s))'
+ ( g ( t ,s ) sir1 ;$(t, s ) ) =~ g ( t , s
) ,~
5.5 Orthonormal Bases
i.e.
f
( s ,t ) = f ( t ,s) ,
f
(s, t ) # 0 ===+ p ( s , t )
g(.%t ) = g ( t , s ) , g(s, t ) # 0
+~

0
(mod 4) ,
**(s, t )  *(t,s) = 2
(mod 4) .
( ts ),
d ) By Proposition 5.5.7.10 a ) , Dim F = 4(Card T)' . P u t n := Card T . By b 2 ) ,
By c l ) and Proposition 5.5.7.10 a),b),
C XI: LET
is a unit o f C ( H ) .
5. Hzlbert Spaces
286
5.6 Hilbert right C*Modules Hilbert C* modules are generalizations of Hilbert spaces in the sense that the scalar product takes its values in a C*algebra rather than in the ground field. In this section, E is always a C*algebra. 5.6.1 Some General Results
Definition 5.6.1.1 ( 0 ) A weak semiinnerproduct right Emodule is a vector space H endowed with a bzlinear map
H xE
+
H,
(<, x ) c,(x
(right multiplication)
such that
(e.1~ = F(xY) for all
<E H
and x, y E E and wzth a sesquihnear m a p
H x H
+ E , (<, q) * (<1q)
(innerproduct)
such that
for all < , q E H and x E E . H is called a semiinnerproduct product) right Emodule if i n uddition
(and ((I<) = O for all (,
E
==+ < = 0)
H , where E denotes the complexzfication of E .
(inner
5.6 Hzlbert right C*Modules
287
The above notions were introduced by Kaplansky (1953) for E unital, complex, and commutative and by Paschke (1973) and Rieffel (1974), for the general case. If E is not unital and E is the unital C*algebra associated to E then, by setting f 1 = f for every E H , H becomes a weak semiinnerproduct right Emodule. If E is unital then under a very weak hypothesis, which will be always fulfilled in this book the axiom
<
is automatically fulfilled (Proposition 5.6.1.2 f)). Let E be a complex C'algebra. If Eo denotes the underlying real C*algebra of E (Theorem 4.1.1.8 a)) then restricting the scalars of H to R we get a weak semiinnerproduct right Eornodlile. The axiom
is exactly the assertion that the above sesquilinear map is Hermitian. Proposition 5.6.1.2 ( 0 ) Let H be a weak semiinnerproduct right Emodule. Then for all x , y E E and f , 7, ( E H :
f) The map
is a semz norm. If it is a norm then for every mate unit 5 of E ,
<E H
and every approxi
5. Hilbert Spaces
288
g) If K = C then H is a semzinnerproduct right E module. h) For e v e y of E .
< E H , {x E El<x = [x* = 0 )
i) {x E El< E H j)
+ [x
is a hereditary C'subalgebra
= 0 ) is a closed ideal of
E.
The vector subspace of E generated by {((Iq) IC, 0 E H ) is an ideal of E.
= x'(SI0x
+ 2re(~*(Flv)x)+ Y*(V\V)Y.
It follows (F + VIE
+ 0) I (F + v1F + 0) + (F  vIF  71) = 2((FIF) + (017))) .
c) By b) and Corollary 4.2.2.3, 0I ((5  v1Fx  0) = xl(
Ill(~l~)l l 2re((~l71)~) ~*~ + (~10) If II(
:= (q(<),we scc
=
(
that
+ (~llrl),
5.6 Hilbert right C ' M o d u l e s
289
d ) follows from c) and Corollary 4.2.1.18. e) follows from b). f) By b) and d),
= X*(
(
for every x E E so that lim((x

2,3
(I(x  <) = 0
(Proposition 2.3.4.9), lim(x = ( . z,3
g) Take
[,TI
E H . By b) and Proposition 4.2.2.15 b),
(([I<)
+ (vlv)
= ((<
?
(vl<)  ((17)) =
+ ivI< + iv), O) E +;
.
h) Put
G := {x E El(x
= <x* = 0 ) .
I t is obvious that G is ax1 involutivc subalgebra of E . By e), it is closed, i.e. it is a C*subalgebraof E . Take x E E and y, z E G . Then
5. Halbert Spaces
290
so that yxz E G . By Proposition 4.3.4.6 b subalgebra of E . i) follows from e) . j) follows from a).
3
a , G is a hereditary C*
Proposition 5.6.1.3 Every weak semiinnerproduct right Mmodule (Example 4.1.1.31) is a semi znnerproduct right M module. Let H be a weak semiinnerproduct right Mmodule and let <,7 E H . Put
Then
(Proposition 5.6.1.2 d ) ) . We identify M with a unital real involutive subalgebra of C2,2as in Example 2.3.1.46. Then
By the above,
=
(m
2 0.
0
C2.2
and M with
5.6 Hilbert right C*Modules
By Example 4.2.3.5,
Definition 5.6.1.4 ( 0 ) A Hilbert right Emodule is a n innerproduct right E module for which the n o r m
(Proposition 5.6.1.2f ) ) is complete. If E is not speczfied, then we simply speak of a Hilbert right C*module. A (unital) Hilbert Emodule is a Hilbert right Emodule H endowed with a bilinear map E x H + H ,
(x,I) c,XI
(left multiplication)
such that
for all x,y E E and
I,17 E H
By Corollary 4.2.1.18,
so that by Proposition 5.6.1.2 e), a Hilbert Emodule is an Emodule with respect to its right and left multiplications.
292
5. Hzlbert Spaces
Let E be a complex C*algebra, F its underlying real C*algebra (Theorem 4.1.1.8 a)), and H a Hilbert right Fmodule. If E is unital, then H endowed with the scalar multiplication
becomes a complex vector space and so a Hilbert right Emodule. But if E is not unital the above extension of the structure of H is not always possible. Take
Then H endowed with the right multiplication
H x F ,H ,
( ( 2 ,a ) ,Y
) I+
(XY
+ cry, 0)
and with the inner product
is a Hilbert right Fmodule. Assume it is possible to extend the structure of H to a Hilbert right E module and put ( x ,a ) := i ( 0 , l ). Then ( 0 ,1) = i ( z ( 0 , l ) )= z(x,a ) = ( i x ,0 ) + ( a x ,a') and we get the contradiction
Example 5.6.1.5 ( 0 ) Let F be a nght zdeal (closed ideal) of E . Then F endowed with the maps

FxE+F,
(<,z)++<x,
F x F 4 E ,
(I,7l)
ExF+F,
(x,O++x<,
(and with the maps
F+F,
v*<
<++<*,
(Corollary 4.2.6.2)) is an innerproduct right Emodule (and an involutive unital Emodule).
293
5.6 Halbert right C' Modules
The first and the third map are bilinear, the second one is sesquilinear, and the last one is an involution such that F is an involr~tiveEmodule whenever F is a closed ideal of E . Take <,7 E F and x E E . Then
(x
5
11x112<*<= 11~11~(<1<)
(Corollary 4.2.2.3)).
Proposition 5.6.1.6
(0) 0
Assume IK = R . Let H be a (semi ) inner0
product right Emodule and E , H the complexifications of E and of the underlying real vector space of H , respectively. Then H endowed wzth the maps
0
is a (semi) inner product right Emodule called the complezification of H . 0
If H is a Hilbert right Emodule then H is a Hilbert nght Emodule such that s ~ P { I I < I1I1~71?12} 5 11(<>7)112 = ll((
for all ( 5 , ~ 6 ) H and the canonical map H x H real Banach spaces.
+H
5
(ll
is an isomorphism of
5. Hilbert Spaces
294
By Lemma 2.1.5.1 a),b), the map
is bilinear and
for all (<, v) E
fi
and (x, y), ( u ,v ) E
For all (<, v), (<, A) E
fi
i
0
Take (<, A ) E H . Then (.I<) and (.IA) are linear. By Lemma 2.1.5.3, the map
is linear. It follows, by the above result, that the map ((C, A)I.) is conjugate linear. Hence the map
is sesquilinear. We have
for all <, 7 E H and x E E . By Lemma 2.1.5.1 b),
h.
is a and (x, y) E By Proposition 5.6.1.2 g), for all (5, v), (C, A) E (semi) innerproduct Hilbert right Emodule. Now suppose that H is a Hilbert right Emodule. Then, by the above relation,
5.6 Hilbert right C'Modules
(Proposition 5.6.1.2 d)). By Proposition 4.3.6.1 c) , s ~ P { I I < Illvl12} I ~ ~ 5 Il(cl0 + (vlv)ll 5 ll(<7v)l12 0
It follows that the canonical map H x H t H is an isomorphism of real Banach 0 W spaces and H is a Hilbert right Emodule. Definition 5.6.1.7 ( 0 ) Let G, H be Hilbert right E modules. W e define LE(G, H ) t o be the set of maps u : G + H for which there zs a map u* : H + G such that (u
G and H are called isomorphic if there is a bzjective u E LE(G, H ) such that
for all (, 7 E G
Let F be a C*subalgebra of E such that
for all <,7 E G and <,q E H . Then G, H are Hilbert right Fmodules and
Proposition 5.6.1.8
(0)
Let F, G , H be Hilbert right Emodules.
a ) If u E LE(G, H ) , then there is a unique map u* : H
+G
such that
for all (<, 7) E G x H . u* is called the adjoint of u . It belongs t o LE(H, G ) and u** = u .
5. Hilbert Spaces
296
b ) LE(G,H ) zs a closed vector subspace of L(G,H ) and llull = IIu'II for every u E LE(G,H ) . c ) The map LE(G,H ) + C E ( H G , ) , u w U* is conjugate linear
g) Assume that F and G are (unital) Hilbert Emodules and for x E E
and u E L E ( F ,G ) put xu : F
+G ,
ux : F
+G ,
< H x(u<) < * u(x<)
Then xu, ux E L E ( F ,G ) and ( x u ) *= u*x*,
(ux)*= xfu*
for every ( x ,u ) E E x L E ( F ,G ) , and L E ( F ,G ) endowed with the maps
{
E x L E ( F ,G ) + L E ( F ,G ) , ( x ,U ) eX U L E ( F ,G ) x E
+ L E ( F ,G ) ,
( u ,X )
ct
ux
is a (unital) Emodule. In particular L E ( F ) is an involutive (unital) Ealgebra with respect to the involution L E ( F )+ C E ( F ) , u HU* . h ) Assume that H is a Hzlbert (right) Emodule and put
( Z :H
+H ,
[ct<x)
for every x E E . Then
for every x E E (z E E C ) and the map E (resp. E c )
+ L E ( H ) , x e%
is an involutive algebra homomorphism (c) and e)).
5.6 Hilbert right C*Modules
a) Take v : H
+G
with (utlo) = (FIvv)
for all ( t , q) E G x H . Then for all q E H , 11u'q
SO

vq11' = (u'q

vq1u'q

vq) =
u* = 21. For all (E, q) E G x H ,
Hence u* E L E ( H ,G) and u** = u . b) Take U E L ~ ( G , H )a , , P ~ K , t , q ~ G , a < n ~d H . T h e n
at + 1817)IC) = ( a t + PqIu'C) = 4uEIC)
Since
<
= a(EIu*C) + P(vb*C) =
+ P(.LL~IC)b u t + PuqIC) .I
is arbitrary, we get u(aC + P7) = o u t + P w ,
i.e. u is linear. For q E H # , put
v, is linear and contiriuous and
for all F E G (Proposition 5.6.1.2 d)). By the Theorem of BanachSteinhaus, a := sup ((v,J(< cm VEII#
5. Hilbert Spaces
298
ll(u<117)11= Ilvtl
IIuFII 5 aI1tII for all
< E G , and so u E L(G,H ) . From r
I I ( F I U * V ) I I= I I ( U ~ I V ) 1 I I I I U ~ III IV I I
II
IIUII
for all (<,7) G x H (Proposition 5.6.1.2 d ) ) ,we get
for all 17 E H , so that JJu8(l 1 llull. By a), IIu'II = llull. L It is easy to see that E ( G ,H ) is a vector subspace of L(G,H ) . Take u E L(G,H ) and let (un)nEn.be a sequence in LE(G,H ) converging to u . Then ( u ; ) , ~is~a Cauchy sequence in L ( H ,G ) , so it converges to a v E L ( H ,G ) . We get
for all (<,q) E G x H (Proposition 5.6.1.2 d)). Hence u E L E ( G ,H ) and L E ( G ,H ) is a closed vector subspace of L(G,H ) . c) Take a , /3 E M and u , v E LE(G,H ) . Then for all (<, 7 ) E G x H ,
so that
d) For all
EH
,
5.6 Hilbert right C*Modules
.u(<x)= (u<)x. e) For all (<, 7) E F x H , ((v o ~ ) < I v= ) (v(uO1v) = (utlvfv) = (
0
u*
.
f ) We have llu
cEG
l l ~ * ~ lltl12? ~ll
(Proposition 5.6.1.2 d)), so llu1I2 I IIu'ouII .
Since
IIu' o uII
I IIu*II IIull r
we get by b), Ilu1l2 = lIu*ouIl . It follows by a) and b), lluou*ll = IIu**ou*II= IIu*Il2= Ilu112 g) For
( F , 77)
E
F x G,
( ( x ~ ) E I= v )( X ( U O I=~ (U
E
(ux)* = xtu* .
E and u E LE(F,G) . Then for any
< € F,
((xY)u)<= (xY)(u<)= x(Y(u<))= X((YU)<) = (X(YU))<,
299
300
5. Halbert Spaces
((lu)< = l(uF) = u t ,
( u l ) < = u(1<) = 4
,
so that (XY)U= ~ ( Y u ) ,~ ( X Y=) (ux)y,
(xu)y = ~ ( u Y ) ,
Since the above maps are bilinear, t E ( F , G) is a (unital) E module Similarly it can be proved that
for all x E E and u, v E L s ( F ) h) For < , v E H ,
((:
= (<xIv) = (t1v)x = x(t1v) = (
so that

I E C E ( H ) , ? = x' Take x , y E E (resp. EC). For every
.
< EH,
Gj<= (xy)< = x(y<) = z(g<) = %g<
(G<= E(xY) = <(Yx)= ( E Y )=~ I(?<) = z g < ) , so that
5.6 Hilbert right C*Modules
Hence the map
E (resp. EC)
+ L E ( H ),
x
H
Z
is an algebra homomorphism. By the above, this map is involutive.
Remark. The inclusion map co + 5.6.1.5). Proposition 5.6.1.9 0
0
(. 0 )
em does not belong to Lem(co,em) (Example
Assume IK
,
=
R . Let G , H be Hilbert right
Emodules, G , H the complexification of G and H , respectively (Proposition 5.6.1.6) and ;(G, H ) , ;(H, G ) the complezification of the underlying real vec0 tor spaces L ( G ,H ) and L ( H , G ) , respectively. Given ( u ,v ) E L ( G , H ) (resp. ( u ,V ) E & H , G )), define

0
(resp. ( u ,v ) : H +
6,
(<, 7 ) cf (u< ~ 7217: + ~
Let p be the isomorphism of complex vector spaces c~ : ;(G, H )
t
~ ( 2i f ,) , ( u ,V )

0. )
(21,~)
(Proposition 2.1.5.6). Then
and
.
(11,
~ ~ i(f ) 8 .

v ( i E ( ~H. ) ) =
v ) = (u*,v*)
for all ( u ,v ) t ;,(G, H ) (hence ~ ~ h ( )2may , be identified with the complexification of L E ( G ,H ) ) . Take ( u ,v ) E i ( G ,H ) 0
Step 1
( u ,v ) E L E ( G ,H ) +

(z) ~~ ( 2 , t
*
if),
( u ,V ) = ( u f ,v*) Take (<, 7) E
6 and
(C, A)
E
i f . We have
5. Hilbert Spaces
302
= ((f, v)I(u*C

Hence ( u , u ) t
L;(;, i ( )
step 2
Put
For
(<,v) E
Gx H.
+ u*X,u*X  u * ( ) ) =
and
(G)~ ~ i()* 2 (.u , u ) t J?E(G, H ) )E
5.6 Hilbert right C*Modules
303
Definition 5.6.1.10 ( 0 ) A (unital) E X *  a l g e b r a is a (unital) involutive E algebra which is at the same time a C*algebra. A n isomorphism o f E C'algebras is an isomorphism of C'algebras which is at the same time a71 isometry of involutive Emodules. Theorem 5.6.1.11
(0)
Let H be a Hilbert right Emodule.
a ) L E ( H ) endowed with the Banach algebra structure induced by L ( H ) and with the involution
(Proposition 5.6.1.8 a), b)) is a C*algebra. b) Let IK = C and u E L E ( H ) (let
for every
IK = R and u E R ~ L E ( H ) )If.
< E H , then u E Re L E ( H )
(U
=0)
c ) If IK = C ( I K = IR) then the following are equivalent for every L E ( H ) ( u E R ~ L E ( H ): )
u E
If these conditions are fulfilled then
and (u
for every
ll~ll(
<E H .
d ) If H is a Hilbert Emodule then L E ( H ) is an EC*algebra (Proposition 5.6.1.8 g)). e ) The following are equivalent for all p, q E Pr L E ( H ) el) p < q .
e2) I m p C I m q .
304
5. Hzlbert Spaces
f)
The followzng are equivalent for every p E L E ( H ) with pZ = p .
g)
The following are equivalent for all p, q E Pr L E ( H ) :
a ) In the complex case, the assertion follows from Proposition 5.6.1.8 a),b),c),e),f). In the real case, it follows from Proposition 5.6.1.9 that the com0
plexification of ,Cti(H) is the C*algebra L & ( H ) , so L E ( H ) is a real C*algebra. b) The map
is sesquilinear. By Proposition 2.3.3.8 b
+a
(by Proposition 2.3.3.7 a
which implies
for every cl c2
< = H , i.e. u = 0 ) .
+ c p . (u
c 1 . If IK = C then
By the Polarization Identity (Proposition 2.3.3.2 d)),
+ b),
5.6 Halbert right C*Modules
t, q E H , so that u = u* , i.e. u E Re C E ( H ) . It follows (for IK E {IR,C) ),
for all
( a ) and Theorem 4.2.2.9 a)), (u[It)
I (u+tlt).
Hence ( ( ~  ) ~ t l= t ) ( ( ~  ) ~ r l u  tI) (ufutIu<) = 0
BY cl
=+ c2,
((u)3t10 = 0 so that by b),
Now assume that u is positive. Then
= sup{(luf
From c2) and u
H#} =
IIU?II'=
llull.
I I(u)(l(Corollary 4.2.1.17 a + b) it follows II~II( ~(utlr) I ~ ) = ( ( I I ~I uI )~t ~ < E ) E+,
d ) follows frorn a ) and Proposition 5.6.1.8 g) el =+ e2. For every E H ,
<
305
5. Halbert Spaces
306
PE (Corollary 4.2.7.6 a
= qpE
+ b), so that imp^ Imq.
e2
+ el.
For every
< E H, pt E Imp C Imq
so that
Hence
(Corollary 1.2.7.6 d + a ) . f, + f2. We have
f2
+ fl.
For all
< , vE
H,
so that
Hence p = p* and p E Pr C E ( H ) . g, +g2. Since
5.6 Halbert right C*Modules
92 3
61. For every
< E H,
b q f l p q f )= (qllp'pql) = (qflpqf)= 0
so that
Hence
( 0 ) Let G ,H be Hilbert right Emodules. Then for every u E LE(G,H) and v E L E ( H ) +,
Corollary 5.6.1.12
In particular,
(uFIuO 5 ll~112(f10 for every
<E G.
For every f E G ,
so that
by Theorem 5.6.1.11, c2
3
c , (and Proposition 5.6.1.8 a),e)). It follows
(Proposition 3.6.1.8 e)). By Theorem 5.6.1.11 c) and Proposition 5.6.1.8 f),
Proposition 5.6.1.13 Let H be a Hilbert right Emodule. For every x E E , define
a)
The following are equivalent for all x,y E E :
5. Hilbert Spaces
308
all
J,v E H =+ (t17)x = ~(t117)

a?) Z E L E ( H ) , z' = y * . b ) The followzng are equivalent for every x E E :
a) For all J , 77 E H ,
(ZJIv) = ( t x l v ) = (J1v)x
1
(Proposition 5.6.1.2 a ) ) .al a a2 follows. b ) follows from a ) .
Example 5.6.1.14 Let I be a finite set, a E (E\{O})', and b E ( E # ) ' . Then E' endowed wzth the right and left multiplications
E x E' +E' ,
( x , J )++(x<,),cI,
and with the innerproduct
is a Hilbert E module. First we remark that
for every J , 17 E E1 . By Corollary 4.2.2.3, a:b:b,a, for every
< E E'
and
L
I ala, ,
J:a:b:b,a,E,
< <:ala,J,
t I . Put
cL := (a:aL a:b:bLaL)
5.6 Hilbert right C*Modules
for every L E I . By the above, for [ E E l ,
Take <, E E l . Then
( ( F I E ) + ( v l v ) , (710  ((17)) =
We have
and for
L
E
I, O5
(CLCL?
cL~]L)*
=
(cL
(E;C;S~
cLvL)
= (sL*~L
v:cL)
(cLcL?
cLvL)
+ 17:cTv~,<:cTa v:cTFL) .
Hence
((
+ ( V I ~ , ) (VIO ,
The other axioms are easy t o verify.

((171)) E
i+.
=
5. Hilbert Spaces
310
5.6.2 Selfduality Proposition 5.6.2.1 every E H put
<
(8 )
Let H be a Hilbert right Emodule and for
< E H * (.I<) E L E ( H ,E), (.I<)* = F,II(.I<)II
a)
IItII
( E z a m ~ l e5.6.1.5).
b) If E is unital then the map
is conjugate linear and an isometry of real Banach spaces with inverse m aP
If H is a Hilbert Emodule then for every x E E and ( E H ,
C)
x((.IO)
=
(.lCx*)
1
((.l<))x = (.lx*<)
(Proposition 5.6.1.8 g), Ezample 5.6.1.5). a) By Proposition 5.6.1.2
d),
ll((.l<))vll = 1l(v1C)11 5 lltll llall for cvcry 9 E H and therefore
(.I<)
E C ( H ,E) and
II(.IOII 5 llrll. From
ll(.lOIl IICII 2 ll((.l<))
ll(~l<)ll2 ll
3
ll(.l<)ll = ll
For (9,x) E H x E , (((.1<))171.)
=
((17101.)
= .*(VIE)
=
5.6 Hilbert right C*Modules
= (v1Fx) = (v1Fx)
(Example 5.6.1.5, Proposition 5.6.1.2 a)) so that
(.I<)
E CE(H, E ) 1
b) For u E CE(H, E) and
(.IS)* =
F.
<E H,
(
The assertion now follows from a ) c) By Proposition 5.6.1.2 a ) ,
(((.lS))x)v = (xvlF) = (vIx.0 for every 17 E H so that x((.lE)) = (.I<%*):
((.IS)). = (.lx*E).
Remark. If E is not unital then b) may fail. Indeed, if H := E := co then CE(H, E) is isomorphic to em.
Definition 5.6.2.2
&(G, H )
(8)

For G , H znnnerproduct right E modules define
:= {U E L(G, H)I(<,X) E
GxE
u(<x) = ( u E ) ~ ) ,
i? := . Z E ( ~E, ) . H is called selfdual (Paschke, 1973) if I(.I<)IE E H I = fi (Proposition 5.6.2.1) (this zmplies that H is complete). By Proposition 5.6.1.8 d), .CE(G,H ) C .C^E(G,H) . If F, G , H are Hilbert right Emodules then u E &(F, G) arid v E &(G, H ) implies vou E E
E ( ~H ,) .
By the Theorem of FrgchetRiesz, every Hilbert space is selfdual.
5. Hilbert Spaces
312
Proposition 5.6.2.3 ( 8 ) Let F be a closed rzght ideal of E . The innerproduct right E module associated to F (Example 5.6.1.5) is selfdual iff there i s a p E P r E such that F = p E .
First assume F is selfdual.Denote by u the inclusion map F there is a p E F such that u=
+ E . Then
(. I P)
W e deduce successively p = u p = @Ip) = p 8 p € R e E ,
p~ P r E , F =u(F)=pE. Now assume there is a p E Pr E with F = p E . Take u E y
:= p(up)* E
Z(F,E ) and put
pE = F
Then for every x E F , (xly)= ( ~ I P ( u P ) * )
=
(UP)PX
= ~ ( P x=)
so that u = (.IY). Hence F is selfdual. Proposition 5.6.2.4 selfdual then
(8)
Let G , H be Hilbert right Emodules. If G is
Let u ~ z ~ ( G , ~ ) . Tqa~ kHe. T h e n f o r a l (l t , x ) ~ G x E ,
5.6 Hilbert right C*Modules
Since G is selfdual, there is an element vq E G such that
It follows
<
for every E G , i.e. u E L E ( G ,H ) . Hence J ? ~ ( GH, ) C L E ( G ,H ) . The reverse inclusion follows from Proposition 5.6.1.8 d). Remark. If G is not selfdual then the above equality may fail. Indeed, the (cO,em) but not to Lc (co,em) . inclusion map co 3 em belongs to Proposition 5.6.2.5 ( 8 ) Let F be a C*subalgebra of E and cp : E + F an involutive continuous lznear map such that:
a ) If H is a Hilbert rzght Emodule, then H endowed with the right multiplication
and with the inner product

is a Hilbert right Fmodule, which will be denoted i n the sequel by H . The norms of H and fi are equivalent so that fi is reflexive iff H is reflexive (Proposition 1.3.8.8). b) If G and H are Hilbert right Emodules then
cE(c, H) = .cF(G,H ) n &(G, H ) and the adjoint defined in LE(G,H ) coincides with the adjoint defined in
L,(C, H ) .
5. Hilbert Spaces
314
c ) L E ( H ) is a unital C'subalgebra of d ) If
H
LF(H)
is selfdual then H is also selfdual.
e ) If E is the real C'algebra C or IH (Example 4.1.1.31) and F := R then
has the above properties 14. T h e same holds for the m a p
f ) If E is the real C'algebra C or M then every Hilbert right Emodule H is selfdual and H and fi have the same norm. g)
Let G be a C ' a l g e b r a and n E
IN . If E 2s the C*direct product of the
family ( G ) ] E N " and
then
&,asthe above properties 1  4 , where we identified F and G i n a nati~ral way. h) Let (E,),,I be a finite family of C*algebras and for each L E I let F, be a C'subalgebra of E L and ip, : ELt F, a n involutive continuous linear map with the above propertzes 1 ) 4 ) . If E and F are the C* direct , and i f product of the famzlies ( E l ) r E Iand ( F L ) l E Irespectively,
then 1)

ip
is a n involutive continuous linear map with the above properties
4).
i) If G , H are Hilbert right Emodules and u E L E ( G , H ) then the norm of u i n L E ( G ,H ) coincides with the n o r m of u i n cF(2;, H). j)
For every x E E put
Then
(1.11,
is a norm o n E equivalent to its initial norm.
5.6 Halbert right C*Modules
a ) is an easy verification. b) Take u E c F ( G , H) n Z E ( ~H, ) . Then for (<, 7, x) E G x H x E ,
If we put x := ((u
then by 4), ( ( u ~ I v) ( < I u * ~ ) ) ( ( u
The inverse incl~isionfollows from Proposition 5.6.1.8 d). Now take u E LE(G, H ) and (<, 7) E G x H . Then
so that
Hence the adjoint of u in LE(G,H ) and L ~ ( GH, ) coincide. c) follows frorri a) and b) (and Theorem 5.6.1.11 d ) , Corollary 4.1.1.21). d ) Take u E Z(H, E) . Then cpou E L(H, F) and for (<, y) E H x F ,
Since
fi
is selfdual, there is an q E H such that
31 6
For
5. Hdbert Spaces
(t,x ) E H
x E
, ( ~ ( ( ut ( t 1 q ) ) x ) = v ( ( u t ) x  ( t l d x ) =
Putting
it follows by 4 ) ,
Hence H is selfdual. e) Only 3) needs a proof. First assume E = C . Take ( x ,y ) E & + . Let x 1 , x 2 ,y l , y2 E
IR
with
By Proposition 4.2.2.24,
Hence
so that ( P X , ' P Y )= ( 2 1 Y, I ) E a + .
Kow assume E = M . We identify M with a real subalgebra of C2,2 and M 0
with C2,2 as in Example 2.3.1.46. Takc (x, y ) E M+ and put
5.6 Hilbert right C*Modules
Then x E M + = IR+ (Proposition 4.2.2.13 a)) and
By Example 4.2.3.5,
Hence
in the case F = IR and
in the case F = C . f) Let H be a Hilbert right Emodule. By e) and a), space and so it is selfdual. By d ) , H is selfdual.
I?
is a real Hilbert
g) 1) and 2) are trivial. 4) follows from Corollary 4.2.1.18. In order to prove 3) we may assume IK = C . Take (x, y) E E+ . By Proposition 4.2.2.24, 0
x, f iy, E G + for every j E IN,. By Proposition 4.2.2.24, again, (x,, y,) E G+ for every j E IN, so that
h) is easy to see. , ). i) Denote by 11ull (by I I I u I I I ) the norm of u i n LE(G, H ) (in L ~ ( Gfi) By c) and Proposition 5.6.1.8 f),
j) )I.)), is obviously a norm on E with
5. Hilbert Spaces
91 8
ll.llG
i llvll 11. 1 1
Takr x E E\{O} and put
Then y E E# and so
where
a := inf{llpzII I z E E+,llzll = 1 ) > 0 .
¤
Hence Il.llG and 11.11 are equivalent.
Proposition 5.6.2.6 ( 8 ) (W. Paschke, 1973) Let E be a C*algebra, G, H innerproduct right Emodules, and u E Z E ( ~H, ) Then (uSIu0 I llu112(
< E G . In particular, if
u is an isometry then
(.
=
(El<)
< E G.
We may assume E unital. For every n E IN put 1 x := ( ( I ) + 1 , <,, :=
on,
Take n E I N . Then (
i1
(Proposition 5.6.1.2 b)), so 1l
(Corollary 4.2.1.17 b
I1
+ a). It follows
~ l( ~ F n l ~ t= n )( u ( < ~ n ) I ~ ( < x n=) ) 11~11212 l l ~ < n I I L = ( ( ~ < ) x n I ( ~ O x= n )xi(~
(Corollary 4.2.1.17 a
+ t) , Proposition 5.6.1.2 b)),
5.6 Hzlbert right C*Modules
Proposition 5.6.2.7 XI E E: , and
(8 )
Let H be a n innerproduct right Emodule,
a) G,! is a vector subspace of H ; denote by qzl : H
+H/G,l
the quotient map.
b)
t E G i , v E H * x1((Flo))= xl((vl<))= 0 .
c ) Take S,Y E H/G,! and
tl,t2E X , v1,q2 E Y . T h e n
5'((<1l71))= x1t(<21v2)). Define
(XIY),e
:= x1((
dl HIG,. endowed with the m a p
( H I G i ) x ( H I G i ) + IK :
(X, Y)
(XIY),'
is a preHilbert space and q,, is continuous with
ll9z~ll5 llxlll~. Denote by Hz, the completion of the preHilbert space H/G,l
e) For every u E H there is a unique u,, E Hz, such that
xl(u<)= ( q , r t I ~ ~ ~ ) ~ ~ for every ( E H . Moreover,
lluz,ll I llull llxlll~. f)
< E H * (.I<)=!= q i < . a) Take 1 , E~G,c . By Proposition 5.6.1.2 b), (< + v1< + v) I 2((<10+ (vlv))
so that
31 9
5. Hilbert Spaces
320
Hence G!, is a vector subspace of H . b) By Proposition 5.6.1.2 c) ,
so that
x1((vlO(
By Proposition 2.3.4.10 b ) , 1x'((E1v))I2 5 11x'11x'((vI<)(Elv)) = 0 ,
= xl((tl

<21v1)) + x1((<21v1 772)) = 0 .
d) It is easy to see that HIG,,
is a preHilbert space. For every
lIqiEll2 = (qr~EIqiE) xl((EIE))
so that q,
is continuous and llqr~ll5
e) Take
IIX'II' .
< E H . By Proposition 5.6.2.6 (and Example 5.6.1.5), (~<)'(.t)
so that
I lI~'11lIEII2
= (u
I lI~1I2(tl<)
< E H,
5.6 Hzlbert right C*Modules
5 llxtll llul12xr((cl<))5
ll~tl1211~l1211
(Proposition 2.3.4.10 b)). It follows that xrou factorizes through H/G,t to a contiriuous linear form. By the FriichetRiesz Theorem (Theorem 5.2.5.2), there is a u , ~E Hz! such that xr(uc) = ( q z ~ < I ~ z ~ ) z ~ for every
< E H . By the above, IIUZ~II
I<
= ~u~{I(qz~
= s.p{lxt(uf)ll(
E H , xr((
5
I I X ' I I ; I I U I.I
The uniqueness follows from the fact that q,t(H) is dense in Hz,. f ) By Proposition 5.6.2.1 a) (and Proposition 5.6.1.8 d)), (.I<) E
fi. For
BEH, = xr((.1<)~) = (q=1~I(.I<)z1)=1
Since q,,(H) is dense in Hz!, (.IE)zl
Proposition 5.6.2.8
(8)
= qi<.
W e assume IK = IR, use the notation of Pro0
positzon 5.6.2.7, and extend it i n a natural way t o the complexification H of H.
b)
T h e algebmic isomorphism
associated to the quotient map 0
q(zl,o): H + H/G(z~,0) is a n isomorphism of complex preHilbert spaces. W e identify the above 0
A
preHilbert spaces as well as their completions Hz# and (H)(,J,o) using this isomorphism.
5. Hilbert Spaces
322
C)
d)
fi =+ ( u , = (u,,, 0) u , v € fi 3 ( ( u ,O ) ( , ~ , o ) l ( ~o,) ( z ~ , o ) ) ( ~ t=, o() u ~ f l v ~ ' ) r ' U €
o)(rl,O)
By Proposition 4.3.6.2 c3
* cl ,
0
(XI,
0) E (El;
so that

1
= (9(x1,0)(<> V ) 9(rr,0)(<, A))(rt,O).
Hence the map
0
r
+
0
I
,
(qrj5,qzlv)
is an isomorphism of complex preHilbert spaces. .) For
( 5 ,v ) E A , ((sz,S,
q,v)l(.zr,
0))Zl =
q(r',o)(C> 7)
5.6 Hilbert right C*Modules
= (q(zfl,o)(t, 17)
1 (21, O)(z~,o))(z~,o) .
Since (<,g) is arbitry (u, 0)(2',0) = (1111,O) . d ) For
<EH, (qz~
Since H/G,, is dense in Hz,, there is a sequence By b) and c) : ( g z ~ < n ) nconverges E~ to ~
~
1
in H such that
.
(u, O)(zl,o)= ( u ~0), = nlim + m (gz~Fn, 0) = nlim + m q(,#,o)(tn,0) It follows ( U ~ ~ I=Ulim ~ ~(q,t<,lv,~),, ) ~ ~ = n+CC
Proposition 5.6.2.9 ( 8 ) We use the notation of Proposition 5.6.2.7 and take x', y' E E'+ with x' 5 y' .
b)
There is a unique
c) For every u E
cp,r,,t
E .L(Hyc,HZ))' such that
fi , cpz',,'Uy'
= u,,
.
324
5. Hilbert Spaces
d) For every u E
fi
there is a filter
5 o n H such that
lim q,l< = u,! F3
for every x' E EI, and 1im q ~ z ~ , y ~0))(= < ,( u 7O)(xl,yq C,3
for every ( x ' , y') E (E): a ) is obvious. b) follows from a). c) Since H/G,l is dense in Hy, , there is a sequence that (qyc
Take
H such
)itc ~ z ~ , y ~ q=~ ) < nit qz' En
< E H and n E N . By Proposition 2.3.4.10
=
(
I l ~ ' l l ~ ' ( ( < I < n ) ( < n lO
b),
(uO(
By Proposition 5.6.2.7 e),
= (qy'(<(uO8)Iuy' qy'
and so lim y1((u<)(u<)* (
n+w
Again by Proposition 5.6.2.7 e),
5.6 Hilbert right C*Modules
325
so that
= llqyf
11<11211~y~
(Proposition 5.6.2.7 e), Proposition 3.6.1.2 b),c), Corollary 4.2.2.3). It follows )) 0 lirn yl((
n+w
0 = lirn x'((
= nlirn ((qzl
Since
< is arbitrary, pz,,ytu
1
 lim  n+w
qztFn = u,, .
1
5. Halbert Spaces
326
d ) For every A E y f ( E i ) and
E
> 0 put
B ( A , E ):= {< E Hlx' E A
* Ilq,~t  u,rII
<E)
B ( A ,E ) is nonernpty. Indeed, put
Then y' E E; and x' 5 y' for every x' E A'. Since H/Gyl is dense in H d , there is a E H such that
<
ll9Y't  ~ Y l l l< E By b) and c),
<
for every x' E A . Hence E B ( A ,E ) and B ( A ,E ) is nonempty. Denote by 5 the filter on H generated by the filter basis
It is obvious that lirn q,,( = ur, t,$
for every x'
EL.
Now take (x',y') E (E)'+. By Corollary 4.3.6.2 cl
(x',Y ' ) 5 ( 2 x f 0, ) . By b),c), and Proposition 5.6.2.8 b),c),
Il9(z~.y~)(J, 0 )  ( u >O ) ( z ~ . y ~ ) = II
By the above, lim q(r,,yl)(t> 0) = ( es
~
O)(rl.vt) 1
*
c4,
5.6 Hzlbert right C*Modules
Proposition 5.6.2.10
(8)
327
Let H be an innerproduct right Emodule
0
0
and H its complexification (Proposition 5.6.1.6). W e identify L ( H , E ) and 0
0

L ( H , E ) using the isomorphism of complex vector spaces
(
HE)

( h)
( u ,v ) ++ ( u ,v )
defined in Proposition 2.1.5.6 (Proposition 4.3.6.1 f), Proposition 5.6.1.6)
0
b) H is selfdual 2ff H is selfdual. 0
a) Take ( u ,v ) E
i?.For every
(<, 77) E
2 and
( x ,y) E
k,
Hence
and
0
Take w 6 H . By the above identification, there is a ( u ,v ) E C ( H ,E ) with
( u , v )= W For every
.
(I,x ) E H x E , ( u ( f x ) v, ( < x ) )= ~ ( ( ( 50 ), ) = w ( ( f ,O)(x:0 ) ) =
328
5. Hilbert Spaces
...
b) First assume H is selfdual. Let w E such that
Since H is selfdual, there are
to,go E H
k . By a), there is a
with
0
For every (<, g) E H ,
= ((C>v)I(CO> 70))
Hence
0
and H is selfdual. is selfdual. Take u E Now assume
k
0
fi

and put
w := (u, 0) . 0
By a), w E H . Since H is self dual, there is a (<, g) E H with
0
(u,v) E
fi
5.6 Hilbert right C*Modules
For every
<E H,
=
((
Hence
u = (.It) and H is selfdual.
Theorem 5.6.2.11 ( 8 ) (Paschke, 1973) Let E be a Wealgebra and H a n innerproduct right E module. For eve y E H put
<
?:= (.It): H 4E ,
7 ++
(7710
and identify H with a subset of f i uszng the znjective map
Hfi,
6?
(Propositton 5.6.2.1 a)). For every ( u ,x ) E f i x E put
ux:H
+ E ,
<x*ut.
W e shall use the notation of Proposition 5.6.2.7. a ) u ~ f i : x , Ey ~~ U X E (~u x?) y, = u ( x y ) . b)
(C,X)E
H X E*?X=G.
c) For K = C there is a unique map
fixj?+E,
(u,u)e(uIv)
such that
((ulv),a ) = (uaIva)a for all u , u E f i and a € E,. d ) Assume K = IR and take u, v E f i . T h e n
((uto)l(vl0 ) ) E E x ( 0 ) and we define (ulv) E E by
5. Halbert Spaces
330
((.I.)>
0) = ((% O)I(v, 0))
Then ((21, v), a ) = (.uolva),
for every a E E+ . e)
t>v.cH
(?I$
=+
f ) ( t , U) E H x g)
E?
= ((I?).
E? + (Flu) = uC.
endowed with the nght multiplzcation E?x~ifi,
(U,X)HUX
(whzch contains also a scalar multiplication) and with the innerproduct
zs a selfdual Halbert right Emodule. h)
The norm of the Hilbert right Emodule fi defined in g) coincides with the initial norm of E? (Definition 5.6.2.2). a ) For every ( E H , (UX)(
so that ux E
E?
and
b) For every 17 E H ,
(Proposition 5.6.1.2 a)), so that
c) CVc use the notation of Proposition 5.6.2.9. Takc u, v E
fi. Define
5.6 Hilbert right C*Modules
Step 1
cp may be extended linearly to E
Let (a,),El be a finite family in C and (a,),,I a family in E+ such that
Put
Then a E E+ and a, 5 a for every
L
E
I . For (, q E H ,
(Proposition 5.6.2.9 b)). It follows
(Proposition 5.6.2.9 b)). Hence cp may be extended linearly to E . We denote this extension of cp also by cp. Step 2
cp is continuous
Take a E E . By Theorem 4.4.3.9 (and Corollary 4.4.2.10), there is a family (a,),,N, in E+ such that
332
5. Hilbert Spaces
We get
(Proposition 5.6.2.7 e)). Hence cp is continuous Define
Uniqueness follows from Theorem 4.4.3.9 (and Corollary 4.4.2.10). d ) By Proposition 5.6.2.9 d ) , there is a filter 5 on H such that
for every a E E+ and lim q(a.6)(tr0) = Cj3
O)(a,b)
for every ( a ,b) E E+ . Put
,.( Y) :=
(
(
~
O)l(v, 0))
9
and take ( a ,b) E E+ . By c) (and Proposition 5.6.2.7 e)),
((x,Y), ( a , b)) = ( ( ( wO)l(v,O)), ( a ,b)) =
5.6 Hzlbert right C*Modules
= lim ((v, O)(<,O), (a, b)) = lim ( ( v l ,O), (a, b)) F>3
333
.
( 3 3
By Proposition 4.4.4.23, y = 0 . By c) and Proposition 5.6.2.8 d),
for every a E E+ . e) First assume
IK = C . For every a
E
E+ , by c) and Proposition 5.6.2.7
c),f),
so that
If IK = IR then, by d ) and the above,

( ( ? 1 3 > 0= ) ((?>0)1(?,0))= ((6,0)1(770)) =
and so
f) First assume IK = C . For every a E E + , by c) and Proposition 5.6.2.7 e),f),
so that (Flu) = u s . If
IK = IR then, by d) and the above, 0) = ((?, O)l(u,0))
=
(u, O)(<,0)
=
(uE, 0)
5. Hilbert Spaces
334
and so
Step 1
u, u E
fi,x E E + (uslv) = (ulu)x
Take a E E+ . Since xa E E (Corollary 4.4.2.10), there is a family (a,)JEn, in E+ and a family (a,),EN, in C such that
(Theorem 4.4.3.9). Put
Then b is an upper bound of a and 5.6.2.9 c),
in E,. By c ) and Proposition
By a)$), and Propositions 5.6.2.7 e) and 5.6.2.9 b),c), for every
Since qb(H) is dense in Hb , it follows
<E H ,
5.6 Hilbert right C*Modules
Hence
A
Step 2
u,v E H
+ (ulv) = (V[U)*
For every a E E+ ,
((UIV),~)= (uaIva)a = (valua), =
so that
Skp3
u~fi=+(uIu)~E+
For every a E E+ ,
((.I.)>
a)
=
(uaIua)a E JR+
so that
(ulu) E E+ by Corollary 4.4.1.6 a). Step 4
u E H , ( U ~ U=) 0 + u = 0
For every a E E+ ,
(ualua)a = ( ( u I u ) , ~ = ) 0 so that
5. Halbert Spaces
336
By Proposition 5.6.2.7 e), aou = 0 so that by Corollary 4.4.1.6 b), u Step 5
=
0.
fi is selfdual
, with Let U E ~ ( j ?E) ( v , ~E )
5x
E
U(vx) = ( U V ) ~ .
Put
<
iri for every E H . Take v E 5 and a E E+ . There is a sequence converges t o v, . For every n E I N , by c), Proposition H such that 5.6.2.6, and Proposition 5.6.2.7 f ) ,
= IIVI12(va  QoInIva  qa
It follows successively
((Vv)*(Vv),a)
=0,
(Vv)*(Vv)= 0 (Corollary 4.4.1.6 b)), vv=o,
5.6 Halbert right C*Modules
IK=R
Case2
(u, x) E fi x E + (u, O)(x, 0) = (UX,0)
Step 6 For (E, 17) E
b,
= (xtuE1x*u17) = ( ( u x ) ~(ux)17) , = (ux)(I,17)
so that (u, O)(x, 0) = (ux, 0)
Step 7
u, v E
+
fi, x E E + (uxlv) = (u1u)x
By d) arid Steps 1 and 6 ,
((.XI.)>
0) = ((ux, O)l(v,0)) = ((u, O)(x,O)I(v, 0)) =
= ((u10)1(u1O))(x,0) = ((ulv),O)(xr0) = ((uIu)x,0)
and so (uxlv) = (uIv)x. Step 8
u, v E fi + (ulv) = (V[U)*
By d) and Step 2,
(.I) Step9
= ( ( 7 4 0)1(v,0)) = ((v, O)I(u; 0)); = (uIu)*
u~fi=+(u(u)~E+
By d) and Step 3,
((ulu),0) = ((u,O)l(u,0)) E so that
i+
5. Hilbert Spaces
338
(ulu) E E+ by Proposition 4.2.2.15 b). Step 10
..
u E H, ( U ~ U=) 0
3
u =0
BY 4, ((u, O)I(u, 0)) = ((uIu),O) = 0 and so, by Step 4,
step 11
+
u.u E fi + ( ( u ~ u ) (VIV), (v/u)  (ulu)) E
i+
By d) and Case 1 (and Proposition 5.6.1.2 g ) ) , (((ulu),0) + ((vlv), 0) ((vlu): 0)  ((ulu), 0)) = 7
so that ( ( 4 4 + ( ~ I v ) (2114  (uIv)) E 7
i+
by Proposition 4.2.2.15 b). Step 12
fi
is selfdual ...
k.
A
2
By Proposition 5.6.2.10 a), fi may be identified with By Step 5, is selfdual, so by Proposition 5.6.2.10 b), fi is selfdual. h) Take u E fi and denote by 1 1 ~ 1 1 the norm of u in fi in the sense of Definition 5.6.2.2. For every a E E + , by c),d), and Proposition 5.6.2.7 e),
By Corollary 4.4.3.16 c),
5.6 Hilbert right C*Modules
Let a €10, J J u J JThere [. is a
5 E H# ff
with
< Ilutll . 
#
By Corollary 4.4.3.16 c), there is an a E E+ such that
a
< (utla).
By c),d), and Proposition 5.6.2.7 d),e), ff2
< (q,clua): i llqat1l211u,ll 2 I
Hence
Proposition 5.6.2.12 Let F be a closed ideal of E and H a selfdual Hilbert right F module. a)
For every ( t , x ) E H x E there is a unique element of Ex, such that
/or e u e y r/ E H
b) H endowed with the right multiplication
and with the innerproduct
H xH
t
E,
((1
7)

is a Hilbert right Emodule. a) P u t u : H 4F ,
9
I+
x*(ql[)
u is linear and continuous and for (7, y) E H x F ,
((17)
N , denoted
by
34 0
5. Hilbert Spaces
417~) = x'(17yIO = x'(171J)y = ~ ( 1 7 ) ~ . Hence u E ~ , c ( HF, ) . Since H is selfdual, there is a J s E H with u =(.I~X).
For every 17 E H (JzIv) = ( ~ l J x )= * (117). = (1117)~. The uniqueness is easy to see. b) is easy to see.
5.6 Hilbert right C*Modules
5.6.3 Von Neumann right W*modules Proposition 5.6.3.1
(8)
Let H , K be Hilbert right E modules. For all
(x',1,7 ) E E' x H x K , put
Take (x1,71,5)E E' x H x K .
f) If A' is an involutive set of E' then the closed vector subspace of ( L E ( H ) ) 'generated by
is an involutive unital LE(H)submodule of (L,q(H))' Take u E L E ( H ,K ) . a ) By Proposition 5.6.1.2 d ) ,

Hence (XI,1,7 ) E ( L E ( H K , ) ) ' and
b) By Proposition 5.6.1.8 a ) , 
I
(stl'!, 9 )

(4= ( X I , 5, V ) ( U * ) = x 1 ( ( u * f I q )=)
5. Hilbert Spaces
34 2
so that
c) By Theorem 5.6.1.11 cl
+ cz ,
for every u E C E ( H ) + . Hence by b) (and Corollary 4.2.2.10), h 
(x1751 e ) e (CE(H)): and by Corollary 2.3.4.7,
d ) We have
(Proposition 5.6.1.2 a ) , Proposition 5.6.1.8. d)), so that
so that
f) follows from a),b),e)
5.6 Hilbert right C*Modules
34 3
Definition 5.6.3.2 ( 8 ) Let E be a W * algebra and H , K Hilbert (right) Emodules. For every a 6 E and ((, 7) E H x K define

(a,<): H
+
M, C
ct
((ClO,a ) ,
and denote by H (by H ) the closed vector subspace of H' (of L E ( H x K)') generated by
(Proposition 5.6.3.1 a)). H is called a von Neumann (right) Emodule if it is self dual. If E is not specified then we simply speak of von Neumann (right) W *module. If E 6 { R , C ,M) then every Hilbert (right) E module is a von iY eumann (right) Emodule (Theorem of FrbchetRiesz for 5.6.2.5 f) for the real C*algebras C and M ).
IR and C and Proposition
Proposition 5.6.3.3 ( 8 ) (Paschke, 1973) Let E be a W*algebra and H a Hilbert right Emodule. T h e following are equivalent: a) H is selfdual.
b)
HH#
is compact.
If these condztions are fulfilled then the map
is a n isometry of Banach spaces. a
+ b.
Let
5
be an ultrafilter on H # . Put
for every ( E H . For ( 6 H and a E E ,
5. Hilbert Spaces
344
(Proposition 5.6.1.2 d)). Hence Put
EE (E)' = E and (:I
5 ((J((for every f
EH.
Then u E C(H, E ) # . Moreover, for (J, x) E H x E and a E E ,
so that
Since H is selfdual, there is a (a,<) E
<
E
H # such that u = (.I<). For every
ExH,
=
Hence
(F,a*) = lim (((17)) a*) = lirn ((TI<), a) = v93 v.3
5 converges to
*
< in
HE
and H# is compact. H
b a and the final assertion. Identify H with a subset of injective map
fi
using the
(Proposition 5.6.2.1 a ) ) and denote by G the closed vector subspace of generated by

2 ~ )E
{(=)((a,
where (a, u ) is defined with respect to
E x H) , (Theorem 5.6.2.11 g)).
+ b (and Theorem 5.6.2.11 g)), fi# H gG is Hausdorff (Theorem 5.6.2.11 f)), #  fi# H I ?  ..
By a
since
fi
A
(6)'
is compact. Since G C
fi
and
5.6 Hilbert right C*Modules
Assume IK = C and let ((a,,
34 5
be a finite family in E+ x H . Put
Take u E fi# and E > 0 . In the sequel we use the notation of Proposition 5.6.2.7. Since qaH is dense in Ha there is an 17 E H# such that IIua9
II
E
< (1 + Card I ) ( l + sup IILII)(l + a l ~ i ) '
LEI
By Proposition 5.6.2.9 b), c), for every
L
EI,
By Theorem 5.6.2.11 c),e) and Proposition 5.6.2.7 d ) (and Corollary 4.2.1.18),
It follows
5. Halbert Spaces
34 6
Since
E
and u are arbitrary,
The reverse inequality being obvious, we have
Hence the above Banach spaces H and G may be identified. By complexification, this assertion holds also for IK = IR . By Proposition 1.3.6.27, the maps
are isomorphisms of Banach spaces. By the above identification of H and G , we get H = fi, i.e. H is selfdual. Proposition 5.6.3.4 Let E be a W'algebra , G , H Hzlbert right E modules, and u E C E ( G ,H ) . Y
a) ( a ,q ) E E x H
h 
+ u r ( a ,7) = ( a ,u e ~ )
c) The m a p
zs continuous.

d ) If G and H are selfdual then the map
H
+G , yr
ury'
is the pretranspose of u (Proposition 5.6.3.3).

5.6 Hzlbert right C' Modules
<E G,
a) For every
so that

h 
u' ( a , 7 ) = ( a , u*q) . b) follows from a ) . c) follows from b) . d) By Proposition 5.6.3.3, G and H may be identified with the duals of G and H , respectively, and the assertion follows from c) and Corollary 1.3.4.9 a + b.
Theorem 5.6.3.5 ( 8 ) (Paschke, 1973) Let E be a \I/'algebra and H , K von Neumann right E modules. a) ( L E ( H ,K)):
is compact and the m a p (
HK)
+ ( H ) u ct (u, .)I H
is a n isometry of Banach spaces. b) L E ( H ) is a Wealgebra wzth H as predual. c)

<
h 
Take a E Re E and E H and let (u, I(a,<, <)I) be the polar representatzon of ( a , t ,5 ) . T h e n
+

( a ,O F ) = ( a ,u+F, u + e ) ,


,
( a ,<,C) = ( a ,u<, u  0 ,
5. Hilbert Spaces
34 8
d ) Let F be a W*algebra,

an involutive algebra homomorphism,
L := { ( a ,t, 77) I ( a ,t,q ) E E x H x H I , and
a map such that
for every y E F and (a,<,q) E E x H x H . Then cp is a W * homomorphism and
where cp denotes the pretranspose of cp (Corollary 4.4.4.8 a) ( L E ( HK , ) ) : is obviously Hausdorff.Let
c)).
5 be an ultrafilter on ( L E ( H K , ))#
For a € E , ( ( , v ) E H x K , a n d u 6 L E ( H , K ) ,
so that the map (LE(HK , )); is continuous for every
t
/ I t l l ~ K # , 2~ ++ 14
< E H . By Proposition 5.6.3.3 a + b , the maps
where the limits are taken in K , and H, , respectively, are well defined and belong t o L ( K , H ) # and L ( H , K ) # , respectively. For all a E E and (<,q)E H x K ,


( ( v t l v ) ,a ) = (a, v)(vO = h $( a ,r/)(uO =
5.6 Hilbert right C*Modules

h 
= lim ((u*qlJ), a*) = lim (a*,<)(u*q) = (a*,<) (wq) = u.3
u,5
It follows

By the above, for a E E and (e, q) E H x K , h 
(a, E , 01 (v) = ((vEIv), a) =: :1
((~Elq),a) = l i y (a, E, II) (u)
Hence 5 converges to v in (LE(H,K)): By Proposition 1.3.6.27, the map LE(H, K )
+( H ) ' ,
and
u
SO
(LE(H.K)):
is compact.
++(u, . ) ( H
is an isometry of Bariach spaces. b) By Proposition 5.6.3.1 f ) , H is an involutive unital LE(H)submodule of (LE(H))' . By a) , L E ( H ) is a W*algebra with H as predual. c) By Proposition 5.6.3.1 b), (a, E, e ) E Re H . By Theorem 4.4.3.9,

and the assertion follows from Proposition 5.6.3.1 e). d) For every y E F and (a, <, q) E E x H x H ,
5. Hilbert Spaces
350
so that
( ~ ) Since c p ' ~ j ~is~continuous,
Hence cp is a W'homomorphism (Proposition 4.4.4.6) and
Definition 5.6.3.6 ( 8 ) Let F be a norrned space and A a subset of F' such that Fa is Hausdorff. A family (
for every a E A . [ is called the sum of the family (tL)rEl in FA and is denoted by
Proposition 5.6.3.7 ( 8 ) Let E be a W*algebra and (
{ Ct, I J E p I ( I ) )is upper bounded. L EJ
b)
The famzly (J,),,,
is summable zn EE .
If these condztions are fulfilled then
where the limzt is taken i n E, The proposition follows from Theorem 4.4.1.8 b)
5.6 Hzlbert right C*Modules
Proposition 5.6.3.8 right Emodule, and
for all distinct
1,
(8 ) (
351
Let E be a W*algebra, H a von Neumann
a family i n H such that
X E I and such that
is upper bounded i n E a)
(
is summable i n H I , . Denote by
<
its s u m zn H , .
For every J E P , ( I ) ,
so that
is a bounded set in H . Let
5 , 6 be ultrafilters on g l ( I ) finer than 5, and
Put
where the limits are taken in H , (Proposition 5.6.3.3 a Take X E I and a E E . Then
so that
It fc)llows
+ b).
5. Hilbert Spaces
352
and so (Proposition 5.6.3.7)
We deduce
C=v. Hence (
< is its sum
H,
Proposition 5.6.3.9 ( 8 ) Let E be a C*algebra and H a n innerproduct right Emodule. Then the following are equivalent for every E H :
<
a
+ b.
By Proposition 5.6.1.2 a),b),
= ((10

( [ I c ) ~ ( ( I < ) ~+ ( < l o 3 = 0
so that <(
b
3
c . By Proposition 5.6.1.2 a ) , (
c 3 a . Putting 77 :=
< , we get (<1<)(<10 = (
so that (
5.6 Hilbert right C*Modules
359
Proposition 5.6.3.10 ( 8 ) Let E be a C order acomplete C*algebra, H a n innerproduct right Emodule, E H , and A a Bore1 set of IR+ with 0 $ A . Put
<
d ) If we put
5n
:= fn((
for every n E K then lim <(<xnI<xn)= < .
n+co
e ) If E is finitedimensional and
<+0
then
(
<(EY~
for some y E E . a) By Propositiori 5.6.1.2 b) and Corollary 4.3.2.5 c),
(<xI<x) = x(
5. Hilberl Spaces
354
b) By a ) and Proposition 5.6.1.2 b),
(<  <(<xl<x)l< <(<xl<x)) =
c) follows from b) d ) BY b) >
for every n E DJ so that
e) By Proposition 2.2.1.17, o((
for every
<EA
and
for all dzstznct <,7 E A . A maxzmal Founer set is called a Fourier basis of H. By Zorn's Lemma, every innerproduct right Emodule has a Fourier basis. Proposition 5.6.3.12 ( 8 ) Let H be a n innerproduct right E module, A a finzte Fourier set of H , and 7 , ( E H .
5.6 Hilbert right C*Modules
a) By Proposition 5.6.1.2 b) and Proposition 5.6.3.9 a
* C,
b) By a ) and Proposition 5.6.1.2 a),
(
17  C<(171<)
Theorem 5.6.3.13 ( set of H , and I ) , C E H
SEA
8)
1
I)
)
 ~ < ( I ) I < )= SEA
Let H be a Hilbert right Emodule, A a Fourier
.
n
~ ~ is such that ( X ; X ~ is) ~summable ~ ~ i n E then a ) If ( x ~ E ) ~(<(<)E FE.1
( < X < ) F ~isAsummable i n H . Moreover,
SEA
implies xF = 0 for every dimensional.
<EA
and so A is finite if H is finite
b) If E is Corder acomplete, A a Fourier basis of H , and
for every
<EA
then
71 = 0
.
c) If E is a LV*algebra then
d ) If E is finitedimensional then ( < ( 1 ) 1 < ) ) is ~ ~summable ~ in H .
5. Halbert Spaces
356
e) If E zs a W * algebra and H zs selfdual then ( < ( v ( < ) ) is ~ ~summable ~ zn H H . f) If E is a W'algebra, A is a Fourier basis of H , and H is selfdual then
a) Take s > 0 . There is a B E ??,(A) such that
for every C E P,(A\B) (Proposition 1.1.6.6). It follows
for every C E P,(A\B) (Proposition 5.6.1.2 b)). Since H is complete, the family ( < X ( ) < ~isAsurnmable (Proposition 1.1.6.6). From
it follows
for every 71 E A . b) Assume 17 # 0 . By Proposition 5.6.3.10 a),d), there is an z E E with
5.6 Hilbert right C*Modules
Since
for every J 6 A , A U {qx) is a Fourier set of H , contradicting the maximality of A . c) By Proposition 5.6.3.12 b), the set
C (J17)(71C) €
I
B E (Pf(A)
I
is upper bounded by (717) so that by Proposition 5.6.3.7,
d ) By c) and Minkowski's Theorem, the family
is summable. By a) and Proposition 5.6.3.9 a e) By Proposition 5.6.3.12 b), the set
3
c, (J(71J))et~ is summable
is upper bounded so that by Proposition 5.6.3.8 a), the family (<(V(J))~EA is summable in H , . f) P u t (by el)
For every Jo E A and a E E ,
5. Hilbert Spaces
358
(Proposition 5.6.3.9 a
+ c). Hence
for every J E A . By b), { = 0 , i.e.
For every a E E ,
Hence
Proposition 5.6.3.14 ( 8 ) Let E be a W*algebra , G and H von Neumann right Emodules, and A a Fourier set of G . For each J E A let be a n element of H such that
for all J , q E A . For every x E em(A) , put
(Theorem 5.6.3.13a), Proposition 5.6.3.8 a), Proposition 5.6.3.9 a a ) For 17 E G , { E H , and x E em(A) ,
+ b).
5.6 Hilbert right C*Modules
b) If x E em(A) then t
E LE(G,H ) ,
+*
+
x =5,
11211 = 1 1 ~ 1 1 ~
a) For a E E ,  +
((2171C)? a) = ( ( a , I),x v ) =
so that
b) By a), for all (7,< ) E G x H ,
( 7 1 5 ~=) ( $ C I ~ ) *=
so that +
x E
+
y
L E ( G ,H ) , x = x
359
5. Halbert Spaces
360
For every
<E A, 1;
2
1<;1
=
Idol
so that + 11x11 2 11x1100 ,
For 7 E G ,
tEA
==$ (&IF)
= x(t)(vIE)
so that by Theorem 5.6.3.13 e),f)
1 1 ~ ~ 51 llxll~lla1I2 1 ~ (Corollary 4.2.1.18),
Proposition 5.6.3.15 ( 8 ) Let E be a W*algebra, H a von Neumann right Emodule, and A a Fourzer basis of H . For every x E em(A) put
(Theorem 5.6.3.13 e), Proposttion 5.6.3.8 a), Proposition 5.6.3.9 a + b)) a)
The map
is an znjective unital W *homomorphism (Theorem 5.6.3.5 b)) with
for every (a,q,C) E A x H x H .
5.6 Hilbert right C*Modules
b)
If x
E lm(A) and B := {x
# 0) then
&
is the carrier of
a) By Proposition 5.6.3.14 b), cp is welldefined, injective and involutive. Take x,y E ["(A) . For 7 , E H ,
<
(Proposition 5.6.3.14 a)), so that +
++
xy = x y . Hence cp is an algebra homomorphism and by Theorem 5.6.3.13 f ) , it is unital. For x E ["(A) , by a),

so that, by Theorem 5.6.3.5 d ) , cp is a W*homomorphism and @((al7 , O ) = ( ( ( J I O ( ~ l 0~, ) ) C EE A['(A). b) By a) ,
E
Pr L E ( H ) and + +
+
egx = x. Let u E L E ( H ) with
Then for ( E B ,
5. Halbert Spaces
562
and so
for every 7 E H (Proposition 5.6.3.4 c)). Hence
and
&
is the right carrier of
4
2 . By Corollary 4.3.3.9, e e
+ is the carrier of z .
rn Proposition 5.6.3.16 ( 8 ) Let E be a finitedimensional C'algebra, H a Hilbert right Emodule, and B the set of Fourier sets A of H such that (((1)zs a mznimal element of Pr E\{O} for every ( E A . Then every maximal element of M zs a Fourier baszs of E . In particular, H has a Fourier basis belonging to B . Let A be a maximal element of B and assume A is not a Fourier basis of H . Then there is a E H such that
<
By Proposition 4.2.7.22, there is a finite set B of minimal elements of P r E\{O) such that
Take p E B . Then
(Corollary 4.2.7.6 a +e, Proposition 5.6.1.2 b)) and
for every 7 6 A . Hence A A.
U {
B and this contradicts the maximality of
5.6 Hzlbert right C*Modules
Proposition 5.6.3.17 in it W*algebra E :
The following are equzvalent for every family ( x , ) , ~ ~
a ) ( x , ) , , ~is summable zn EE b) For every a E E and
for every K E
y f (I\J)
E
> 0 there is a J
E
qf( I ) such that
.
If these conditions are fulfilled then:
e ) ( x , ) , , ~zs summable zn E , for every J , the map
is continuous, where p ( I ) is identified with (0, I ) ' , and
is a compact set of E E and a bounded set of E .
a
+ b . There is a
J E p f ( I ) such that
for every K E p f ( I ) , J c K . T h e n for every K E
so that
363
p f (1\J),
364
5. Hilbert Spaces
b
+a.
Take a E E . By b), there is a J E p f ( I ) such that
for every K E p J ( I \ J ) . Put
Then for K E Q f ( I ) ,
Since a is arbitrary, it follows from the theorem of BanachSteinhaus (Theorem 1.4.1.2) that
is a bounded set of E . Hence, by the AlaogluBourbaki Theorem, it is a relatively compact set of E, . Let x be a point of adherence in E, of the image of the filter 5, with respect t o the m a p
Let a E E and
E
> 0 . By b) , there is a JoE pf( I ) such that
for every K E Y I ( I ) , K that
Then for K E Y j ( I ) , J
c I \ J o . Further
c K,
there is a J E p , ( I ) , Jo c J , such
5.6 Halbert right C ' M o d u l e s
Hence ( x , ) , is ~~ summable in EE and x is its sum. c) follows from a) and Proposition 4.4.1.3. d ) For every a E E ,
Since a is arbitrary,
Similarly,
and
e) By a H b , ( x , ) , , ~is summable in EE for every J C I . Take a E E E > 0 . By b), there is a J E p f ( I ) such that
for every K E
for every K
p f (I\J)
c I\J.
. By continuity,
Let K l ,K Z E p ( I ) with
5. Hzlbert Spaces
366
Then
Hence the map
is continuous. Since a is arbitrary, the map
is continuous. It follows that
is a compact set of E E . By BanachSteinhaus Theorem (Theorem 1.4.1.2), it is a bounded set of E .
Corollary 5.6.3.18
Let E be a W*algebra, (x,),,,
a summable family in
E E , and
(Proposition 5.6.3.17 e)). For every a E E and J C I put  ( a , J ) := sup
{
(gX.,
.)I
K
c I\.J)
5 ~llall.
5.6 Hilbert right C*Modules
a ) If a E E , J
cI,
K E q f ( I \ J ) , and a E l m ( I ) then
b) For every a E E and
E
> 0 , there i s a J
for every a E e m ( I ) and K E C)
For every
CY
E
pf(I)such
that
(I\J) .
E e m ( I ) , ( a , ~ , ) ,i ,s ~summable i n E, and
d) T h e maps
are continuous and
as a compact set of E,.
a ) follows from Proposition 1.1.1.5. b) By Proposition 5.6.3.17 a b , there is a J E y j ( I ) such that
5. Hilbert Spaces
368
for every (Y E em(I) arid K E p f ( I \ J ) . c) By b) and Proposition 5.6.3.17 b =. a , ( ~ , x , ) , ~isIsummable in EE for every a E !"(I). The inequality follows from a ) . d) By c) , the map em(I) + E , is continuous. Let a E E and
E
a

E
a,xL LEI
> 0 . By b) and c ) , there is a J E p f ( I ) such that
for every a E (IK#)' and K
c I\J.
Let a ,0 E (IK')' E
1%  PtI < 3 ( 1 + 87) for every
L
E J . By c) ,
Hence the map
is continuous. Since a is arbitrary, the map
is continuous. It follows that
is a compact set of E,
such that
5.6 Hzlbert right C*Modules
36.9
Let E be a W*al.qebra and let (x,),,~, ( Y X ) A E L be sum~nablefamilies i n E E .
Proposition 5.6.3.19
5
a)
XEL
( ( eL XE IL ) Y X )
(..
= E (5 ")) (5") ( f Y X ) . LEI
=
XEL
XEL
LEI
b) T h e map
(Proposition 5.6.3.17 e)) is continuou.~,where p ( I ) and p ( L ) are identified with { O , l ) ' and {0,1)', , respectzvely. a ) For a E E ,
( j (eYA), (yA.atxL) ((ezL) = ( eX yE LX , a = xL ELI )
=
LEI
A €1,
XEL
LEI
= L E I (z.,
(5
X E L .A)
= XEL
a) =
XEL
LEI ( 2 .
=
yX,a)
(2
,A€ t, y X )
,
.a)
and t,he assertion follows from the fact that a is arbitrary. b) By Proposition 5.6.3.17 e), ( X ~ ) ~ ~and . J ( Y , ) , , ~ are summable in E, for all .I C I and K C L . The assertion follows from a ) and C. Constantinescu, Spaces of Measures, Theorem 3.5.2 c). Proposition 5.6.3.20
Let ( x , ) , ~be~ a fa~nily of the W * algebra E such
that
for all distinct
L,
X E I . Then the following are equivalent:
5. Hzlbert Spaces
3 70
a) (x,),,~ zs summable zn E, b ) (x:x,),,I
.
zs summable zn E,
If these condztzons are fulfilled then c)
LEI
LEI
a
LEI
* b & c . By Propositions 5.6.3.17 c),d) and 5.6.3.19 a),
( ) ( ) (5.:)(5 5 5 LEI
LEI
=
LEI
E
= ):
LEI
E
xX) =
€ 1
(x:
AEI
xA) =
E
x:xA
=
X~X,
LEI
rE1 XEI
*
b a . Let a E E and let (y, l a [ ) be its polar decomposition (Theorem 4.4.3.5 a)). For every J E T J ( I ) , b y Proposition 2.3.4.6 c ) ,
5 (Y*
(p)(5
XL)
By Proposition 5.6.3.17 a =+ b , for every that
E
Y? l a , ) ( l , , a l ) =
> 0 , there is a J
E T J ( I ) such
5.6 Hilbert right C ' M o d u l e s
for every K E g f ( I \ J ) . By the above,
for every K E p , ( I \ J ) . By Proposition 5.6.3.17 b in E,.
+a,
( x , ) , ~ Iis summable
Remark. The above conditions imply that ( ~ ~ , l is ~ sumrnable ) ~ , ~ in E, for every cr 2 2 , since
, ~ ~summable in is a cofinite subset of I . But it may happen that ( I X , ~ ~is) not EE for every (Y E ]0,2[. A counterexample in this sense is given by the sequence (x,),,~ in L(e2),where
for every n
N
Corollary 5.6.3.21 such that
for all distznct
L, X
Let E be a W*algebra and (p,),El a family in Pr E
E I . Then (p,),,,
is summable i n E, and
The set
is upward directed and bounded (by 1) so that (P,),,~ is summable in E, (Theorem 4.4.1.8 b)). By Proposition 5.6.3.17 c) and Proposition 5.6.3.20 a =+ b&c,
5. Halbert Spaces
372
Proposition 5.6.3.22 such that
for all distinct
L,
Let E be a W*algebra and ( P . ) [ ~ a~ famzly zn Pr E
X E I and
Then the map
LEI
(Corollary 5.6.3.18 d)) is a unztal injective W *  h o m o m o r p h i s m with pretranspose
For a , B E e r n ( [ ) ,by Propositions 5.6.3.19 a ) and 5.6.3.17 c),
Hcnce the map is an ir~volutivealgebra homomorphism. It is obviously unital and injective. Moreover, for a E t m ( I ) and a E E ,
so that the map is a Whomomorphism with pretranspose
5.6 Hilbert right G' Modules
5.6.4 Examples Proposition 5.6.4.1 ( 0 ) Let ( H l ) , , I be a family of (weak) semiinnerproduct right Emodules and
If
:=
@ H,
:= {
fE
LEI
n
the family ((<,Ifl)),,l is summable in E ) .
LEI
a ) H is a vector subspace of
fl H,
and H endowed with the maps
LEI
as a (weak) semiinnerproduct right Emodule. b ) If H, is an innerproduct right Emodule for each innerproduct right E module and
for every f E H . If <,71 E
n H , such that
i
E I , then H is an
E H and
g
LEI
for every
L
E
I , then f E H and
< 11q11. H is a Hilbert
c ) If H , is a Hilbert right Emodule for each right E module.
L
E I , then
d) If H, is a (unital) Hilbert Emodule for each Emodule H endowed with the map
L
E I , then the Hilbert right
is a (unital) Hilbert E module. Moreover if ( x , ) , , ~zs a bounded family in E then
for everg
<E H
3 74
5. Halbert Spaces
e) Take
<EH
and for every
L
E I put
is summable i n H and
Then
f ) If (IX)XEL is a parlition of I then a)
a) H ,
isomorphic to a) H, i n
X€L LEI*
LEI
a natural way. a ) It is easy to see that H is a weak semi innerproduct right Emodule whenever I is finite. . Proposition 5.6.1.2 d),e) Take <,71 E H and x E E . Let J E V f ( I ) By applied to the above construction with respect to J ,
By Proposition 1.1.6.6, it follows that the families ((<'I~,)),EIand (((Lxl~,x)),E, are summable. In particular (
for every
<+
71
L
E I . By the above, the family ((5,
E H and H is a vector subspace of
product right F module for each
+ 7/,1<, + qL))lE1is summable, so
n H,
. Assume
H, is a semiinner
LEI
L
E
I . Then for <,17 E H ,
= (1((CLlCL)+ ( a l e ) ) . x ( ( a l I , )  (61%)) LEI
LEI
Hence H is a semiinner product right Emodule.
5.6 Hilbert right C*Modules
11) By Corollary 4.2.1.18,
IlrLI12 = II(E~I<~)II I 11 C(E~IE~)II = II(SIE)II XEI
for every
L
E
I . Hence H is an innerproduct right Emodule and SUP
IlELll
~ € 1
I IIEII .
The other inequality follows from
The final assertion follows from Corollary 4.2.1.19. c) Let (<(n))n,N be a Cauchy sequence in H . For every m, E IN such that
For all n,p E N 4.2.1.18,
E
> 0 , there is an
, T L 2 m, , p 2 m, . Let J be a finite subset of I . By Corollary
for every
< E H . It follows that
for every
E
(
is a Cauchy sequence for every
> 0 and every n,p E I N , with n 2 m, , p 2 m, . In particular,
tL:= lirn
n+m
i
E I . For each
L
E I , put
Then for every finite subset J of I ,
for all
E
> 0 2nd every n E IN with n > m, . Hence
<(")

< E H and
5. Hilbert Spaces
376
for every
E
> 0 and n
E I N , n 2 m, . Thus f E H and ( c ( ~ ) ) , , ,converges N to
f in H . In particular, H is a right Hilbert Emodule.
d) The map E x H Then
+H
is obviously bilinear. Take x , y E E and f , 7 E H .
LEI
LEI
By Corollary 4.2.1.19 and Proposition 5.6.1.2 e), ( x , J , ) , ~ E IH and
e) Let
E
> 0 . By the definition of H , there is a .I
for every L C I \ J . Hence, for K E
E
P f ( I ), such
that
P , ( I ) ,J c K ,
LEI\K
Thus ( f { , l ) , , l is sun~mablein H and
X
((1)= C
LEI
f ) follows from a).
Remark. Take
< E fl HL. If LEI
then the family ((f,lf,)),,I is absolutely summable, so it is summablc. The converse is not true as thc following example shows: E := co , I := N , HL:= E for every L E I , En := &en for every 71 E N .
5.6 Hilbert right C*Modules
Example 5.6.4.2
(0)
377
Let I be a set, F a closed right ideal of E , and is summable} .
e 2 ( I ,F ) := { J E F'lthe family
a ) e2(1,F ) is a vector subspace of F1 and e 2 ( I ,F ) endowed with the maps
t 2 ( 1 ,F ) x e 2 v ,F )
4E
,
( I ,7 )

7;E' iEl
is a Hilbert right E module.
C)
If F is an ideal of E and ( x , ) , , ~is a bounded family in E then for every E E e 2 ( I ,F ) ,
In partzcular, t 2 ( 1 ,F ) endowed with the map E x e2(1,F )
('(1,F ) ,
t
( x ,E )
(xEL)LEI
is a (unital) Hilbert Emodule. d) If F is an ideal of E and
I* :=
(<:)LEI
E e 2 ( I ,F )
<
whenever E e 2 ( I ,F ) (Corollary 4.2.6.2) (this is the case when F is finitedimensional or I is finzte) then there is an a > 0 such that
IIE'II r ~ I I I I I for every J E e2(1,F ) and (x<)*= <*x*,
((x)' = XI'$*
for all ( x , J )E E x e2(1,F ) . e ) Take
(1,
E2,53, I 4
E
:
E
IR' and put
+,
L
* I I ( I , )+~G ( L +) ~h ( ~ )+j ( q ( ~ ) k .
5. Hilbert Spaces
378
Then the following are equivalent:
b,(4
E e2(I, R),
el)
e2)
< E P2(I,M) ,
e3) <* E e2(I,M) ,
e2u,R).
e4) ( I I < ( L ) I I E ),~I
If these conditions are fulfilled then
a), b), and c) follow from Example 5.6.1.5 and Proposition 5.6.4.1. d ) We may assume that I is infinite. Suppose that the map
is not continuous. Then there is a sequence (
for every n E IN. By continuity, for every n E I N , there is an InE p J ( I ) such that
We may assume that (In)nENis a disjoint sequence. Put
( <,, if L E In for some n
By Corollary 4.2.1.18
E IN
5.6 Hilbert right G*Modules
i.e.
< E 12(1,F ) . On the other hand
for every n E IN, so that t*$! 12(1,F ) , which contradicts the hypothesis. Thus the above map is continuo~isand there is an a > 0 with
<
for every E 12(1,F ) . The other relations are easy to see. If F is finitedimensional, then by Corollary 4.2.1.20, never t E t 2 ( I ,F ) . e) By the Examples 2.3.1.46 and 4.1.1.31,
for every
L
t* E
12(1,F) whe
E I.
Remark. The hypothesis of d ) may fail (sez Remark of Proposition 5.3.2.13). Even if it holds, a may be different from 1 . Indeed, put
Then
so that
5. Hilbert Spaces
380
, ) be the Proposition 5.6.4.3 ( 0 ) Take n E IN and let En := t 2 ( N n E Hilbert Emodule defined in Example 5.6.4.2 c). For <'E E l n , put
Then
F E (En)' and
(g
l
2
)
5
ill 5
2 llc:ll 1=1
for every <' E Eln and the map
is a homomorphism of Emodules and an involutive isomorphism of Banach spaces (Proposition 2.2.7.2, Example 2.2.7.21). We have
for every
< E En
(Example 5.6.4.2 b)),so that
Take 6' E ] O , l [ .For every z E E n , there is a
Given i E IN,, put
Then
and so
F E (En)' and
<, E
E\{O) such that
5.6 Hzlbert right C*Modules
Since 0 is arbitrary, we see that
By Example 5.6.4.2 b), the norm of En is equivalent to the Euclidean norm. By Proposition 1.2.2.13, the map
is an isornorphism of Banach spaces. Take I' E Eln. Then
for every J E E n , so
and the above map is involutive. Moreover, for every x E E ,
for every
I E E , so that
Proposition 5.6.4.4
Let R, S, T be sets and let
be a conjugate involution. For each u E LE(e2(S, E ) , e2(T,E ) ) define
Ti : t 2 ( S E , ) 4.t2(T,E ) ,

ct vJ
.
5. Hilbert Spaces
382
a)
(1
17 E P2(T,E ) =+
(?I$
=
G)
b) T h e Banach space P2(T,E ) endowed with the znvolution
is a n involutzve Banach space. c) If u E LE(P2(S,E ) , P2(T,E ) ) and x E E then
  i z = C5
xu = xu.
(Example 5.6.4.2 c), Proposition 5.6.1.8 g)). d ) The Banach space LE(e2(S,E ) ,e2(T,E ) ) endowed with the involution
LE(P2(s,E ) , P 2 ( ~ E, ) ) 4LE(P2(s,E ) ,['(T, E ) ) , u ct ii zs a n involutive Banach space.
e ) u E L E ( e 2 ( S , E ) , P 2 ( T , E )v) E, L E ( P 2 ( R , E )P, 2 ( S , E ) )3 ~02)=COG
f ) The map
is a conjugate znvolutzon of the C*algebra LE(P2(T,E ) ) (Theorem 5.6.1.11 a)). g)
Assume IK = C . Let F be a Csubalgebra of L E ( P ~ (E I ,) ) such that 6 E F whenever u E F and let F denote the complexification of the underlying real C ' a l g e b r a of F . T h e n the map 0
is a n isomorphism of complex C*algebras.
a ) (?I$ =
C ;i;G LET
=
b) follows from a ) . c) h is linear and
c 2E c te'r =
LET
cv ; ~ h 
=
V
= (117)
t ET
5.6 Halbert right C*Modules
so that
Take ( E e2(S,E ) and 17 E E2(T,E ) . By a) ,
so that

ii E C E ( ! ~ ( E S ,) , t 2 ( ~ , ~ 2 ) )=, u * . Moreover,
so that
iz = iiz, (1) The involution is conjugate linear, so that the assertion follows from c). c ) For all ( E e2(R,E ) ,
f ) follows from c),d), and e). g) follows from f ) and Theorem 4.1.1.8 e) Proposition 5.6.4.5
(0)
Let E be a finitedimensional C*algebra and
(e,),N, an algebraic basis of E consisting of invertible elements of E . For ever9 x E E define ( x , ) , ~ N , E IK" such that
384
5. Hilbert Spaces
and put
For S , T sets and
*
:= (
a
)
E
( L ( P ( T )@ . (s)))~,
:EN.
where for
E
e2(S,E ) and j
E
N,, I1E e 2 ( S ) is defined by
Then €

LE(e2(s,E ) , ~ ' ( TE, ) ) , ii* = u*
for every u E ( L ( e 2 ( S )e,2 ( T ) ) ) n(Definition 5.6.1.7) and the map
zs bzjective and linear. For every z E N, L
so that
For
5 E t 2 ( S ,E )
and
71 E
t 2 ( T ,E ) ,
5.6 Hilbert right C*Modules
Hence
ii E ,c,(e2(s, E ) , e 2 ( ~E :) ) , ii*

= ,u*.
It is obvious that the m a p is linear. Take
u E ( , c ( e 2 ( s )e, 2 ( T ) ) ) " and assume ii = 0 . Fix k E ISnrtake
and we obtain successively
< E e 2 ( S ) ,and define 7 E e2(S,E ) by
5. Halbert Spaces
386
the map is injective. Take
v E L E ( e 2 ( sE, ) , e2(T,E ) ). Since E is finitedimensional, it is unital. Denote by 1 the unit of E . For every i E IN,, put
IL :=
For every
(zL,),E ~ ~( L, ( e 2 ( s )e ,2 ( ~ ) ) .) ,
< E e2(S.E ) ,
(Proposition 5.6.1.8 d)), so that ii = v and the map is surjective.
Proposition 5.6.4.6 ( 8 ) Suppose E is a Wealgebra and let ( H l ) r Ebe~ a family of Hilbert right Emodules. Put
and
for every
< E n H, LEI
and J
cI.
38 7
5.6 Hilbert right G*Modules
a ) If J , 17 E H then J EE . Put
+ 1) E H
and the family
((JLI~L))LE~
is summable i n
b) H endowed with the right multiplication
H xE
*H ,
((,I)
:= ( < L X ) L E /
and with the innerproduct
is a Hilbert right Emodule c)
For every a
E
E , J E H , and
E
> 0, there is a J
E
P I ( I ) such that
( ( ( ~ I < I \ K ) , ~ ) (< E
for every 17 E H # and K E P 1 ( l ) ,J
c K . In particular,
for every J E H , where for every H i n a natural way.
I we identijed H , with a subset of
L
E
d) If H , is a (unital) Hilbert Emodule for each L E I then the Hilbert right Emodule H endowed with the left multiplication
is a (unital) Hilbert Emodule. e ) If E zs finitedimenszonal then
f ) Assume there is a famzly ( P ~ ) i~n ~PrI E\{O)
for every
L
E I . Put ft
for every
L
such that
:= ( ~ L X P L ) X E I
I T h e n ( f L ) r E I is a Fourier basis of H .
5. Hilbert Spaces
388
a) We may assume IK = C (Theorem 4.4.1.2, Proposition 5.6.1.6). By Proposition 5.6.1.2 b),
<
for each L E I SO that + 7 E H (Proposition 5.6.3.7 a + b). The family ((
4). b) Take <,7 E H and x E E . Then
for every J E q f ( I ) (Proposition 5.6.1.2 b)) so that <x E H (Proposition 5.6.3.7 a 3 b). Moreover,
(Proposition 4.4.1.3),
(Proposition 4.4.1.3),
389
5.6 Hilbert right C*Modules
be a Cauchy sequence in H . For every
Let
E
> 0 , there is a
p, E K such that
for all m, n E I N , m 2 p, , n 2 p, . Since for every
L
E I , the map
is linear and continuous, (
For every
E
> 0 , m E N , m > p, , and J E y f ( I )
l <$m) Hence
<(rn) 
 fJll = Iim n+m
<E H
l <$m)

<,y)ll5
Iirn
n+m
Il<("') 
5E
and from
it follows
(Proposition 5.6.3.7 and Corollary 4.2.1.16 a e b ) . By a ) , f E H and lim f(") = 6
m+w
Thus H is a Hilbert right E module. c) Assume the contrary. By recursion, we construct an increasing sequence (Jn)nEn in v f ( I ) and a sequence (77(n))nEN in H# such that
for every n E N . Take n E IX and assume that the sequences were constructed up to n  1 . By the above assumption, there is a J2n1E Q f ( I ) and an E H # such that J 2 n  2 c J2nL (JO:= 0) and
5. Hilbert Spaces
390
By a ) , there is a J2, E
P f ( I ), J2n1c JZn such that
This finishes the recursive construction. P u t
for every n E N Define
i V ( " ) if
7/:I+E,
0 By Corollary 4.2.1.16 a
for every J E
P I( I )
L
E Kn for some n E N
1
SO
otherwise.
=+ b ,
that 7 E H . For every n
Hence
which is a contradiction. d) Take <,17 E H and z, y E E . Then
N,
5.6 Hilbert right C*Modules
for every
L
E
I
SO
that
for every J E p j ( I ) . It follows x< E H and
( x t I x 0 5 llxll2(tIO. T h e relations
are easy to verify. e ) follows from Minkowski's Theorem. f) For L , X E I ,
Let
<E H
for every
L
srich that
E
I . Then for every
L
E
I,
t;
:b,xp,lx
0 = (
= P,<, =
<,
A€ 1
Hence
< = 0 and
fL)LElis a Fourier basis of H .
Theorem 5.6.4.7 ( 8 ) W e use the notation and the hypotheses of Proposition 5.6.4.6 and assume i n addition that H , is selfdual for each L E I . a ) H is selfdual.
b)
a ) H , is a Banach subspace of H dense i n H,, LEI
c) For every J
c I , the map
belongs to Pr L E ( H ) and for every u 71
E
LE(H)
= lim 7r~u7r.~ , J , ~ I
where the limit as taken i n ( L E ( H ) ) ,
392
5. Hilbert Spaces
d ) Put G : = { " ‘ E L E ( H ) I u ( ~L) EHIL ) C La)H., EI
U * ( ~ H . )
OH.}
c
LEI
and
for every u E G . Then G is a unital C*subalgebra of L E ( H ) , dense i n (LE(H)), , generatzng t E ( H ) as W*algebra ( a ) and Theorem 5.6.3.5
a)),
ii E
(
tE 0 H ~ )for every u E G , and the m a p LEI
is a n isometry of C'algebras.
a) By Proposition 5.6.3.3 b + a , it is sufficient to prove that pact. Let 5 be an ultrafilter on H # . Put
where the limit is taken in (H,),, (Proposition 5.6.3.3 a J E p J ( I ) . Then
HZ
+ b) . Let
is com
a E E and
IIvJI/
I((V.I~VJ)~~)~ I((<~VJ)>~)~ 5 llall
(Proposition 5.6.1.2 d ) ) so that
1 1 ~ ~ =1 1 II(VJIVJ)~~ ~ = SUP ~((vJIvJ),~)~ I1 1 ~ ~ 1 1 ~ a€E*
Since .I is arbitrary, 7 E H # (Proposition 5.6.3.7, Corollary 4.2.1.17 a a b). Take a E E , ( E H , and E > 0 . By Proposition 5.6.4.6 c), there is a J E PJ(I)such that
for every
< E H# . It follows
5.6 Hilbert right C' Modules
5 I((v  C I t ~ ) , a ) l+ 2~ for every
Since
E
< E H#
and so
is arbitrary,
'jB (=)(o = E ) ( V )
HH#
<
arc also arbitrary, 5 converges t o 71 in H'. Hence is Since a and H compact. b) follows from Proposition 5.6.4.6 c) . c) It is easy t o see that T J E P r L E ( H ) for every J c I . Take (a,<,71) E E x H x H and E > 0 . By Proposition 5.6.1.6 c), there is a J E y f ( I ) such that, for every E H # and K E Vf ( I ) ,J c K ,
<
l ( ( < l ( ~ * ~ ) r \ t i )< . ~2(1 )l Hence for every K E (Pf ( I ) ,J C K ,
E +
l l f l l .)
394
5. Hilbert Spaces
so that
Hence
in ( L E ( H ) ) ~ . is a unital C'subalgebra of L E ( H ) . By c) , G d ) It is easy to see that is dense in (LE(H))" , SO that G generates C E ( H ) as W* algebra (Theorem 5.6.3.5 b) and Corollary 4.4.4.12 a)). Let u E G . For [, 7 E a ) H L , LEI
so that ii E LE(
a> H,)
and ii* = i*. In particular, cp is involutive. It is easy
LE l
to see that cp is an injective unital algebra homomorphism. We want to show that cp is surjective. Let u E LE( H,) and let 3 be an ultrafilter on v f ( I ) finer than 3 1 .
a>
LEI
Then
for every ( E H and J E ?,(I). By a) and Proposition 5.6.3.3 a define
* b , we may
where the limits are considered in HH . Take <, 7 E H . Then for every a E E ,

= (a*,O(wv) =

1jy ( a * ,E ) ( u * v ~=)
5.6 Hilbert right C*Modules
lim ( ( u * v ~ \ ) ,
J,3
= lim lim(((JIu8qK),a ) = l i m ( ( f J I w v ) ,a ) = 5,s
K,3
J,3
Hence
and so v E L E ( H ) and v* = w . For every
<E
a) H , , LEI
so that
T h u s v E E and
Therefore cp is surjective and so it is an isometry o f G*algebras. Proposition 5.6.4.8 P r E (in P r E ) . Put
for every
L
H
Let E be a W*algebra and ( P , ) ~ ~a, family i n Ec r
E I.
a ) H , is a von Neumann (right) unztal Emodule for every
L
E I
396
5. Hilbert Spaces
W
b) H :=
a)H, is a uon Neumann (right) unital Emodule. For every x E E
LEI
(resp. x E E C )put
Z :H
4 H
c ++xE
,
(resp. lx)
c) Z E C E ( H ) for every x E E (resp. x E EC). Put q : E (resp. E C )+ C E ( H ) , x ct Z
d ) cp is a W*homomorphism (Theorem 5.6.3.5 b), Corollary 4.4.4.12 e)) and
for every ( a ,[, 17) E E x H x H , where i : EC + E denotes the inclusion map and @ the pretranspose of cp .
e) Let
c,11 E H
such that
for all distinct and
L,
X E I . Then
(<,),,I
f ) Assume p,px = 0 for all distinct
L,
and
( T / , ) ,are ~ ~summable
X E I and put
(Corollary 5.6.3.21). Then the map
is an isomorphism of uon Neumann (right) Emodules.
in E,
5.6 Hilbert right C*Modules
397
a) By Example 5.6.1.5, H, is a Hilbert (right) unital Emodule and by Proposition 5.6.2.3, it is selfdual. b) By a) and Proposition 5.6.4.6 b),d) , H is a Hilbert (right) unital Emodule and by Theorem 5.6.4.7 a), it is selfdual. c) was proved in Proposition 5.6.1.8 h ) . d ) By Proposition 5.6.1.8 h), cp is an involutive algebra homomorphism. For every x E E , (resp. x E EC),
By Theorem 3.6.3.5 d ) , cp is a W*homomorphism and
e) By Proposition 5.6.3.20 b + a, (<,),,, and (qL)lElare summable in E, . For J E
Y J ( ~ 3
By Proposition 5.6.3.19 a) (and Proposition 4.4.1.3),
f ) For every <, 7 E H and distinct
L,
XE I,
398
5. Hilbert Spaces
By e ) ,
and
(<,),,I
( v ' ) , are ~ ~summable in
(5 (5L ) LEI
Let
CE
%) *
LEI
p E . Then pLC E HL for every
for every J E
yr( I ). Hence
L
E, and
=
E I and
(pLC),EI E H . Moreover
(Proposition 5.6.3.17 d ) ) , so that the map
LEI
is surjective. Example 5.6.4.9
Let I be a n infinite set. Put
(Proposition 5.6.4.8 a), b)), := { U E
L E ( H ) I u ( t 2 ( IE , ) ) C e2(1,E ) ,
u * ( t 2 ( 1E, ) ) C t 2 ( 1 ,E ) ) ,
a n d f o r e v e y < E H and i , A E I ,
Fix a poznt
for every
LO
i n I and define for every
<EH
and A, p E I .
L
E I a m a p p, : H
+ H
by putting
5.6 Hilbert right C*Modules
a) If L ,X E I \ { L ~ ) then p, E Pr G and
b) The family ( P , ) , , ~ is unbounded i n G
c ) C E ( e 2 ( IE , ) ) is not a W *algebra. a) For < , V E H and p~ I ,
=
1 
)
(
+
"El
=
1

"El
<",
(&LO
tLL)
6', (6UL0 + 6UL)Siu, =
+ 6"') 6',
(v',,' + Sill) =
so that
Hence p, E C E ( H ) and p, = pt . It is easy to see that p E For E H and p , v E I ,
<
G for every
1
E I.
4 00
5. Hilbert Spaces
so that
P~E P r G .
b) Let u be an upper bound of the family (p,),,~ in
< := (6,,,er),,, Then for
L
E I \ { L ~ )and A, j~ E
1 (P~<)A, = 2(
G. Put
E e2(1,E ) .
I, 1
+ 6~1)= 26L11(6~L0 +~ A I )
so that 1


(PL<~<)F = ~ ( P L < ) A P ~ X5 , 6 (
.
XE I
By Theorem 5.6.1.11 c,
+ c2 ,
Hence the family ((u
and this is a contradiction. is c) By a), (p,),,,\(,,) is a family in Pr G . By b) and Theorem 4.4.1.8 i), not a Wealgebra. By Theorem 5.6.4.7 d), L ~ ( l ' ( 1 ,E)) is not a W*algebra.
5.6 Hzlbert right G*Modules
401
Proposition 5.6.4.10 ( 8 ) Let E be a W*algebra, H a von Neumann right E module, and A a Fourier basis of H . For every E A denote by
<
the con Neumann right E module (Proposition 5.6.2.3, Proposition 5.6.4.8 a ) ) and put
a) For every 17 E H
,
and the map
zs a n zsomorphism of von Neumann right Emodules (Theorem 5.6.4.7
a)).
b) If @ denotes the pretranspose of cp then
for every ( a , C) E E x G
c) For every p E Pr E put
and define
If E is finitedimensional then H is isomorphic t o
a) By Propositiori 5.6.3.9 a
+ b , for every 6 E A ,
(vl<) = (vl<(
4 02
5. Hilbert Spaces
is surnmable in EE , SO that
47 is obviously linear and injective (Theorem 5.6.3.13 b).
Take
BE
C
E G . By Corollary 4.2.2.3 (and Proposition 5.6.1.2 b)), for every
Pf(A),
By Proposition 5.6.3.8 a), the family ({(F)FEAis summable in H H . P u t
Then
for every toE A . Hence
and p is surjective. Take 9, E H . By Theorem 5.6.3.13 f),
<
Hence p is an isomorphism of von Neurrlarin right Emodules. b) For every q E H ,
5.6 Hilbert right C*Modules
so that
) . Theorem 5.6.3.13 a ) , ( < X ~ )
in H . Put
Let 17 E H and put
By 'Theorem 5.6.3.13 d),f), y E
a) e2(Tp,pE) and
p€Pr S
so that the map
is surjective. By Thcorerri 5.6.3.13 f ) ;
for all x, y E
a) e2(TP,pE). Hence the above map is an isomorphism of p€Pr E
Hilbert right Emodules.
Corollary 5.6.4.11 Fourier basis of H . a ) For every 17 E H
(8 ) ,
Let H be a Hilbert right Mmodule and A a
4 04
5. Hilbert Spaces
b) The map
is a n isomorphzsm of Hzlbert right Mmodules.
c) All Fourier bases of H have the same cardinality and H m a y be endowed wzth the structure of a unztal Hilbert Mmodule. d) T w o Hilbert right M  modules are isomorphic iff their Fourier bases have the same cardinality. e) If H denotes the real Hilbert space obtained by endowing the underlying real vector space of H wzth the scalar product
(Proposition 5.6.2.5 a),e)) then A U ( A i )U ( A j )U ( A k ) is a n orthonormal basts of H .
a ) By Proposition 5.6.2.3 f), H is selfdual so that the assertion follows from Theorem 5.6.3.13 d),f) . b) follows from a ) and Proposition 5.6.4.10 c). c) and d) follow from b) . e) is easy to see. Proposition 5.6.4.12 ( 8 ) Take E E {R,C, M} , n E I N , and H a Hilbert right En,, module. For every z E IN,, put
a ) pl zs a mznzmal element of Pr E,,,\{O}
for every i E IN,.
b ) If 2L denotes the set of Fourzer sets A of H such that
for every ( E A , then every mazzmal element of 2L (U ordered by incluszon) is a Fourzer basis of H .
c) There zs a unzque cardznal number N such that H is isomorphic to e2(N,P I E,,,,) .
5.G Hilbert right C*Modules
a) It is easy to see that p, E Pr E,,,\{O}. Then
(Corollary 4.2.7.6 a
Let q E Pr E,,,\{O}
4 05
with q 5 p, .
+ f) so that
for all j , k E INn. Hence q = p and p is a minimal element of P r E,,,\{O}. b) Let A be a maximal element of 2L. Assume A is not a Fourier basis of H . Then there is an Q E H such that
and
Since
there is an i E Nn with q p , # 0 . It follows ~ 1 ( 0 1 1 7 ) ~=1 ( V P I I V P I )
so that there is an
Put
and
For j , k E N, ,
(Y
> 0 with
f 0,
4 06
5. Halbert Spaces
so that
and
for every ( E A . This contradicts the maximality of A . c ) The existence follows from b) and Proposition 5.6.4.10 c). The uniqueness is obvious.
Remark. N is a kind of Hilbert dimension for H Proposition 5.6.4.13 ( 8 ) Assume E is the C'direct product of a finite famzly ofunital C*algebras and for every L E I , denote by e, the unit of E, . Further let G , H be Hilbert right Emodules. a) For every L E I , H e , is a Hzlbert right Emodule and a Hzlbert right ELmodulein a natural way. b)
The map
is an isomorphzsm of Hilbert right Emodules and
is its inverse c) For every
5.6 Hzlbert right C*Modules
Then ii E LE(G,H ) and i' = 71' for every
d) The map
is an isomorphism of Banach spaces e) The map of d) is an isomorphism of the C*direct product of the family (Ls(He,))LEI of C*algebras (Theorem 5.6.1.11 a)) onto L E ( H ). a) and b) are obvious.
c) ?or
((,I))
EGx H,
so that
d) For every v E L E ( G ,H ) and V , : Gel
Then for
(<,TI
E
L
E I , put
+ He,,
< 4 (v()e,
(Ge,) x ( H e , ) ,
= (
so that v, E LE,(Ge,,He,). Putting 7' := ( v J L E I ,
it follows
4 08
5. Halbert Spaces
for every
<
G , i.e.

u=v and the map is surjective. The other assertions are easy t o see. e ) follows from a),c),d).
Example 5.6.4.14 ( 0 ) Let G , H be Hilbert right Emodules. Then L E ( G ,H ) endowed with the bilinear map
and with the sesquilinear map ) L E ( G ,H ) x L E ( G ,H ) + L E ( G ) , ( u , ~cf
(uI~)
:= V*OU
is a Hilbert righ,t L E ( G )module (Theorem 5.6.1.11 a)). Take u , v E L E ( G ,H ) and w, w l , w2 E L E ( G ) . Then
(Proposition 5.6.1.8 a ) , e ) ) , (ulu) = u'ou E LE(G)+ (Corollary 5.6.1.12),
and
(Proposition 5.6.1.8 f)). B y Proposition 5.6.1.2 g ) ; L E ( G :H ) is a Hilbert right Emodule in the complex case.
5.6 Halbert right C*Modules
4 09
Assume now IK = R . In the sequel we use the notation o f Proposition 5.6.1.9. By this proposition, L o ( G ,H ) and may be identified with the E coniplexifications o f L E ( G ,H ) and L E ( G ) ,respectively. Let u , v E L E ( G ,H ) . By the above, 0
*
0
~ ~ ( 8 )

 
0 5 ( u ,v ) o ( u ,v ) = ( u , v1)0(u, v ) =
(Proposition 5.6.1.9, Lemma 2.1.5.3). Hence L E ( G ,H ) is a Hilbert right LE(G)module. Definition 5.6.4.15 ( 0 ) Let F := ( F t ) t E rand G := ( G s ) s E Sbe families of Hilbert right Emodules. We denote by NGSFthe vector space of families (u.~,~)(S,~)ESXT
such that
for every ( s ,t ) E S x T and such that
is finite. For every u E N G , F , define u* E NF,G by
for all ( t ,s ) E T x S . If H := ( H r ) t E Ris a third family of Hilbert right E modules then for all u E N G , F and v E N H , G define vu E NII,Fby
for all ( r ,t ) E R x T . It is easy t o see that the above u* and vu are welldefined. If S := N, , T := N, for some m, n E IN and
for all ( s ,t ) E S x T then
5. Hzlbert Spaces
410
and U * and v u coincide with the corresponding notation in the matrix case (Definitions 2.1.4.23, 2.3.1.30). Hence NG,F has to be seen as a generalized matrix. The more natural notation will be reserved for the completion of N G , F (Definition 5.6.4.18).
Proposition 5.6.4.16 ( 0 ) Let F := (Ft)tET and G := (Gs)sESbe families of Hilbert right Emodules and for every u E N G , F define
LET
SES
a ) For every u E N G , F ,
LET
~ E S
(Proposition 5.6.4.1 c)), and
where for every s E S , us,. is the element of
NG,Fdefined by
for all (s', t ) E S x T
b) The map
zs linear and injective (bijective zf S and T are finite).
C) If H := (Hr)rER is a famzly of Hilbert right Emodules then for all u E NG,Fand v E N H , ~ ,
5.6 Hilbert right C*Modules
d) If T is finite then
NF,F endowed with the norm
is a C*algebra and the map
is an isomorphism of C'algebras.
a) Take
E a) Ft and 77 E
=
)x(~~ttt1strtl.l~) L E T s€S
Hence
Moreover,
a) G, . Then
SES
LET
=
1) ( F L I ~ ~ V ~=) LET sES
5. Hilbert Spaces
41.2
(Proposition 5.6.1.2 d), Proposition 5.6.4.1 b)). Hence
Ilu112 5 C
C lIustII
llusr
ll
=
SES t.tlET
C SES
From
it follows
Fix (s, t ) E S x T and take
tt E F? . Define <'E a) Ftc by t1ET
for every t' E T . By Proposition 5.6.4.1 b ) ,
<'E
a) Ftl (t.ET
,"
and so
It follows
( aF ~ ). #
Now fix s E S and take ( 6
tET
By Proposition 5.6.4.1 b),
5.6 Hilbert right C*Modules
so that
It follows
b) It is obvious that the map is linear. By a ) , it is injective. Assume S and T finite and take
uE
CE
( ~ aE T l ~ ~ ~ S@ E SG S )
<
.
Fix (s, t) E S x T . For every E Ft and q E G , denote by Fp and by q' the element of a)G,, defined by
a
L'ET
<' the element of
s'ES
for every t' E T and s' E S . Define
Then for ( E Ft and
E G,,
Hence us, E L E ( F tG , , ) and it is easy to see that 6 = v . Therefore the map
LET
is surjective. c) Take
<E
a) Ft . Then tET
SES
5. Hilbert Spaces
414
so that
H
d) follows from a),b),c).
Corollary 5.6.4.17 Let H be a n infinitedimensional Hilbert space and n E I N . T h e n the C'algebras C(H) and K ( H ) are isomorphic to the C*algebras L(H),,, and K(H),,, , respectively (Proposition 5.6.4.16 d)).
Let A be an orthonormal basis of H . Since A is infinite, there is a bijection A t A x IN,, so H is isomorphic to a)H (Proposition 5.5.1.22). Hence kEN"
C ( H ) (resp. K ( H ) ) is isomorphic to C
( m )( m P
K( QH)
kEN,
which is isomorphic to C(H),,,
kEN,
)
(resp. K(H),,, ) (Proposition 5.6.4.16 d)). H
Proposition 5.6.4.18 ( 0 ) In this proposition we use the notation of Proposition 5.5.7.10. For every E H , put
<
and zdentify H and e2(T) x e2(T) via the m a p
H
4
e 2 ( ~x) e 2 ( T ) ,
< cf (FlrEz)
For each v E C(e2(T))2,2put :H
+
H,
< cf (UIIFI+ ~12F2,
UZI
+ ~22<2)
a) For every u E C(e2(T))2,2,
and the map L(e2(T))2,2+ L ( H ) , v
cf
11
is a n isomorphism of 6' algebras (Propositzon 5.6.4.16 d)).
5.6 Hzlbert right C*Modules
b) For every
(ull+~2i+u~j+u4k)(<~l+~i+~j+<~k).
If we define c~ : ( L ( ~ ' ( T , I R ) ) +) ~ F ( L ( H ) ),
4
u 4u ,
then cpo$' is an isomorphism of real C*algebras (Example 5.6.4.2 a), Theorem 5.6.1.11 a)) and
p o $  1 ( L M ( e 2 ( ~ , Mn) K ( 1 2 ( T ,I H ) ) ) = F ( K ( H ) ) . c)
The complexification of L s ( e 2 ( T , M ) ) (of L H ( e 2 ( TM , ) ) n K ( e 2 ( T .M))) is isomorphic to L ( H ) (to K ( H ) ) .
a ) By Proposition 5.6.4.16 d),
ii E L ( H ) for every u E L ( e 2 ( T ) ) 2 , 2and the map
is an isomorphism o f C*algebras. Take u E L ( ! 2 ( T ) ) 2 , 2and
< E H . Then
5. Hilbert Spaces
Since 6 E F ( C ( H ) ) iff Ga: = sz,it follows G E F ( C ( H ) ) iff
b) By a ) and Proposition 5.6.4.5,
for every u E (C(e2(T,R ) ) ) h n d cp and 4 are bijective and linear. Hence cp 0 I/' is bijective, linear, axid involutive. Take a, b E Lm (12(T,El)) and put
+
c := cp(u)cp(v)=
with c E C(e2(T))2,2.Then
c22 = ~ 1 v 1 u2v2

u3v3  uqv4
+ i(uIv2
 UZVI 
u3v4
+ u4v3)= w1

iw2.
Hence
and cp o is an algebra homomorphism, i.e. an isomorphism of real C*algebras. The last relation is easy to see. c) follows from b) and Proposition 5.5.7.10 a ) .
5.6 Hilbert right C*Modules
417
Definition 5.6.4.19 ( 8 ) Let F := ( F t ) t E TGr := ( G s ) s E Sand , H := ( H r ) r Ebe ~ families of Hilbert right Emodules. W e endow the vector spaces of the J o n N G , F with the norms
(Proposition 5.6.4.16 b)) and denote by we define the maps
MG,F
M G . ~ + L E ( tET Q F ~ .~c.7 0
the completion of N G , F . Further
~
~C ~ ).
U
by extending continuously the corresponding maps of Proposition 5.6.4.16 a)).
N (Definition 5.6.4.15,
T h e first m a p is linear and norm preserving, the second one is conjugate linear and norm preserving, and the third one is bilinear and continuoub.
Proposition 5.6.4.20 ( 8 ) Let F := ( F I ) I , =G~ ,:= ( G s ) s , = s ,H := ( H r ' ) r E Rbe families of Hilbert right E  m o d ~ l e s .
~ is a unzque famzly a ) For each u E M G , there
( ~ , , t ) ( ~ , t ) ~ . such $x~
for every (s,t ) E S x T and such that
for every
<E
b) For every u
a) Ft . LET
E MG,F
and with the notation of a ) ,
that
5. Hilbert Spaces
418
~ by where for every s E S , us,. is the element of M G , defined
0Ft
+
LET
0Gsc , F
((~F)s'~s.s')~'Es
s'tS
Moreover,
for all ( s ,t ) E S x T and
c ) For all u E M G , F and v E M H , ~ ,
COG = Z . d) M F , endowed ~ with the multzplzcation
M F , Fx M F , F+ M F , F , ( u , v )H uv and wzth the involution M F , F+ M F , F
uH
U*
is a C*algebra. If T is finite then the map M F , F + C ~ ( ~ $ F LI ) u  6 is as isomorphism of C*algebras.
e ) M G , endowed ~ wzth the bilinear map M G , FX M F , F+ MG.Fr
(u?)
UV
and with the sesguzlznear map
is a Halbert right MF,Fmodule.
f ) If we put 0
0
F := ( F L ) L E TG, := ( G s ) s ~ s (Propositzon 5.6.1.6) then the complexi~catzonof the C*algebra M F , F is isomorphic to M;,;.
5.6 Halbert right C*Modules
g)
For all u E M G , F A, such that
c S , and
B
c T , there is a unique
419
UA,B
E
MGVF
if ( s ,t ) E A x B (uA,B),,~
ij(s,t)$AxB
0
Moreover, the map
is a n operator (of n o r m 1 i f A and B are nonempty). h ) Put
order Then
by inclusion, and denote by
5 the upper setion filter of
1)32.
u = lim u A , ~ AxB,S
for every u E M G , ~ . i) Assume E is a W*algebra and for every A E p f ( S ) and B E ' P f ( T )
put
Then
TA
E Pr LE
(5
G.) ,
TB
E ~r
( kt)for every A c s LET
and B c T ,
A W
for every u1 E a) Ft i n the topology of pointwise convergence o n LET
W
LE ( % LETF L
s@ ES G S )
,
5. Hilbert Spaces
and u = lim
for euery u E CE
+ . .
( a)Ft , W
W
LET
SES
a)G,)
TAOUOTB
in the topology of pointwise conver
gence on a ) Ft . LET
j) Let (S,),,I and ( T L ) , E Ibe disjoint families in v ( S ) and v ( T ) , respec
tively, and put
for every
L
E I . Let
such that (llIL1l)tclE % ( I ) and define vSt :=
u ~ if, ( s~,t )~E S , x T , for some L E I 0 if ( s ,t ) E S x T\ U ( S , x T,) LEI
for every (s,t ) E S x T , Then there is a unique v E MG,Fsuch that
for every
<E
a) Ft . Moreover, LET
a) The uniqueness is obvious. By Proposition 5.6.4.16 a ) , the m a p
is coritinuous for every ( s ,t ) E S x T . We may extend this m a p continuo~~sly to M G , ~getting , in this way a family ( T L , , ~ with ) ~ ~ ~ US.^ E L E ( F ~G,) ,
for every ( s ,t ) E S x T and
5.6 Hilbert right C*Modules
< E t a) Ft with { t E T I EL # 0) holds for arbitrary < E a) Ft . for every
finite. By continuity, this relation
ET
LET
b),c), and d) follow from a) and Proposition 5.6.4.16. e ) follows from c),d), and Example 5.6.4.14. f ) is easy to see.
g) Take v E MG,E.and assume
UA,J
exists. We want to prove the inequa
lity
Take
< E (t$Ft)X
and define q E
n Ft by
LET
By Proposition 5.6.4.1 b), 11 E ( t $ ~ t ) w
The linearity of the rnap
is obvious. By the above, its norm is one if A and B are nonempty. Now we prove the first assertion. The uniqueness and the case u E &,F are trivial. Let be a sequence in NG,F converging to v . By the above, ( ~ l ; l ? k is) ~a ~Cauchy ~ sequence in Nc,F. Put
v := lim u F b E M,,F n+m
BY b),
4 22
5. Halbert Spaces
for all (s, t) E S x T . It follows
Vs,t
=
us,,
if (s, t) E A x B
0
if (s, t ) $ A x B
for all (s, t ) E S x T . h) Take E > 0 . There is a v E NG,F such that IIu

vII
E
< . 2
Take A E Y f ( S ) and B E Y f ( T ) such that
Then
""d by g) ,
Hence lim
UA,B
=u.
AxB,3
i) It is easy to see that n~ E Pr C E
( &c,) and
n~ E Pr CE
sFS
for every A
c S and B c T . For the assertion concerning
where
By Proposition 5.6.3.1 e) , h 
<, 7 ) " ~= ( a ,T B < , T A V )
m ~ ( a ,
for every A
c S and B c T , so that
we have to prove
. tFT .
U'
, we may assume
5.6 Hilbert right C*Modules
4 23

T 9) .
lim (a, <e,, 9e*)  (a,
AxB,S
Take v E LE
(5F ~5, c ~ )
and
LET
E
> 0. Since the family
SES
(((ti I ( v * ~ ) Q ~ ) ), ~ E T is summable, there is a Bo E p j ( T ) such that
for every B E p j ( T ) , Bo c B . For every B c T ,
(Corollary 4.2.1.18). By Proposition 5.6.4.6 c), there is an AIJ E p j ( S ) such that I((v<ei I 9e2\*). a) I <
E
5
for every B c T and A E p j ( S ) , A. C A . Take A € p j ( S ) , A 0 c A , a n d B € p j ( T ) , B o c B . T h e n
4 24
5. Hilbert Spaces
so that
in the topology of pointwise convergence on
(
T h e assertion concerning u follows. j) For every L E I , we identify a) FL and LET,
a subspace of a) Ft and LET
W
W
IET
sES
LE @ Ft. O C , ) . a) G, in a natural way with ~ES,
a) G , , respectively. For
L
E
I and
<,
E
sES
(Corollary 5.6.1.12). Moreover, for every X E I\{&) and
E a) Ft , IET*
(GLCL
<E
a) Ft and for every
L
E
I , define
LET
2 := ( < L ) L E TE, LOFt ET Then for every J E Y , ( I ) ,
a) Ft ,
LET,
5.6 Hilbert right C*Modules
For every A E
pf ( S x T ) and v ~ , $:= t
( s ,t ) E S x T , put
vst if ( s ,t ) E A if ( s ,t ) E S x T\A
0
Since (llll,ll),El E c O ( I ) it , follows from the above that
exists, where
for every
5
denotes the upper section filter of p f ( S x T ) ,
< E a) Ft , and tET
Since the reverse inequality is trivial, we have
ll;ll
= SUP Il'iiLll . LEI
Example 5.6.4.21 Assume that E is the C*direct sum of a family (E,),,I of C*algebras and for every L E I let 5, be an approximate unit of E L .Let H be a Hilbert right Emodule and for each L E I put H,
:= {< E
H
I < = lim
Define
a ) G endowed with the right multiplzcation
and with the innerproduct
is a Hilbert right Emodule such that
for every
<EG
426
5. Hzlbert Spaces
b)
E H for every (' E G and the map
LEI
G+H,
<Ccl LEI
is an isomorphism of Hilbert right Emodules. c ) H is selfdual iff H, is sevdual for every
Proposition 5.6.4.22
i
E I.
Let E be a C*algebra and x
E such that
p:=x*x~PrE, q:=xxg€PrE. Define cp:pEtqE,
Y++XY,
$:qE+pE,
zx'z.
Then 9 and i are isomorphisms of Hilbert right Emodules (Example 5.6.1.5), and E
L E ( P E ,q E ) , i E L E ( Q EP, E ) , P*= @ = c p  I .
First remark that
for every y E pE (Corollary 4.1.2.22 a + c), so that cp (and For y E pE and z E q E ,
i )is welldefined.
so that cp
L E ( P E ,q E ) ,
+ E L E ( Q EP, E ) ,
P* = * .
Moreover,
and so
E yE E, For y1,y2 E ~ and
( V Y II'pY2) =
Thus
ip
(XYI 1 ~ ~ = 2 Y;x'xYl )
= Y;YI = (91ly2)
and @ are isomorphisms o f Hilbert right Emodules.
5.6 Hilbert right C*Modules
Proposition 5.6.4.23 such that
for all distinct
1,
427
Let E be a W8algebra and ( p J r E l a family i n Pr E
X E I and
Put
and
for every x 6 E (Proposition 5.6.3.20 b + a). a) 5 E L E ( H ) for every x E E and 9:E
k' = 2.Define
+L E ( H ) ,
x
*+ Z
b) y is a n isomorphism of W*algebras (Theorem 5.6.9.5 b), Theorem 5.6.4.7 a), Proposition 5.6.2.3) such that
for every ( a ,t,17) E E x H x H (Proposition 5.6.3.17 d)), where (;j denotes the pretranspose of p . Moreover, 27 we put P := ( P ~ ) ' E I
then
for every u E L E ( H )
ff
4 28
5. Hilbert Spaces
c) If there is a family ( x , ) , ~in~ E such that
for every
L
E I , then the W *algebras E and
are isomorphic. a) For <,q E H ,
=
5 (5 E l
LEI
17:)
5 $
X ~ A= ~ A XEI
%) *
5. Htlbert Spaces
51 6'
(Propositiorl 5.6.1.8 d)). Hence, by b), u ( K ) C K . For <, 7 E K ,
d) By a) and Theorem 5.6.3.5 b), L E ( H ) and Lc.(K) are W* algebras with preduals H and K t rcspectivcly. By c), the map
y : L E ( H ) +L F ( K ) , u
ILK
is welldefir~ed and is involutive. It is obviously an algebra homomorphism
For every
<E H
and J
cI
T J :
For every
L
put
Ht H ,
E I put
* ~ A P ;,
fl : I + E , obviously f, E K . Take u E Kcr p and
< E 11. Then
uf,=uKf,=O and so uJ(c) = tl(fr<') = (~fL)
L
E
I (Proposition 5.6.1.8 d)). It follows IL<.,
=
0
5.6 Hilbert right C*Modules
for every J E
PI([).By Theorem
5.6.4.7 c),
u = lim TJUT,, = 0 . 5,31
Thus cp is injective. Step 2 cp(LE(H)) is a W*subalgebra of C p ( K ) Take u E L E ( H ) , u E F , and ( , T I E K . By Corollary 4.4.4.12 c), F is a W*subalgebra of E and so there is a b E E with blF = a (Corollary 4.4.4.9). From

(cpu, ( a . 7)) ~ = ( ( ( v u ) E ~ ~a) I ) ,= (((cpu)
it follows by Theorem 5.6.3.5 d), that v is a Wehomomorphisnl. By Corollary ) a Mpsubalgebra of L p ( K ) . 4.4.4.8 b), ~ ( C E ( H ) is Stcp 3 p is surjective Put
for every J
c I . Take u
E L F ( K ) and
L
E I . The map
. a) and Proposition 5.6.2.4, it belorlgs to C E ( H ) . For belongs to Z E ( ~ )By CEK,
so that
Moreover, P{A)uP{')= ( T { A ) ~ )EK'P(LE(H)) for every X E I (Theorem 5.6.4.7 c)). It follows
5. Hilbert Spaces
518
for every J E p j ( I ) . By Theorem 3.6.4.7 c),
u =lirnp~up~ J,3,
in (LF(K))K. By Step 2,
u E v(CE(H))
1
i.e. cp is surjective.
Theorem 5.6.7.2
for all
1, X
Let p E Pr E and let ( x , ) , ~be~ a family i n E such that
E I and
For every L E I denote by H , the won Neumann rzght Emodule associated to p E (Proposition 5.6.2.3) and put
a) C E H , (
b) For every x E E the map
belongs to LE(H) with Z* = 2' . Define
c) For every u E L E ( H ) put
G := (uCl() and define
Then cp and
)I
are isomorphisms of W*algebras and $I = cp'
5.6 Hilbert right C*Modules
51 9
+
and 11, denote the pretranspose of cp and q h , respectively (Theorem d) If 5.6.4.7 a), Theorem 5.6.3.5 b)), then

$ ( a , <,v ) = (IIOa((1v) ,
for all a E E and [,v E H .
f ) If
L,
X E I and
<EH
then
g) If p E p is commutative then the map
is an isomorphism of W*algebras (Corollary 4.4.2.5). a ) By Corollary 4.1.2.22 b
for every
L
E
+c,
I . Since
it follows ( E H arid
(
= 1 . Moreover, for E
((<(.I<))J)L = <'(I10 = 5:
1
XX<X
XEI
b) By a), Z is welldefined. For
=
5E
H and
L
E
E
Xtl
A€ I
E
I,
C X:XXIX = C ~ ~ X P=< IX r,
I,7 E H ,
5. Hilbert Spaces
520
Thus 5 E L E ( H ) and Z' = 2 . c) For x , E~E and E H , by a),
<
= C ~ ( C I C ) Y ( < I C )= Cxy(
so that
and p is an algebra homomorphism. By b), it is involutive. For X E E , b y a ) :
= (CIC)x(CIC)= x .
For u E L E ( H ) and
< E H , by a ) ,
= C(.(C(
= <(u
so that
Hence p and 4 are isomorphisms of W8algebras and 4 = cp' . d ) For x E E h 
( p x , (a.<, 7 ) ) = ((Z
Thus
For
IL E
LE(H)
5.6 Hzlbert right C*Modules
0.
Thus
* a = (a, <,
f ) We have
g) It is easy to see that the nlap is an involutive algebra homomorphism. Take x E Ec with xp = 0 . Then
for every
L
E
I , SO that
Hence the map is injective Put
For every L E I denote by ICL the von Neunlanrl right Fmodule associat,ed to F (Propositiori 5.6.2.3) and defino
5. Hilbert Spaces
522
Let y E F and put
<ey<=[y.
u:K+K, Then
(uFlv) = (FyIv) = (FIP)Y = ~(Flv) = (thy*) for all
<, 7
E K , so that u E L F ( K ) . For every
v E L F ( K ) and
EK,
By c) and Proposition 5.6.7.1 d ) , there is an x E Ec with
z< = u< for every ( E K . By e), Z<
for every
< E K . Take
L

= x< = UF = Y<
E I and
< :
(~,AP)XE~
Then
Hence the map is surjective. By the above, it is an isomorphism of LV*algebras. H
Proposition 5.6.7.3 Let p, q E Pr E such that pEp is commutative and 0 < q <_ p . Further let H be a von Neumann right Emodule, A a Fourier basis of H , and B a Fourzer set of H such that
for all
<EA
and 7 E B .
5.6 Hzlbert right C*Modules
C) Card B 5 Card A a) By Proposition 5.6.3.9 a
+ b and Proposition 5.6.1.2 a),
(tlv) = (<(tlOIv(vlv))= (vlv)(tlv)(tlt) =
Since pEp is commutative, it follows
(
c) Case 1 A is finite Let K be the vector subspace of H generated by A U B . By a),
(
E
K . Take x' A'
E
u(pEp) with (q, x') # 0 and put
:= {< E
A I ( ( t l t ) , ~ '#) 0 ) .
The map
K
X
K +K ,
(<,,<2)* ((<1I<2),x1)
being a positive sesquilinear form, we get by Schwarz inequality,
l((
<E K
and
< E A\A1.
Define
?: A' + IK, t 6 ((CIt),x1) for every
< E K . Then C E ['(A')
for every
<EK
and
524
5. Hilbert Spaces
for all (I, C2 E K (Theorem 5.6.3.13 f)). In particular,

for all '71,% E B . Thus (inqGBis linearly independent. Since (<)c€A,generates ['(A') (in fact, it is an orthonormal basis of ['(A') ), Card B 5 Card A' Case 2
5 Card A .
A is infinite
Take a E E+ with (g, a) # 0 and put
f
: A t P ( B ) ,
t
I+
{D E B I ((FV)(171C),a) f 0)
By a ) and Theorem 5.6.3.13 c), for every
<
A,
so that f (<) is countable. Take 7 E B . By Theorem 5.6.3.13 f),
Hence
vc Uf(<)> B c Uf(<), €€A
CtA
Card B 5 Card (A x No) = Card A .
5.6 Hilbert right C*Modules
525
Corollary 5.6.7.4 Let p E P r E be such that pEp is commutative. Further let H be a von Neumann right Emodule and A, B Fourier bases of H such that
(771 1771) = (7721772)
for all
&,(2
1P
E A and q1,q2 E B . Then
Card A = Card B
and
for all (<,q) E A x B .
Card A
= Card
B
follows from Propnsition 5.6.7.3 c ) . By Proposition 5.6.7.3 a),b) and Proposition 5.6.3.9 a + b , for all 77 E B and 5 E A ,
so that (77177)
5 (CIC)
By symmetry,
((10= (1717) for all (<, 17) E A x B .
526
5. Halbert Spaces
Proposition 5.6.7.5 Let p E Pr E such that pEp is commutative. Further let H be a von Neumann right Emodule, A a Fourier basis of H such that
for every
b)
< E A , and
( u l A ) ~ l , A ~a, ~family ,l in L E ( H ) such that
There is a Fourier famzly ( q ) l EznI H such that
for all
L,
XE I.
c ) Card 1 5 Card A
.
a) By 1) and 2 ) ,
b) Take p E I and Proposition 5.6.3.4 c ) ,
Hcnce there is a
<EA
C
E Imn,,\{O).
By a), Theorem 5.6.3.13 f ) , and
with upp<# 0 . From
(<  ? ~ p p < l ? ~ p p <= )
 up,<)lt) = 0
5.6 Halbert right C* Modules
it follows
(ufip
then (17117) E Pr E\{O) .
Moreover,
(17117) = x*(u,,
E PEP
Put 171
for every
L
E I . Then for all
L,
:= U',17
XEI,
= ~ l x ( ~ l u , ,=~ 6,~(17117) ) 5 P.
c) follows from b) and Proposition 5.6.7.3 c).
Proposition 5.6.7.6
Let p, q , r E P r E such that p E p is commutative and O
O
Further, let ( x , ) , , ~ ,( Y , , ) , ~ ~ be families i n E such that
5. Hzlbert Spaces
528
X:XA
for all
L,
= 6,,4g,
Y>Y" =
X E I and p, Y E M and
Then
Card I = Card M , q = r . Put
By Theorem 5.6.7.2 c), E is isomorphic to CE(H) and C E ( K ) . Put U L := ~ xLx;
for all
L,
X E I . Then
for all
1,
A, p, v E I . By Proposition 5.6.7.5 c),
Card I 5 Card M . By symmetry, Card I
= Card
M
By Theorem 5.6.7.2 g), qEq and r E r are isomorphic, so that
Proposition 5.6.7.7 Let I be a set and p E I . Put
and define
for evenj
L
E I
5.6 Hilbert right C*Modules
a) uLE LE(H) f o r every
I.
E I and
for every 7 E H .
a) For
<,TI
E H,
Hence u, E LE(H) and
for every
< E H , so that
c) follows from a) and b) . d ) By a ) , for L E I , ( , ~ H E , and a~ E ,
529
5. Hilbert Spaces
530
Hence
Definition 5.6.7.8 T h e W'algebra E zst called homomogeneoua if there ezist a p E P r E and a family ( x , ) , , ~i n E such that p E p is commutative,
for all
L,
X E I , and
Every commutative W' algebra is homomogeneous. We have
for every
L
E
I , SO that p # 0
Theorem 5.6.7.9 W * algebra F put
For every cardznal number N and for every commutative

c
a ) ( N , F ) is a homogeneous W*algebra and F is isomorphic to ( N , F )
b) For every homogeneous WWalgebra E there is a pair ( N , F ) , s&that N is a cardinal number, F is a commutatzve W *algebra, and ( N , F ) is isomorphic to E . C)
If N , N' are cardinal numbers and F , F' are commutative W*algebras such that
 
( N , F ) = (N', F')
then N = N' and F and F' are isomorphic
5.6 Hilbert right C*Modules
531
d) Let F be a commutative W8algebra, N a cardinal number, n E N and


T h e n (N, F),,, is isomorphic to (N', F ) . In particular En,, is homogeneous for every homogeneous W*algebra E . a) follows from Proposition 5.6.7.7 and Theorem 5.6.7.2 g). b) There exist a p E Pr E and a family (x,),,, in E such that F := p E p is commutative.
for all
1,
X E I , and

By Theorem 5.6.7.2.c) and Proposition 5.6.7.1 d), E is isomorphic to Card I, F )

c) By Proposition 5.6.7.7, there is a family ( u ~ ) , in , ~ (N, F ) such that for all L , X E I ,

U:UA
= ~ L X: P
where p E Pr (N, F)\{O) . Put
for every
1,
X E N . Then for
I,,
A, p, v E N ,
By Proposition 5.6.7.5 c) (and Proposition 5.6.4.6 f ) ) ,
5. Hilbert Spaces
532
By symmetry,
H = N'. By a ) , F and F' are isomorphic. d ) It is easy to see that W
a>
F and
a)
W
a) F
are isomorphic, so that the assertion follows from Proposition 5.6.4.16 d).
W
P r o p o s i t i o n 5.6.7.10 Assume E is the C *direct product of the family (Ex)x€r, of Wealgebras and let I be a set. Put
and
for el~ery X E L and denote by L the C*direct product of the family
(LE,(Hx))xEL. a ) The elements of H may be zdentified wzth the functions [ defined on I x L such that
for every X E L and such that
c ) For every u E L put
Then ii E L E ( H ) for every u E L and the map
is an isomorphism of W*algebras
5.6 Hilbert right C*Modules
5,?9
Definition 5.6.7.11 E is called a type I W*algebra if it is isomorphic to the C*direct product of a family of homomogeneous W *algebras. This is not the usual definition of a type I W*algebra (in the complex case), but is equivalent to it. Theorem 5.6.7.12 Let 5 be a class of commutative W*algebras such that any commutative W*algebra is isomorphic t o exactly one element of 5 . Let further 3 be the class of families (FN)NEIi n 5 for which I is a set of cardinal numbers. For every F = (FN)NEI E 3 denote by F the C* direct product of the family
( L F ~ ( ~L EFNN ) )
a)
F
is a type I IV'algebra for every F
N ~ I
E
3
b) For every type I W  *algebra E there is exactly one F E 3 such that E is isomorphic t o F . c) En,,is a type I W*algebra for every type I W*algebra E and for every
EN. a ) follows from Theorem 5.6.7.9 a). b) By definition, E is isomorphic to the C*direct product of a family ( G x ) A ~ofL ho~nogeneotis IV*algebra. By Theorem 5.6.7.9 t)), for every A E L , there is a cardinal number NA and an Fi E 5 such that Gx is isomorphic to
Put
I := {NA / X
E
L) .
For every N E I put
IN:= { A E L I Nx = N) and denote by FN the element of 5 which is isomorphic to the C*direct product of the family (Fi)hEI,,. By Proposition 5.6.7.10 c):
5. Hilbert Spaces
is isomorphic to the C'direct product of the family
Hence, if we put
then F E
G
F and F is isomorphic to E .
Let F := ( F N ) N ~and I G := (GN)NELbe elements of F such that are isomorphic. Then there is a bijection rp : I + L such that
are isomorphic for every
N
E
I . By Theorem 5.6.7.9 c),
for every N E I , i.e. F = G . c) follows immediately from Theorem 5.6.7.9 d)
F
and
Name Index
Name Index Alaoglu, L. 1.2.8.1 Arens,R.F. 1.5.2.10,2.2.7.13 Arzela, C. 1.1.2.16 1.1.2.14, 1.1.2.16 Ascoli, G. Atkinson, F.V. 3.1.3.7, 3.1.3.11, 3.1.3.12, 3.1.3.21, 5.3.3.16 2.3.1.3 Autonne, L. Banach, S. 1.1.1.2, 1.2.8.2, 1.3.1.2, 1.3.2, 1.3.3.1, 1.3.4.1, 1.3.4.10, 1.4.1.2, 1.4.2.3, 1.4.2.19 2.2.5.4 Beurling, A. Bourbaki, S . 1.2.8.1 1.3.5.14 Branges, L. de Calkin, J.W. 5.4.3.5 3.1.3.1 Carlernan, T. Cauchy, A. 1.3.10.6 Choquet, G. 5.4.3.5 1.7.2.1 Dcdekind, R. 1.2.8.2, 3.1.3.9 Dicudonnb, J. Dixrnier, J. 4.4.4.4 4.3.2.13 Drewnoxski, L. 1.1.6.14 Dworetzky, .A. 4.1.3.7 Dye, H.A. Eberlein: W.F. 1.3.7.15 4.2.4.15 Effros, E.G. Enflo,P. 3.1.1.7 Ford, J.W.hl. 2.4.2.4 2.4.6.2 Fourier, J.B.J. Frkchet, IM. 1.1.1.2, 1.1.2.13, 5.2.5.2 Fredholm, E. 3.1.6.23 Frobenius, G.F. 2.1.4.21 Fuglede, B. 4.1.4.1 4.2.1.1 Fukarniya, M. 1.4.1.9, 2.2.5.4, 2.2.5.3, 2.4.1.2, 2.4.5.7, 4.1.1.1, 4.1.2.5 Gelfarid, 1.M. 4.1.3.1, 4.2.6.6, 5.4.1.2, 5.4.2.5 3.1.3.12 Gohberg, I. 1.3.6.8 Goldstine. H.H.
Goodearl, K.R. 2.2.1.19. 4.1.1.1, 5.4.2.13 1.2.1.12 Gowers, W.T. Gram, J.P. 5.5.1.18 1.6.1.1,3.1.6.25,4.2.8.13 Grothendieck,A. Hahn, H. 1.1.1.2, 1.2.1.3, 1.3.3.1, 1.3.6.1, 1.3.6.3, 1.3.8.1, 1.4.1.4 Hamilton, W.R. 2.1.4.17 Hellinger, E. 5.2.5.4 1.1.1.2, 1.3.3.13 Helly, E. 2.1.3.1 Hilbert, D. Hirschfeld, R.A. 2.2.5.6 2.1.3.10 Jacobson, N. James, R.C. 1.3.8.1 Jordan, C. 5.1.1.6 Kadison, R.K. 4.3.3.20 Kaplanski,I. 4.1.2.1,4.2.6.5,4.4.2.24,5.6.1.1 Kelley, J.L. 4.2.1.1 1.2.3.11 Kojirna, ?? Kolrnogoroff, A. 1.1.1.2 Kottrnan, C.A. E 1.3.5 Krein, M.G. 1.3.1.10, 1.3.7.3 Laguerre, E.N. 2.2.3.5 Laurent, P.A. 1.3.10.8 5.3.1.3 Lax, P.D. Lt Page, C. 2.2.3 8, 2.2.4.3, 2.2.5.6 Lindenstrauss, J. 1.2.5.13 Liouville, J. 1.3.10.6 3.1.5.10 Lornonsov, V.I. 5.1.1.1 Lowing, H. .Mackey, G.W. 1.3.7.2 .Vazur, S. 2.2.5.5 Mihlin, S.G. 3.1.3.12 Milgrarn, A.S. 5.3.1.3 Milrnan, D.P. 1.3.1.10 1.1.1.2, 1.1.3.4 Minkowski, H. Murray, F.J. 1.2.5.8 2.2.1.1 Nagurno, M. Nairnark, M.A. 4.1.1.1, 4.1.2.5, 4.1.3.1, 4.2.6.6, 5.4.1.2, 5.4.2.5, 5.5.1.24 Ncurnann, C. 2.2.3.5
Name Index
1.1.1.2, 3.1.3.1, 5.1.1.1, 5.1.1.6, 5.2.4 Neurnann, J . von 4.3.2.14, 5.2.1.2 Nikodym, 0. 3.1.3.1 Noether, F. Palmer, T.W. 4.1.1.1 5.6.1.1, 5.6.2.2, 5.6.2.6, 5.6.2.11, 5.6.3.3, 5.6.3.5 Paschke, W.L. Pedersen, G.K. 4.2.4.16 2.2.1.15 Peter, F. Pettis, P.J. 1.3.8.4, 1.3.8.5 1.2.5.14, E 1.3.3, 2.1.4.9 Phillips, R.S. Pierce,B. 2.1.1.1,2.1.3.6 Plancherel, M. 5.5.4.1 Putnam, I.F. 4.1.4.1 Rellich, F. 5.1.1.1 Rickart, C.E. 4.1.1.20, 4.1.2.12 Rieffel, 1M.A. 5.6.1.1 1.1.1.2, 1.1.4.4, 1.2.1.1, 1.2.2.5, 2.2.5.1, 3.1.1.1, Riesz, F. 3.1.3.8, 3.1.3.17, 3.1.5.1, 5.2.1.2, 5.2.5.2, 5.3.3.20 Rogers, C.A. 1.1.6.14 4.1.4.1 Rosenblum, M. Russo, B. 4.1.3.7 Sakai, S. 4.4, 4.4.1.1, 4.4.3.5 Schauder, J.P. 3.1.1.22 Schmidt, E. 1.1.1.2, 5.5.1.18 Schur, I. 1.2.3.11, 1.2.3.12, 1.3.6.11 Schwartz, L. 2.4.6.5, 2.4.6.8 4.2.6.2, 4.2.6.5, 4.2.8.2, 5.4.1.2 Segal, I.E. Shirali, S. 2.4.2.4 Sierpinski, W. 1.1.2.17 Silow, G. 2.2.4.27 Srnulian,V. 1.3.7.3,1.3.7.15 Steinhaus, H.D. 1.4.1.2 1.3.4.10, 1.3.5.16, 2.3.3.12, 4.1.2.5 Stone, M.H. 4.2.6.3 Starrner, E. Toeplitz, 0. 1.2.3.4, 2.3.1.3, 5.2.5.4 Vaught, R.L. 4.2.1.1 Vitushkin, A.G. 2.4.3.7 Volterra, V. 2.2.4.22 1.3.5.16 Weierstrass, K.
Weyl, H . 2.2.1.15 2.2.5.8 Wielandt, H. Wiener, N . 2.4.5.7 Yood, B. 3.1.3.11, 3.1.3.12, 4.1.1.13 ~elazko,W. 2.2.5.6
Subject lndex
Subject Index NT means Notation and Terminology
(A, B,C)mliltiplication
1.5.1.1
absolute value of a number
1.1.1.1 absolute value of a measure NT absolutely convex 1.2.7.1 absolutely convex closed hull 1.2.7.6 absolutely convex hull 1.2.7.4 absolutely summable family 1.1.6.9 acts irreducibly 5.3.2.19 acts nondegenerately 5.3.2.19 additive group NT adherence, point of NT atihererit point NT adjoint 2.3.1.1,5.6.1.8 adjoint differential operator 3.2.2.3 adjoint kernel 3.1.6.5 adjoint operator 5.3.1.4 adjoint sesquilinear form 2.3.3.1 adjoint sesquilir~earmap 2.3.3.1 adjoiritable 5.6.1.7 algebra 2.1.1.1 algebra, Calkin 3.1.1.13 algebra, complex 2.1.1.1 algebra, dsgerierate 2.1.1.1 algebra, division 2.1.2.1 algebra, Gelfand 2.4.1.1 algebra, Gelfand unital 2.4.2.1 algebra, involutive 2.3.1.3 algebra, involutive Gelfand 2.4.2.1 algebra, irivolutive uriital Gelfand 2.4.2.1 algebra, normed 2.2.1.1 algebra, real 2.1.1.1 algebra, semisimple 2.1.3.18 algebra, strongly symmetric 2.3.1.26
algebra, symmetric 2.3.1.26 algebra, unital 2.1.1.3 algebra, unital Gelfand 2.4.1.1 algebra homomorphism 2.1.1.6 algebra homomorphism associated to A 5.4.2.3 algebra homomorphism associated to x' 5.4.1.2 algebra homomorphism associated to (x:),,, 5.4.2.2 algebra homomorphism, unital 2.1.1.6 algebra isomorphism 2.1.1.6 algebra isomorphism, unital 2.1.1.6 algebraic dimension 1.1.2.18 algebraic dual 1.1.1.1 algebraic eigenspace 5.3.3.20 algebraic eigenvector 5.3.3.20 algebraic isomorphism, associated 1.2.4.6 algebraic multiplicity 5.3.3.20 algebras, isomorphism of involutive 2.3.1.3 analytic function 1.3.10.1 approximate unit 2.2.1.15 approximate unit of a C*algebra, canonical 4.2.8.2 Arens multiplication, left 1.5.2.10, 2.2.7.13 Arens mutliplication, right 1.5.2.10, 2.2.7.13 associated algebraic isomorphism 1.2.4.6 associated quadratic form 2.3.3.1 associated quadratic map 2.3.3.1 associated unital C*algebra 4.1.1.13 atom 4.3.2.20 atomic 4.3.2.20 atomless 4.3.2.20 Baire function 1.7.2.12 Baire set 1.7.2.12 ball, unit 1.1.1.2 Banach Banach Banach Banach Banach Banach
algebra 2.2.1.1 algebra, involutive 2.3.2.1 algebra, quasiunital 2.2.1.15 algebra, rrnital 2.2.1.1 categories, functor of 1.5.2.1 categories, functor of unital 1.5.2.1
Subject Index
Banach Banach Banach Banach Banach Banach Banach Banach Banach
category 1.5.1.1 category, unital 1.5.1.1 space 1.1.1.2 space, complex 1.1.1.2 space, involutive 2.3.2.1 space, ordered 1.7.1.4 space, real 1.1.1.2 subalgebra generated by 2.2.1.9 system 1.5.1.1 1.5.1.9 Banach system, bidual of a Banach system, dual of a 1.5.1.9 Banach systems, isometric 1.5.2.1 band 1.7.2.1 basis, Fourier 5.6.3.1 1 basis, orthonorrrial 5.5.1.1 basis of l 2 ( T ) canonical , orthonormal 5.5.3.1 Bergman kernel 5.2.5.9 Bessel's ideritity 5.5.1.7 Bessel's inequality 5.5.1.8 bicommutant 2.1.1.16 bidual of a Banach systerr~ 1.5.1.9 1.3.6.1 bidual of a normed space bijective NT bilinear map 1.2.9.1 binomial theorem 2.2.3.12 bitranspose 1.3.6.15 bound, lower 1.7.2.1 bound, upper 1.7.2.1 bounded map 1.1.1.2 bounded operator 1.2.1.3 bounded operator, lower 1.2.1.18 bounded sequence 1.1.1.2 bounded set 1.1.1.2 boundedness, principle of uniform 1.4.1.2 bouridedness theorerri, Nikodym's 4.3.2.14 C* algebra 4.1.1.1 C*algebra, canonical approximate unit of a 4.2.1.2 C*algebra, carionical order of a
4.2.8.2
C*algebra, complex 4.1.1.1 C*algebra, complex unital 4.1.1.1 C*algebra, Gelfand 4.1.1.1 C*algebra, purely real 4.1.1.8 C*algebra, real 4.1.1.1 C*algebra, real unital 4.1.1.1 C*algebra, simple 4.3.5.1 C*algebra, unital 4.1.1.1 C*algebra associated, unital 4.1.1.13 Calkin algebra 3.1.1.13 Calkin category 3.1.1.12 canonical approximate unit of a C*algebra 4.2.8.2 canonical involution of EF 2.3.1.1 canonical metric of a normed space 1.1.1.2 canonical norm of a preHilbert space 5.1.1.2 canonical norm of e ( E ,F ) 1.2.1.9 canonical order of a CWalgebra 4.2.1.2 canonical orthonormal basis of e Z ( T ) 5.5.3.1 canonical projection of the tridual of E 1.3.6.19 canonical scalar product of IKn 5.1.2.4 cardinal number NT cardinality, topological NT 4.3.3.1 carrier carrier, left 4.3.3.1 carrier, right 4.3.3.1 carrier of a function NT carrier of a Radon measure NT 1.5.1.1 category, Banach C*direct product 4.1.1.6 C*direct sum 4.1.1.6 character 2.4.1.1 characteristic function of a set 1.1.2.1 C*hull 4.1.1.22 class NT 1.4.2.19 closed graph theorem closed involutivc subalgebra generated by 2.3.2.14, 2.3.2.15 closet1 invollltive unital subalgebra generated by 2.3.2.14, 2.3.2.15 closed subalgebra generated by 2.2.1.9
Subject Index
closed unital subalgebra generated by closed vector subspace generated by C*module,Hilbert right 5.6.1.4 codimension 1.2.4.1 codomain NT
2.2.1.9 1.1.5.5
cokernel of a linear map 1.2.4.5 commutant 2.1.1.16 commutative 2.1.1.1 commutative monoid E 2.1.1 compact, relatively 1.1.2.9 compact operator 3.1.1.1 compatible, simultaneously 1.5.1.1 compatible (left and right) multiplications complement of a subspace 1.2.5.3 complernented subspace 1.2.5.3 complete, Corder 4.3.2.3
1.5.1.1
complete, order 1.7.2.1 complete norm 1.1.1.2 complete normed space 1.1.1.2 complete ordered set 1.7.2.1 completion of a normed algebra 2.2.1.13 completion of a normed space 1.3.9.1 completion of a preHilbert space 5.1.1.7 complex algebra 2.1.1.1 complex Banach spacc 1.1.1.2 complex C'algebra 4.1.1.1 complex C*algebra, unital 4.1.1.1 complex Hilbert space 5.1.1.2 complex normed algebra 2.2.1.1 complex normed space 1.1.1.2 complex preHilbert space 5.1.1.1 complex unital C*algebra 4.1.1.1 complex universal representation 5.4.2.16 complex Waalgebra 4.4.1.1 complexification of algebras 2.1.5.7 complexification of Banach algebras 2.2.1.19 complexification of Hilbert spaces 5.3.1.8 complexification of involutive algebras 2.3.1.40
complexification of involutive vector spaces 2.3.1.38 complexification of right C* modules 5.6.1.6 complexification of vector spaces 2.1.5.1 composition of functors 1.5.2.1 composition of maps NT compression of a representation 5.4.2.8 cone 1.3.7.4 cone, sharp 1.3.7.4 conjugacy class 2.2.2.7 conjugate exponent of 1.2.2.1 conjugate exponents 1.2.2.1 conjugate exponents, weakly 1.2.2.1 conjugate involution 2.3.1.3 conjugate linear map 1.3.7.10 conjugate number 1.1.1.1 continuous, order 1.7.2.3 convergence, radius of 1.1.6.22 convcx 1.2.7.1 convex, absolutely 1.2.7.1 convcx closed hull 1.2.7.6 convex closed hull, absolutely 1.2.7.6 convex hull 1.2.7.4 convex hull, absolutely 1.2.7.4 convolution 2.2.2.7,2.2.2.10 Corder complete 4.3.2.3 Corder ucomplete 4.3.2.3 C* subalgebra 4.1.1.1 C * subalgebra, unital 4.1 .I .1 C'subalgebra generated by 4.1.1.1 C*subalgebra generated by, hereditary 4.3.4.1 C*subalgebra generated by, unital 4.1.1.1 cyclic element 5.4.1.1 cyclic representation 5.4.1.1 cyclic vcctor 5.3.2.19, 5.4.1.1 cyclic vector associated to x' 5.4.1.2 decomposition, spectral 4.3.2.19, 5.3.4.7 degenerate algebra 2.1.1.1 derivative 1.1.6.24
Subject Index
diagonalization of u 5.5.6.1 differentiable 1.1.6.24 3.2.2.3 differential operator, adjoint differential operator, selfadjoint 3.2.2.3 dimension, algebraic 1.1.2.18 dimension, Hilbert 5.5.2.2 Dirac measure 1.2.7.14 direct integral of E with respect to p 5.5.2.19 direct sum 1.2.5.3 directed, downward 1.1.6.1 directed, upward 1.1.6.1 disjoint family of sets 1.2.3.9 distance of a point from a set 1.1.4.1 division algebra 2.1.2.1 NT domain downward directed 1.1.6.1 dual, algebraic 1.1.1.1 dual of a Banach system 1.5.1.9 dual of a normed space 1.2.1.3 dual space 1.3.1.11 Ealgebra 2.2.7.1 E algebra, involutive 2.3.6.1 Ealgebra, involutive unital 2.3.6.1 Ealgebra, unital 2.2.7.1 Ealgebras, homomorphisni of 2.2.7.1 Ealgebras, homomorphism of involutive 2.3.6.1 Ealgebras, homomorphism of involutive unital 2.3.6.1 E algebras, homomorphism of u~iital 2.2.7.1 E C*algebra 5.6.1.10 EC*algebra, unital 5.6.1.10 EC*algebras, isomorphism of 5.6.1.10 eigenspace 3.1.4.1 eigenspace, algebraic 5.3.3.20 eigenvalue 3.1.4.1 eigenvector 3.1.4.1 eigenvector, algebraic 5.3.3.20 Emodule 2.2.7.1 5.6.1.4 Emodule, Hilbert
Emodule, Hilbert right 5.6.1.4 Emodule, inner product right 5.6.1.1 Emodule, involutive 2.3.6.1 Emodule, involutive unital 2.3.6.1 Emodule, semiinnerproduct right 5.6.1.1 Emodule, weak semiinnerproduct right 5.6.1.1 Emodule, unital 2.2.7.1 Emodule, unital Hilbert 5.6.1.4 Emodule, von Neumann (right) 5.6.3.2 Emodules, homomorphism of 2.2.7.1 Emodules, homomorphism of involutive 2.3.6.1 equicontinuous 1.1.2.14 equivalence class NT equivalence class of a point NT equivalence of GNStriples 5.4.1.2 equivalence of representations 5.4.1.1 equivalence relation NT equivalent G N S  triples 5.4.1.2 equivalent norms 1.1.1.2 equivalent representations 5.4.1.1 essential spectrum 3.1.3.24 Esubmodule 2.2.7.1 Euclidean norm 1.1.5.2 evaluation 1.2.1.8 evaluation functor 1.5.2.1 evaluation operator of a normed space 1.3.6.3 Evalued spectral measure 4.3.2.16 exact set 1.7.2.12 expansion, Fourier 5.5.1.15 2.2.3.5 exponential function exponents, conjugate 1.2.2.1 exponents, weakly conjugate 1.2.2.1 extreme point 1.2.7.9 faceofaconvexset 1.2.7.9 factorization of a linear map 1.2.4.6 faithful, order 4.2.2.18 faithful representation 5.4.1.1 family NT
Subject Index
1.1.6.9 family, absolutely srimmable family, sum of a 1.1.6.2 family, summable 1.1.6.2 family of sets, disjoint 1.2.3.9 filter, lower section 1.1.6.1 filter, upper section 1.1.6.1 NT filter of cofinite subsets finitedimensional 1.1.2.18 Finvariant 3.1.4.4 Fourier basis 5.6.3.11 Fourier expansion 5.5.1.15 Fourier integral 2.4.6.2 FourierPlancherel operator 5.5.4.1 Fourier set 5.6.3.11 Fourier transform 2.4.6.2 FrdchetRicsz Theorem 5.2.5.2 Fredholm alternative 3.1.6.23 Frcdholm operator 3.1.3.1 Fredholm operator, index of a 3.1.3.1 free ultrafilter NT function NT function, Bairc 1.7.2.12 function, step NT functional calculus 4.1.3 functor 1.5.2.1 functor, identity 1.5.2.1 functor, inclusion 1.5.2.16 functor, isometric 1.5.2.1 functor, quotient 1.5.2.17 functor, transpose of a 1.5.2.3 functor of (unital) Banach categories 1.5.2.1 1.5.2.1 functor of (unital) Acategories functor of (left, right) Amodules 1.5.2.1 1.3.2.1 functors, composition of Gelfand, Theorem of 2.2.5.4 Gelfand algebra 2.4.1.1 Gelfand algebra, involutive 2.4.2.1 Gelfand algebra, involutive unital 2.4.2.1
2.4.1.1 Gelfand algebra, spectrum of a Gelfand algebra, unital 2.4.1.1 Gelfand C' algebra 4.1.1.1 GelfandMazur, Theorern of 2.2.5.5 Gelfand transform 2.4.1.2 GNSconstruction 5.4.1.2 5.4.1.2 G N S triple of E associated to x' GNStriples, equivalence of 5.4.1.2 GNStriples, equivalent 5.4.1.2 GramSchmidt orthonormalization 3.5.1.18 graph NT, 1.4.2.18 Green function 3.2.1.2 group, additive NT HahnBanach Theorem 1.3.3.1 hereditary 4.3.4.1 hereditary C*~subalgebragenerated by 4.3.4.1 Hermitian sesquilinear map 2.3.3.3 Hilbert dimension 5.5.2.2 Hilbert Emodule 5.6.1.4 Hilbert Emodule, unital 5.6.1.4 Hilbert right C*module 5.6.1.4 Hilbert right Emodule 5.6.1.4 Hilbert right Emodules, isomorphic 5.6.1.7 Hilbert space 5.1.1.2 Hilbert space, complex 5.1.1.2 Hilbert space, complexification 5.3.1.8 Hilbert Hilbert Hilbert Hilbert Hilbert Hilbert
space, involutive 5.5.7.1 space, real 5.1.1.2 5.4.2.3 space associated to A 5.4.1.2 space associated to x' 5.4.2.2 space associated to (x:),,, 5.1.2.3 space of square summable sequences 5.1.3.1 Hilbert sum of a family of Hilbert spaces 5.4.2.1 Hilbert sum of a family of representations
Hiilder inequality 1.2.2.5 homogeneous W*algebra 5.6.7.8 homomorphism of C*algebra 4.1.1 .2O honiornorphism of Ealgebras 2.2.7.1
Subject Index
homomorphism honiomorphism hornomorphism homomorphism honiomorphism
of of of of of
E modules
2.2.7.1 involutive Ealgebras 2.3.6.1 involutive Emodules 2.3.6.1 2.3.6.1 involutive unital Ealgebras uriital Ealgebras 2.2.7.1
hyperstonian space ideal 2.1.1.1 ideal, left 2.1.1.1 ideal, maximal proper
1.7.2.12
2.1.1.1
ideal, maxirnal proper left ideal, maximal proper right
2.1.1.1 2.1.1.1
ideal,proper 2.1.1.1 ideal, proper left 2.1.1.1 ideal, proper right 2.1.1.1 ideal, regular maximal proper 2.1.3.17 ideal, regular maximal proper left 2.1.3.17 ideal, regular maximal proper right 2.1.3.17 ideal, right 2.1.1.1 ideal generated by 2.1.1.2 idempotent 2.1.3.6 identical representation 5.5.1.23 identity functor 1.5.2.1 identity map NT identity operator 1.2.1.3 iff NT image of a linear map 1.2.4.5 imaginary part 1.I .1. I , 2.3.1.22 inclusiori functor 1.5.2.16 NT inclusion map index of a Fredholm operator 3.1.3.1 indexofU 3.1.3.21 induced norm 1.1.1.2 infimum 1.7.2.1
infinitedimensional
1.1.2.18 infinite niatrix 1.2.3.1 injective N'r inner multiplication 1.5.1.1 innerproduct 5.6.1.1
innerproduct right Emodule 5.6.1.1 integral of E with respect t o p , direct 5.5.2.19 interior point NT invariant vector subspace 3.1.4.4 inverse of a bijective m a p NT inverse of a morphism 1.5.1.6 inverse of a n element in a unital algebra 2.1.2.4 inverse operators, principle of 1.4.2.4 1.5.1.5,2.1.2.1 invertible invertible, left 1.5.1.5 invertible, right 1.5.1.5 involution 2.3.1.1 involution, conjugate 2.3.1.3 2.3.1.1 involution of E F , canonical involutive algebra 2.3.1.3 involutive algebra, complexification of an 2.3.1.40 2.3.1.26 Proposition involutive algebra, strongly symmetric involutive algebra, symmetric 2.3.1.26 involutive algebras, isomorphism of 2.3.1.3 involutive Banach algebra 2.3.2.1 2.3.2.1 involutive Banach space involutive Banach unital algebra associated t o 2.3.2.9 involutive Ealgebra 2.3.6.1 involutive Emodule 2.3.6.1 2.4.2.1 involutive Gelfand algebra involutive Hilbert space 5.5.7.1 involutive map 2.3.1.1 involutive normed algebra 2.3.2.1 involutive normed space 2.3.2.1 involutive normed unital algebra associated t o 2.3.2.9 involutive set 2.3.1.1 involutivcspace 2.3.1.1 involutive subalgebra generated by 2.3.1.18 involutive unital algebra associated t o 2.3.1.9 2.3.6.1 involutive unital Ealgebra involutivc unital E module 2.3.6.1 involutive unital Gelfand algebra 2.4.2.1 involutive unital subalgcbra generated by 2.3.1.18
Subject Index
involutive vector space 2.3.1.3 involutive vector spaces, isomorphism of 2.3.1.3 involutive vector subspace generated by 2.3.1.18 irreducible representation 5.4.1.1 irreducibly, acts 5.3.2.19 isometric Banach systems 1.5.2.1 isometric functor 1.5.2.1 isometric normed algebras 2.2.1.1 isometric normed spaces 1.2.1.12 isometric normed unital algebras 2.2.1.1 isometry of Hilbert spaces, partial 5.3.2.25 isometry of normed algebras 2.2.1.1 4.4.4.5 isometry of W*algebras isometry of normed spaces 1.2.1.12 isometry of normed unital algebras 2.2.1.1 isomorphic algebras 2.1.1.6 isomorphic Hilbert right Emodules 5.6.1.7 isomorphic normeti algebras 2.2.1.1 isomorphic riormed spaces 1.2.1.12 isomorphic normed unital algebras 2.2.1.1 isomorphic unital algebras 2.1.1.6 isomorphism, algebra 2.1.1.6 isomorphism associated to a linear map, algebraic isomorphism of EC*algebras 5.6.1.10 isomorphism of involutive algebras 2.3.1.3 isomorphism of involutive vector spaces 2.3.1.3 isomorphism of normed algebras 2.2.1.1 isomorphism of normed spaces 1.2.1.I2 isomorphism of normed unital algebras 2.2.1.1 kernel, Bergman 5.2.5.9 kernel of a linear map 1.2.4.5 Kronecker's symbol 1.2.2.6 lattice 1.7.2.1 lattice, vector 1.7.2.1 Laurent series 1.3.10.8, 1.3.10.9 Lax~Milgrarn Theorem 5.3.1.3 left Arens multiplication 1.5.2.10, 2.2.7.13 left carrier 4.3.3.1
1.2.4.6
left ideal 2.1.1.1 left ideal, maximal proper left ideal, proper 2.1.1.1 left left left left
2.1.1.1
ideal, regular maximal proper 2.1.3.17 ideal generated by 2.1.1.2 invertible 1.5.1.5 multiplication 1.5.1 . l , 5.6.1.4
left shift 1.2.2.9, E 1.2.11 left (unital) A  module 1.5.1.10 linear form 1.1.1.1 linear form, positive 1.7.1.9 linear map, conjugate 1.3.7.10 lower bound 1.7.2.1 lower bounded operator 1.2.1.18 lower section filter 1.1.6.1 L2distributions, in the sense of map NT map, bilinear 1.2.9.1 map, map, map, map,
3.2.2.3
bounded 1.1.1.2 conjugate linear 1.3.7.10 identity NT inclusion NT
map, inverse of a bijective map, involutive 2.3.1.1
NT
map, nuclear 1.6.1.1 map, qrlotier~t 1.2.4.1 maps, composition of KT matrix, infinite 1.2.3.1 maximal proper ideal 2.1.1.1 maximal proper ideal, regular 2.1.3.17 maximal proper left ideal 2.1.1.1 maximal proper left ideal, regular 2.1.3.17 maximal proper right ideal 2.1.1.1 maximal proper right ideal, regular 2.1.3.17 mean ergodic theorem 5.2.4.3 measure, Dirac 1.2.7.14 measlrrc, Evalued spectral 4.3.2.19 measure, Radon NT
Subject Index
measure of x , spectral 4.3.2.19 measure space, ufinite 3.1.6.14 metric of a normed space, canonical 1.1.1.2 module 2.2.7.1 module, involutive 2.3.6.1 module, involutive unital 2.3.6.1 module, unital 2.2.7.1 modriles,homomorphismof 2.2.7.1 modules, homomorphism of involutive 2.3.6.1 modulo NT modulus 4.2.5.1,4.4.3.5 monoid E 2.1.1 E 2.1.1 monoid, commutative morphism 1.5.1.1 morphism, inverse of a 1.5.1.6 multipliable sequence 2.2.4.33 multiplication 2.1.1.1 multiplication, (A,f?, C) 1.5.1.1 multiplication, compatible (left and right) multiplication, inner 1.5.1.1 multiplication, left 1.5.1.1, 5.6.1.4
1.5.1.1
multiplication, left (right) Arens 1.5.2.10, 2.2.7.13 multiplication, right 1.5.1.1, 5.6.1.1 m~lltiplicationoperator 2.2.2.22 multiplicity 3.1.4.1 multiplicity, algebraic 5.3.3.20 negative 1.7.1.1 negative part 4.2.2.9, 4.2.8.13 Nikodym's boundcdncss theorrm 4.3.2.14 nilpotent 2.1.1.1 nondegenerate representation 5.4.1.1 nondegenerately, acts 5.3.2.19 norm 1.1.1.2 norm, complete norm, Euclidean norm, induced norm, quotient norm, supremum
1.1.1.2 1.1.5.2 1.1.1.2 1.2.4.2 1.1.2.2, 1.1.5.2
norm of an operator 1.2.1.3 norm of a preHilbert space, canonical
5.1.1.2 norm of L ( E , F) , canonical 1.2.1.9 norm topology 1.1.1.2 normal 2.3.1.3 normed algebra 2.2.1.1 normed algebra, completion of a 2.2.1.13 normed algebra, complex 2.2.1.1 normed normed normed normed normed normed normed norrned
algebra, involutive 2.3.2.1 algebra, quasiunital 2.2.1.15 algebra, real 2.2.1.1 algebras, isometric 2.2.1.1 algebras, isometry of 2.2.1.1 algebras, isomorphic 2.2.1.1 algebras, isomorphism of 2.2.1.1 space 1.1.1.2
norrned normed normed normed normed normed normed normed normed normed normed normed
space, bidual of a 1.3.6.1 spare, complete 1.1.1.2 space, completion of a 1.3.9.1 space, complex 1.1.1.2 space, involutive 2.3.2.1 space, ordered 1.7.1.4 space, real 1.1.1.2 spaces, isometric 1.2.1.12 spaces, isometry of 1.2.1.12 spaces, isomorphic 1.2.1.12 spaces, isomorphism of 1.2.1.12 unital algebra 2.2.1.1
norrned unital algebras, isometric
2.2.1.1
normed unital algebras, isometry of normed unital algebras, isomorphic normed unital algebras, isomorphism of norms, equivalent 1. l . 1.2 nuclear map 1.6.1.1 number, cardinal NT number, ordinal NT object of a Bariach system 1.5.1.1 onto NT
2.2.1.1 2.2.1.1 2.2.1.1
Subject Index
open mapping principle 1.4.2.3 operator 1.2.1.3 5.3.1.4 operator, adjoint operator, adjoint differential 3.2.2.3 operator, bounded 1.2.1.3 operator, compact 3.1.1.1 5.3.4.1 operator, FourierPlancherel operator, Fredholm 3.1.3.1 operator, identity 1.2.1.3 operator, index of a Fredholm 3.1.3.1 operator, lower bounded 1.2.1.18 operator, multiplication 2.2.2.22 operator, order of an 3.1.3.18 operator, selfadjoint differential 3.2.2.3 operators, principle of inverse 1.4.2.4 order complete 1.7.2.1 order continuous 1.7.2.3 order faithful 4.2.2.18 order of a pole 1.3.10.9 order relation of a C*algebra, canonical 4.2.1.2 order summat)le 1.7.2.10 order acomplete 1.7.2.1 order acontinuous 1.7.2.3 order afaithful 4.2.2.18 ordered Banach space 1.7.1.4 ordered normed space 1.7.1.4 ordered set, complete 1.7.2.1 NT ordered set, totally ordered set, acomplete 1.7.2.1 ordered vector space 1.7.1.1 IVT ordinal number orthogorial 5.2.2.1 4.1.2.18, 5.2.3.2 orthogonal projection orthogonal set of A 5.2.2.1 orthogonal sets 5.2.2.1 orthogonal vectors 5.2.2.1 orthonormal basis 5.5.1.1 orthonorn~albasis of e2(T), canonical 5.5.3.1
orthonorrnal farnily 5.5.1.1 orthonormal set 5.5.1.1 orthonormalization, GramSchmidt parallelogram law 2.3.3.2 Parseval'sEquation 5.5.1.15 partial isometry of Hilbert spaces partition of a set NT pnorm 1.1.2.5,1.1.5.2 point, adherent NT point, extreme 1.2.7.9 point, interior NT
5.5.1.18
5.3.2.25
point of adherence NT point spectrum 3.1.4.1 polar 1.3.5.1 polar representation 4.2.6.9, 4.4.3.1, 4.4.3.5 polarization identity 2.3.3.2 pole (of order) 1.3.10.9 positive 1.7.1.1, 2.3.3.3, 2.3.4.1 positive linear form 1.7.1.9, 2.3.4.1 positive part 4.2.2.9,4.2.8.13 power series 1.1.6.22 precompact 1.1.2.9 predual of a Banach space 1.3.1.11 predual of a W'algebra 4.4.1.1, 4.4.4.4 preHilbert space 5.1.1.1 preHilbert space, canonical norm of a 5.1.1.2 pre  Hilbert space, completion of a 5.1.1.7 pre Hilbert space, complex 5.1.1.1 preHilbert space, real 5.1.1.1 prepolar 1.3.5.1 pretranspose of an operator 1.3.4.9, 4.4.4.8 principal part 1.3.10.8, 1.3.10.9 principle of inverse operators 1.4.2.4 principle of open mapping 1.4.2.3 principle of uniform boundedness 1.4.1.2 product 2.1.1.1 product, C'direct 4.1.1.6 product of a farriily of sets NT
Subject Index
product of a sequence 2.2.4.33 product, scalar 5.1.1.1 product associated to f , scalar 5.1.2.9 projection 1.2.5.7 projection, orthogonal
4.1.2.18, 5.2.3.2
projection of the tridual of E , canonical proper ideal 2.1 .l.1
1.3.6.19
proper ideal, maximal 2.1.1.1 proper ideal, regular maximal 2.1.3.17 proper left ideal 2.1.1.1 proper left ideal, maximal
2.1.1.1
proper left ideal, reguar maximal 2.1.3.17 proper right ideal 2.1.1.1 proper right ideal, maximal 2.1.1.1 proper right ideal, regular maximal 2.1.3.17 pure state 2.3.5.1 pure statc space 2.3.5.1 purely rcal C*algebra 4.1.1.8 Pythagoras' Theorem 5.2.2.3 quadratic form, associated 2.3.3.1 quadratic map, associated 2.3.3.1 quasinilpotent 2.2.4.20 q~rasiunital 2.2.1.15 quaternion 2.1.4.17 quotient functor 1.5.2.17 quotient map NT, 1.2.4.1 quotient norm 1.2.4.2 quotient space 1.2.4.2 quotient Acategory 1.5.2.17 quotient Amodule 1.5.2.17 Raabe's ratio test 2.2.3.11 radical 2.1.3.18 radius of convergence 1.1.6.22 Radon mcasure NT RadonNikodym Theorem 4.4.3.15 range of values KT real algebra 2.1.1.1 real Banach space 1.1.1.2
real C*algebra 4.1.1.1 real C*algebra, purely 4.1.1.8 real C'algebra, unital 4.1.1.1 real Hilbert space 5.1.1.2 real normed space 1.1.1.2 realpart 1.1.1.1,2.3.1.3 real preHilbert space 5.1.1.1 real W*algebra 4.4.1.1 reduces u reflexive
5.2.3.11 1.3.8.1
regular maximal proper ideal
2.1.3.17
regular maximal proper left ideal 2.1.3.17 regular maximal proper right ideal 2.1.3.17 relatively compact 1.1.2.9 representation 5.4.1.1 representation, associated to A 5.4.2.3 representation, associated to x' 5.4.1.2 representation, associated to ( x , ) , , ~ 5.4.2.2 representation, complex universal 5.4.2.6 representation, compression of a 5.4.2.8 representation, cyclic 5.4.1.1 representation, faithful 5.4.1.1 representation, identical 5.5.1.23 representation, irreducible 5.4.1.1 representation, nondegenerate 5.4.1.1 representation, unital 5.4.1.1 representation, universal 5.4.2.3 representation, 05.4.1.1 representations, equivalence of 5.4.1.1 representations, equivalent 5.4.1.1 representations, Hilbert sum of 5.4.2.1 residue 1.3.10.8, 1.3.10.9 resolvent 2.1.3.1 resolvent equation 2.1.3.9 Riesz, theorem of 2.2.5.1 right Arens mllltiplication 1.5.2.10, 2.2.7.13 right carrier 4.3.3.1 right C*module, tlilbert 5.6.1.4
Subject Index
right Emodule, Hilbert 5.6.1.4 right Emodule, innerproduct 5.6.1.1 rightEmodule,semiinnerproduct 5.6.1.1 5.6.3.2 right Emodrlle, von Neumann right Emodule, weak semiinnerproduct 5.6.1.1 right ideal 2.1.1.1 right ideal, maximal proper 2.1.1.1 right ideal, proper 2.1.1.1 2.1.3.17 right ideal, regular maximal proper right ideal generated by 2.1.1.2 right invertible 1.5.1.5 right multiplication 1.5.1.1, 5.6.1.1 right shift 1.2.2.9, E 1.2.11 1.5.1.10 right (unital) Amodule, right W*module, vori Neumann 5.6.3.2 scalar 1.1.1.1 scalar product 5.1.1.1 scalar product associated to f 5.1.2.9 scalar prodrlct of IKn , canonical 5.1.2.4 2.4.6.5 Schwartz space of rapidly decreasing Cmfunctions Schwarz inequality 2.3.3.9, 5.1.1.2 section filter, lower 1.1.6.1 section filter, upper 1.1.6.1 selfadjoint 2.3.1.1 selfadjoint differential operator 3.2.2.3 selfdual 5.6.2.2 selfnormal 2.3.1.3 sen~iinnerproduct right E module 5.6.1.1 semiinnerproduct right Emodule, weak 5.6.1.1 seminorm 1.1.1.2 semisimple algebra 2.1.3.18 separating vector 5.3.2.19, 5.4.4.1 NT sequence series, Laurent 1.3.10.8, 1.3.10.9 series, power 1.1.6.22 sesquiliriear form 2.3.3.1 sesquilinear form, adjoint 2.3.3.1 sesquilinear map 2.3.3.1
sesquilinear map, adjoint sesquilincar map, Hermitian set, Baire 1.7.2.12
2.3.3.1 2.3.3.3
set, bounded 1.1.1.2 set, complete ordered 1.7.2.1 set, exact 1.7.2.12 set, partition of a NT set, totally ordered NT set, pnull NT set, acomplete ordered 1.7.2.1 sharp cone 1.3.7.4 shift, left 1.2.2.9 shift, right 1.2.2.9 simple C*algebra 4.3.5.1 simultaneously compatible 1.5.1.1 space, Banach 1.1.1.2 space, bidual of a norrned 1.3.6.1 space, complete rlormed 1.1.1.2 space, completion of a normed 1.3.9.1 space, complex Banach 1.1.1.2 space. complex Hilbert 5.1.1.2 space, complex normed 1.1.1.2 space, complex preHilbert 5.1.1.1 space, dual 1.3.1.11 space, Hilbert 5.1.1.2 space, space, space: space, space,
hypcrstonian 1.7.2.12 involutive 2.3.1.1 involutive Banach 2.3.2.1 involutive normed 2.3.2.1 involutive vector 2.3.1.3
space, normed 1.1.1.2 space, ordered Banach 1.7.1.4 space, space, space, spacr, space, space,
ordered norrned 1.7.1.4 ordered vector 1.7.1.1 preHilbert 5.1.1.1 pure state 2.3.5.1 quotient 1.2.4.2 real Banach 1.1.1.2
Subject Index
space,realHilbert 5.1.1.2 space, real normed 1.1.1.2 space, real preHilbert 5.1.1.1 space, state 2.3.5.1 space, Stone 1.7.2.12 space, subspace of a normed space, vector 1.1.1.1 space, 0.Stone
1.1.1.2
1.7.2.12
space of square summable sequences, Hilbert 5.1.2.3 spaces, isometric normed 1.2.1.12 spaces, isometry of normed 1.2.1.12 spaces, isomorphic normed 1.2.1.12 spaces, isomorphism of involutive vector 2.3.1.3 spaces, isomorphism of normed 1.2.1.12 spectral decomposition 4.3.2.19, 5.3.4.7 spectral rrleasure, Evalued 4.3.2.16 spectral measure of x 4.3.2.19 spectral radius 2.1.3.1 spectrum, essential 3.1.3.24 spectrum, point 3.1.4.1 spectrum of an element 2.1.3.1 spectrum of a Gelfand algebra 2.4.1.1 square summable sequences, Hilbert space of 5.1.2.3 state 2.3.5.1 state, pure 2.3.5.1 state space 2.3.5.1 state space, pure 2.3.5.1 step functiori NT Stone space 1.7.2.12 strongly synimetric invol~rtivealgebra 2.3.1.26 subalgebra 2.1.1.1 subalgebra, unital 2.1.1.3 subalgebra generated by 2.1.1.4 subalgebra generated by, invol~ltive 2.3.1.18 subspace, complemented 1.2.5.3 subspace generated by, closed vect,or 1.1.5.5 subspacc of a normcd space 1.1.1.2 sum, C* direct 4.1.1.6
sum, direct 1.2.5.3 sum of a family 1.1.6.2 5.6.3.6 sum of a family in FA sum of representations, Hilbert 5.4.2.1 summable, absolutely 1.1.6.9 summable, order 1.7.2.10 summable family 1.1.6.2 5.6.3.6 summable in FA support of a function NT NT support of a Radon measure supremum 1.7.2.1 1.1.2.2,1.1.5.2 supremumnorm surjective NT symbol, Kronecker's 1.2.2.6 symmetric involutive algebra 2.3.1.26 symmetric involutivc algebra, strongly 2.3.1.26 5.2.4.3 theorem, mean ergodic Theorem of AlaogluBourbaki 1.2.8.1 Theorem of Banach 1.3.1.2 Theorem of BanachSteinhaus 1.4.1.2 Theorem of closed graph 1.4.2.19 Theorem of FrkchetRiesz 5.2.5.2 Theorem of Gelfand 2.2.5.4 2.2.5.5 Theorem of GelfandMazur Theorem of HahnBanach 1.3.3.1 Theorem of Laurent 1.3.10.8 5.3.1.3 Theorem of Lax14ilgram Theorem of Liouville 1.3.10.6 1.3.1.10 Theorem of KreinMilman 1.3.7.3 Theorem of reinSmulian Theorem of Minkowski 1.1.3.4 1.2.5.8 Theorem of Murray Theorem of Pythagoras 5.2.2.3 4.4.3.15 Theorem of Radon~Nikodyni Theorem of Riesz 2.2.5.1 Theorem of Weierstrass Stone 1.3.5.16 topological cardinality NT topological zerodivisor 2.2.4.24
Sribjcct Index
topology, norm 1.1.1.2 topology, weak 1.3.6.9 NT totally ordered set transpose kernel 3.1.6.5 transpose of a functor 1.5.2.3 1.3.4.1 transpose of an operator transpose unital category of L 1.3.2.2 transposition functor of L 1.5.2.2 triangle inequality 1.1.1.2 tridual of a Banach system 1.5.1.9 tridual of a normcd space 1.3.6.1 type I W*algebra 5.6.7.11 uinvariant 3.1.4.4 NT ultrafilter, frce uniform boundedness, principle of 1.4.1.2 unit 1.5.1.1, 1.5.1.4, 2.1.1.1 unit, approximate 2.2.1.15 1.1.1.2 unit ball unit of an inner multiplication 1.5.1.1 unital algebra 2.1.1.3 unital algebra, norn~ed 2.2.1.1 unital algebra associated to 2.1.1.8 unital algebra associated to, involutive 2.3.1.9 unital algebra homomorphisni 2.1.1.6 unital algebra isomorphism 2.1.1.6 unital algebras, isometric normed 2.2.1.1 unital algebras, isometry of normed 2.2.1.1 unital algebras, isomorphic 2.1.1.6 rlnital algebras, isomorphic normed 2.2.1.1 unital algebras, isomorphism of normed 2.2.1.1 urlital Banach algebra 2.2.1.1 unital Banach algebra associated to 2.2.1.4 unital Banach algebras, isomorphism of 2.2.1.1 unital Banach category 1.5.1.1 rlnital Banach subalgebra generated by 2.2.1.9 unital C*algebra 4.1.1.1 unital C' algebra, complcx 4.1.1.1 4.1.1.1 unital C*algebra,real
unital C*algebra, associated to a C* algebra 4.1.1.13 unital C*subalgebra 4.1.1.1 unital C *subalgebra generated by 4.1.1.1 ur~italEalgebra 2.2.7.1 unital Ealgebra, involutive 2.3.6.1 unital E C*algebra 5.6.1.10 unital E module 2.2.7.1 unital Emodule, involutive 2.3.6.1 unital Gelfand algebra 2.4.1.1 unital Gelfand algebra, involutive 2.4.2.1 unital Hilbert Emodule 5.6.1.4 unital involutive algebra associated to 2.3.1.9 unital involutive Banach algebra associated to 2.3.2.9 unital involutive normed algebra associated to 2.3.2.9 unital left Amodule 1.5.1.10 unital normed algebra associated to 2.2.1.4 unital normed algebras, isomorphisrri of 2.2.1.1 unital representation 5.4.1.1 unital right Amodule 1.5.1.10 unital subalgebra 2.1.1.3 unital subalgebra generated by 2.1.1.4 unital subalgebra generated by, involutive 2.3.1.18 unital W * subalgebra 4.4.4.5 unital IV'subalgebra generated by 4.4.4.5 unital 11 category 1.5.1.14 unital 11rr~odule 1.5.1.12 unital (A, A ) module 1.5.1.12 unitary 2.3.1.3 universal representation 5.4.2.3 uriiversal representation, complex 5.4.2.6 upper bound 1.7.2.1 upper section filter 1.1.6.1 upward directed 1.1.6.1 vector, cyclic 5.3.2.19, 5.4.1.1 vector, separating 5.3.2.19, 5.4.1.1 vector associated to z', cyclic 5.4.1.2 vector lattice 1.7.2.1 vector space 1.1.1.1
Subject Index
vector space, involutive 2.3.1.3 vector spaces, isomorphisms of involutive 2.3.1.3 Volterra integral equation 2.2.4.22 von Neumann (right) Emodule 5.6.3.2 von Neurnann (right) W*module W * algebra 4.4.1.1
5.6.3.2
W* algebra, complex 4.4.1.1 W*algebra, homogeneous 5.6.7.8 W*algebra, predual of a 4.4.1.1, 4.4.4.4 W*algebra, real 4.4.1.1 5.6.7.11 Wealgebra,type I W*algebras, isometry of 4.4.4.5 Mr*homomorphism 4.4.4.5 W* module, von Neumann (right) 5.6.3.2 W*subalgebra 4.4.4.5 W*subalgebra, unital 4.4.4.5 W*subalgebra generated by 4.4.4.5 IV*subalgebra generated by, unital 4.4.4.5 weak semiinnerproduct right E module 5.6.1.1 weak topology 1.3.6.9 weakly conjugate exponer~ts 1.2.2.1 zerodivisor 2.1.1.1 zero divisor, topological 2.2.4.24 Acategories, functor of (unital) 1.5.2.1 A category 1.5.1.14 Acategory, quotient 1.5.2.17 Acategory, unital 1.5.1.14 Amodule 1.5.1.12 Amodule, left (right) 1.5.1.10 A module, quotient 1.5.2.17 11modlile, unital 1.5.1.12 A module, unital left (right) 1.5.1.10 A modules, functor of left (right) Asubcategory 1.5.2.16 Asi~bmodule 1.5.2.16 (A, A) n~odule 1.5.1.12 1.5.1.12 (A, A) module, unital pnull set ST
1.5.2.1
acomplete, aorder 4.3.2.3 acomplete order 1.7.2.1 acomplete ordered set 1.7.2.1 acontinuous, order 1.7.2.3 a faithful, order 4.2.2.18 ufinite measure space 3.1.6.14 aStone space 1.7.2.12 0representation 5.4.1.1
Symbol Index
Symbol Index NT means Notation and Terminology la1 a* A* A' A
4.4.3.5 2.3.1.30 2.3.1.1 5.2.2.1 NT NT "A,A" 1.3.5.1 A", A"", A"" 2.1.1.16 A', A", A"' 1.5.1.9 AAl 1.3.6.9 aa' , a'a 2.2.7.8 ab 2.1.4.23 ax 2.2.7.23 a'z" 2.2.7.11

( a ,(, 7) 5.6.3.2 AIt3 1.5.2.17 A +B 1.2.4.1 A\B NT AAB NT AxB NT A +z 1.2.4.1 B 1.1.2.4 C NT c 1.1.2.3,2.1.4.3 co 1.1.2.3,2.1.4.3 c(T) 1.1.2.3,2.1.4.3 cO(T) C(T) C ( T ,E) Co(T)
1.1.2.3,2.1.4.3 1.1.2.4, 2.1.4.4 1.1.2.8 1.2.2.10, 2.1.4.4
NT
Card Coker
1.2.4.5
dA
1.1.4.1
Det NT Dim 1.1.2.18 D ( k , p, v) 3.1.6.15
&(k, E' E" E"' E e~ et e: er ez
U)
3.1.6.1 1.2.1.3 1.3.6.1 1.3.6.1
2.1.5.1, 2.1.5.7, 2.3.1.38, 2.3.1.40, 5.3.1.8 1.1.2.1 1.1.2.1 1.1.2.1 1.1.2.1 2.2.3.5
Ea(u) 3.1.3.18 Et,(u) 3.1.3.18 Em, 2.1.4.23, 2.3.1.30 En, 2.1.4.24, 2.3.1.31, 5.6.6.1 ~ ' r 1.1.2.1 E(') 1.1.2.1 E" 1.7.2.3 E" 1.7.2.3 E 4.4.4.4 E+ 1.7.1.1,4.2.1.1 E# 1.1.1.2 E,# 1.7.1.4 E ; , E: 2.2.7.15 E(x) 2.3.2.15 E ( x , 1) 2.3.2.15
Symbol Index
f' 1.1.6.24 f 2.3.3.1 3,4 1.2.6.1 3(E) 3.1.3.1 3 ( E ,F ) 3.1.3.1
51
fls f (a, .) f (., b) f(A) f(x) f'
1.1.6.1
NT NT NT NT NT, 4.1.3.1, 4.1.3.2, 4.3.2.5 NT
1
f (B) f (Y)
NT
I
h'T f:X+Y NT f : X + Y , x.tT(x) F [ s ,t ] NT F[t] NT F @G 1.2.5.3 { f =9 ) NT { f # 9) NT if>@) KT g0f NT, 1.5.2.1 4,4 1.7.2.3
EI
6 H .. H irn Im Indu Ind U
NT
2.1.4.17, 2.3.1.46, 4.1.1.31 5.6.1.6 5.6.2.2 5.6.3.2 5.6.3.2 1.1.1.1, 2.3.1.22 1.2.4.5 3.1.3.1 3.1.3.21
n
k
u
k
1.2.3.1
Ker
1.2.3.1 1.2.4.5
C
1.2.1.3, 1.5.1.1, 2.1.4.6
Ln Lr L'
1.5.1.1 1.2.1.3 1.6.1.1,1.6.1.3 1.6.1.13
LE(G,H ) ( H) LE(H)
5.6.1.7 5.6.2.2
5.6.1.7 1.1.2.5 e 2 ( I ,F ) 5.6.4.2 &P
tP(T) t2(T)
1.1.2.5 5.5.7.1
e0
1.1.2.3 P(T) 1.1.2.3
trn
1.1.2.2, 2.1.4.3
&"(T) 1.1.2.2, 2.1.4.3 eys, T ) 1.2.3.2 t r 2 q ( ST, ) 1.2.3.2 log
2.2.3.9, 4.2.4.4
Mb M MI.,^
1.1.2.26
IN
IN,
5.6.4.20 5.6.4.19
NT 1.1.3.3
Symbol Index
N,! No NF,F
NF,G
Pr
p Ip,
2.3.4.1 2.3.1.3 5.6.4.16 5.6.4.15, 5.6.4.19 4.1.2.18 1.1.2.1 1.1.2.1
Q
NT IR NT IR NT re 1.1.1.1,2.3.1.3 Re 2.3.1.1 Re E# 2.3.2.1 ( x )( x ) 2.1.3.1 Sn 2.3.1.3 S(IRn) 2.4.6.5
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