*-Regularity of Some Classes of Solvable Groups

Math. Ann. 261, 477481 (1982) 0 Springer-Verlag1982

*-Regularity of Some Classes of Solvable Groups J. Boidol Fakult~it ffir Mathematik der Universit~t, Universit~tsstrasse 1, D-4800 Bielefeld I, Federal Republic of Germany

A locally compact group G is called *-regular if for every closed set S C ~, the unitary dual of G, and every nEG\S there exists a function f~LX(G) such that )7(a) = a ( f ) = 0 for all a~S and f ( n ) = n(f)4:0. In [2] we classified all connected *-regular groups. Let G be connected with commutator group L and let n~ G. Then there exists a unique closed normal subgroup N~ of G containing L such that n is weakly equivalent to a representation induced from N~ and N, is minimal with these properties. Let K~ = Kern6n. A connected group G is *-regular if and only if NffK, has polynomial growth for all n~ G. For general locally compact groups however there are only the following few results, see [1]: (A) Every *-regular group is amenable. (B) Every group with polynomial growth is *-regular. (C') Every semidirect product G = H ~
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groups in 1"10] that for connected, simply connected Lie groups G with dim G < 6 *-regularity is equivalent to symmetry.

1. A Reduction Theorem In this paragraph we prove a reduction theorem which allows us to reduce in many cases the proof of *-regularity of a group to the case of compactly generated groups.

Theorem 1. Let G be a locally compact group and let {Hi}~ t be a family of closed subgroups of G such that every open, relatively compact set K C G is contained in some H~. I f all Hi, ieI, are *-regular, then G is *-regular. Proof Let rq, r~2 be unitary representations of G with KernLl(~)g 1 C KernL1(~)rc2 . According to 1"2,Sect. 1], this implies KernLllrh/~11m C KernLl(m)re21m for all iel. Let feJg'(G) and K={oeG]f(9)+O}. Then K is open and relatively compact and there exists i oe I w i t h / ( C H~o. But then H~o is also open in G and the Haar measure on H~o is equal to the restriction of the Haar measure of G. Now the *-regularity of H~o gives according to [1, Satz 1],

13 Since n,ln,o(fln~)=rq(f), i= 1,2, this implies [Irr2(f)[I < I{rq(f)l[. Therefore ][rc2(f)ll < II~l(f)l[ for all f e X ( G ) . Again by [1, Satz 1], G is *-regular.

Corollary 1. Let G be a locally compact group such that all open, compactly generated subgroups are *-regular. Then G is *-regular. Proof. Every open, relatively compact set K C G is contained in an open, compactly generated subgroup of G. Therefore one can apply Theorem 1 to the family of open, compactly generated subgroups of G. Remark. A result corresponding to Corollary 1 does not hold with respect to symmetry. In 13] an example is given of a locally finite discrete group which is not symmetric. 2. Some Classes of Metabelian Groups Theorem 2. Every connected metabelian group G is *-regular. Proof By [2, Lemma 9], it is sufficient to prove the theorem for connected, simply connected Lie groups G. Assume that there exist connected, simply connected metabelian Lie groups G which are not *-regular. Let GO be a Lie group of this kind of minimal dimension. Since the roots of go, the Lie algebra of GO, are not all

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pure imaginary by [1, Satz 2], there exists a minimal not central ideal m of go, where go acts by a root q~ with Req~4:0. Let gl = K e r n Req~. gt is an ideal of g which is also metabelian. The stabilizer in G O of every nontrivial functional f e m* is contained in G 1 = exp g r Therefore the irreducible unitary representations of G O either vanish on W = exp m or are induced from G 1 = exp g r N o w G 1 and Go/W are metabelian hence by the minimality of dim G O *-regular. The induction theorem [2, Theorem 1] gives the contradiction that G O is *-regular. We consider now the case of a semidirect product of two abelian groups.

Proposition 1. Let G = H ~
1'

Prim o C*(G) ~~ , P r i m , o LI(G). i. and i are homeomorphic embeddings and ~Po is a homeomorphism according to [1, Satz 40)]. It follows that ~p~ is a homeomorphism. This shows that the G s are *-regular for all ~t. But then G is *-regular by I2, L e m m a 91. Lamina 1. Let G = H D
Proof. Let K E G be compact. Then K E K H x K s where K n and K s are open, symmetric and relatively compact in H resp. N. Consider the set N O= U (KNrr')". rl, m

N O is an open subgroup of N which is K n, hence(KR)-invariant. H o = ( K n ) is an open subgroup of H. It follows that Ho ~ No is an open subgroup of G containing K. It is clear that H o and No are a-compact.

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T h e o r e m 3. Every semidirect product G = H ~
and N is *-regular. Proof. Let {C~}i~x be the family of a-compact open subgroups G O= H o t~ N o of G, where H o is a subgroup of H and N o a subgroup of N. Then C i is *-regular for all i~ I by Proposition 1. Furthermore by Lemma 1 every compact K C G is contained in some C~. Therefore we can apply Theorem 1 and get that G is *-regular. 3, S y m m e t r i c Groups T h e o r e m 4. Every discrete, symmetric, solvable group is *-regular.

Proof. Let G be discrete, symmetric, and solvable. Then every subgroup of G has the same properties, especially the finitely generated subgroups. By Theorem 1 we are reduced to the case of finitely generated groups. But by I-8, (17)] for finitely generated discrete solvable group symmetry implies polynomial growth. Then G is *-regular by [1, Satz 2]. Theorem 4 and the results of I-11, 12] suggest the following conjecture: (C1) Every symmetric locally compact group is *-regular. Proposition 2. I f every symmetric, compactly generated separable group is *-regular, then (C1) is true.

Proof. Let G be symmetric. Then every open, compactly generated subgroup of G is also symmetric. By Theorem 1 it is therefore sufficient to prove that every compactly generated, symmetric group is *-regular. By [5, A 10] a compactly generated locally compact group G has the form G = prolimG~ with separable groups G~. If G is symmetric all G~ are symmetric and by assumption *-regular. By 1-2, Lemma 9], it follows that G is *-regular. Theorem 3 and [6] show that there exist compactly generated, solvable groups which are *-regular but not symmetric. But it seems probable that the following conjecture is true: (C2) Every almost connected *-regular group is symmetric. P r o p o s i t i o n 3. I f every *-regular connected Lie group is symmetric, then (Cz) is true.

Proof. Let G be *-regular and almost connected. Then G = prolim G~, where the G~ are almost connected Lie groups, hence finite extensions of connected Lie groups. The G, are *-regular. But then by the results of I2] it is easy to see that the connected components of the identity Go are *-regular. Thus if we assume that *-regular connected Lie groups are symmetric, we get that the Go are symmetric. By [7, Theorem 3] it follows that the G~ are symmetric and by 1-9,Theorem 2] that G is symmetric. References 1. Boidol, J., Leptin, H., Schiirmann, J., Vahle, D. : R~iumr primitiver Ideale yon Gruppenalgebren. Math. Ann. 236, 1-13 (1978) 2. Boidol, J.: Connected groups with polynomially induced dual. J. Reine Angew. Math. 331, 32-46 (1982)

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3. Fountain, J.B., Ramsay, R.W., Williamson, J.H. : Functions of measures on compact groups. Proceedings of the Royal Irish Academy, Sect. A 76, 235-251 (1976) 4. Gootman, E.C., Rosenberg, J. : The structure of crossed product C*-algebras: a proof of the generalized Effros-Hahn conjecture. Invent. Math. 52, 283-298 (1979) 5. Greenleaf, F.P., Moskowitz, M. : Cyclic vectors for representations associated with positive definite measures: non separable groups. Pacific J. Math. 45, 165-186 (1973) 6. Jenkins, J. : An amenable group with a nonsymmetric group algebra. Bull. Am. Math. Soc. 75, 45-47 (1969) 7. Leptin, H. : On symmetry of some Banach algebras. Pacific J. Math. 53, 203-206 (1974) 8. Leptin, H. : Ideal theory in group algebras of locally compact groups. Invent. Math. 31, 259-278 (1976) 9. Leptin, H., Poguntke, D. : Symmetry and nonsymmetry for locally compact groups. J. Functional Analysis 33, 119-134 (1979) 10. Poguntke, D. : Nicbt symmetrische sechsdimensionale Liesche Gruppen. J. Reine Angew. Math. 306, 154-176 (1979) 11. Poguntke, D.: Symmetry and nonsymmetry for a class of exponential Lie groups. J. Reine Angew. Math. 315, 127-138 (1980) 12. Poguntke, D.: Operators of finite rank in unitary representations of exponential Lie groups. Math. Ann. 259, 371-383 (1982) Received June 2, 1982