EXERCISES ----
(a)
1.
Let A be a Frechet a l g e b r a s a t i s f y i n g t h e c o n d i t i o n (C')
n i t i o n...

Author:
T. Husain

EXERCISES ----

(a)

1.

Let A be a Frechet a l g e b r a s a t i s f y i n g t h e c o n d i t i o n (C')

n i t i o n 3.1).

Show t h a t t h e s p e c t r a l r a d i u s p ( x ) =

sup If(x) f EM( a )

.,

( s e e Defi-

1

=

(see

m

[161)* (b)

Show t h a t a Banach a l g e b r a o r a Q-algebra

cannot s a t i s f y c o n d i t i o n

(C') ( s e e [ 1 6 ] ) . 2.

Show

that

the

class

of

infrasequential

(or

sequential

or

strongly

s e q u e n t i a l ) a l g e b r a s i s c l o s e d under t h e following o p e r a t i o n s : subalgebras ;

(i )

(ii)

f i n i t e products; q u o t i e n t s i f A , i n a d d i t i o n , is m e t r i z a b l e ;

(iii) (iv)

3.

unitization:

L e t h be a TA.

A+ = A $ For e a c h x

C ( s e e [16]). E

A , d e f i n e B(x) = i n f { X

>

0: (X

-1

n

x) ) i s

bounded}. Show t h a t A i s s t r o n g l y s e q u e n t i a l i f f { x

(a)

E

A: B(x)

< 11 i s a neigh-

borhood of 0. If A i s e i t h e r a commutative pseudocomplete LC a l g e b r a o r LMC, then fl

(b)

i s a s u b m u l t i p l i c a t i v e norm on t h e s u b a l g e b r a A" of A , c o n s i s t i n g of a l l

x

E

A with e ( x )

4.

0.

Under t h e h y p o t h e s i s (b) A is s e q u e n t i a l i f f fl i s s e q u e n t i a l l y contin-

(c) uous.

<

A i s s t r o n g l y s e q u e n t i a l i f f (3 i s continuous ( s e e [ 2 9 ] ) .

(a)

If A is a pseudocomplete LC a l g e b r a w i t h continuous quasi-inversion,

then 4 i s s t r o n g l y s e q u e n t i a l i f f A i s a Q-algebra.

5.

(b)

A Frechet a l g e b r a i s s e q u e n t i a l i f f i t is a Q-algebra.

(c)

A metrizable algebra is sequential i f f i t is strongly sequential.

Show t h a t t h e d i r e c t sum of s e q u e n t i a l ( o r s t r o n g l y s'equential) l o c a l l y

77

(a)

1.

Let A be a Frechet a l g e b r a s a t i s f y i n g t h e c o n d i t i o n (C')

n i t i o n 3.1).

Show t h a t t h e s p e c t r a l r a d i u s p ( x ) =

sup If(x) f EM( a )

.,

( s e e Defi-

1

=

(see

m

[161)* (b)

Show t h a t a Banach a l g e b r a o r a Q-algebra

cannot s a t i s f y c o n d i t i o n

(C') ( s e e [ 1 6 ] ) . 2.

Show

that

the

class

of

infrasequential

(or

sequential

or

strongly

s e q u e n t i a l ) a l g e b r a s i s c l o s e d under t h e following o p e r a t i o n s : subalgebras ;

(i )

(ii)

f i n i t e products; q u o t i e n t s i f A , i n a d d i t i o n , is m e t r i z a b l e ;

(iii) (iv)

3.

unitization:

L e t h be a TA.

A+ = A $ For e a c h x

C ( s e e [16]). E

A , d e f i n e B(x) = i n f { X

>

0: (X

-1

n

x) ) i s

bounded}. Show t h a t A i s s t r o n g l y s e q u e n t i a l i f f { x

(a)

E

A: B(x)

< 11 i s a neigh-

borhood of 0. If A i s e i t h e r a commutative pseudocomplete LC a l g e b r a o r LMC, then fl

(b)

i s a s u b m u l t i p l i c a t i v e norm on t h e s u b a l g e b r a A" of A , c o n s i s t i n g of a l l

x

E

A with e ( x )

4.

0.

Under t h e h y p o t h e s i s (b) A is s e q u e n t i a l i f f fl i s s e q u e n t i a l l y contin-

(c) uous.

<

A i s s t r o n g l y s e q u e n t i a l i f f (3 i s continuous ( s e e [ 2 9 ] ) .

(a)

If A is a pseudocomplete LC a l g e b r a w i t h continuous quasi-inversion,

then 4 i s s t r o n g l y s e q u e n t i a l i f f A i s a Q-algebra.

5.

(b)

A Frechet a l g e b r a i s s e q u e n t i a l i f f i t is a Q-algebra.

(c)

A metrizable algebra is sequential i f f i t is strongly sequential.

Show t h a t t h e d i r e c t sum of s e q u e n t i a l ( o r s t r o n g l y s'equential) l o c a l l y

77

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