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• W2004564780 abstract "Let G be a locally compact group, it: G -> £(L2(G)) the right regular representation of G, and C'»(i£ G: the function g ~» it(gxg~]) is norm continuous}. This note is devoted to the study of Gc. In particular, the compactly generated groups for which G = Gc are characterized. 1. Let G be a locally compact group and let it: G -» t(L2(G)) be the right regular representation of G on L2(G) with respect to a right Haar measure. The function it is continuous when £,(L2(G)) is given the strong operator topology, but it is not continuous with respect to the norm topology, except in trivial cases. Nevertheless, there is a middle ground, to which this note is devoted. Following [5], [8], let £G = {T E t(L2(G)): the function g~*it(g)Tit(g)* is norm continuous}. Then £G is a C*-algebra which contains the compact operators and has various pleasing properties (cf. [5, Theorem 2.2]). Let Gc = {x E G: it(x) E eG} = [x E G: the function g ~» it(gxg'x) is norm continuous}. It is easy to see that Gc is a subgroup of G. If G is abelian or discrete, then Gc = G; in general it is much smaller. The relationship between Gc and G is the main subject of this note. We shall denote the identity component and the center of G by t70 and Z(G), respectively. For x,y E G, CG(x) denotes the centralizer of x in G and [x,y] = xyx~xy~x. Definition 1.1. Let fibea Banach space of functions on G. Suppose that there exist constants C, 8 > 0 such that the following conditions are satisfied: (i) If tp E B and x E G, then Btt(x)tp E B, and IIJ»'77'(JC)*JDII ** C|MI> where Bit(x)tp(t) = tp(txx). (ii) Given 0 such that || X(tp) for all p E B such that tp • i/ = 0. (iii) For every neighborhood U of e in G there exists 0 =£ tp E B such that tp = 0 off UandX(tp) > 8tp. Then B will be called a homogeneous separating Banach space of functions on G. Received by the editors March 13, 1980 and, in revised form, June 18, 1980. 1980 Mathematics Subject Classification. Primary 47C15, 47D10. 'Research of both authors was partially supported by the National Science Foundation. 99 © 1981 American Mathematical Society 0002-9939/81/0000-0219/$02.50 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 100 CLAUDE SCHOCHET AND BERTRAM SCHREIBER Examples 1.2. Most of the Banach spaces commonly encountered in harmonic analysis satisfy the conditions of Definition 1.1. The following is a sampling of such spaces. (i) L(G), 1 £(7?) be the regular representation as in (i) of Definition 1.1. Set BGC = {x E G: the function g ~*Bit{gxg~x) is norm continuous}. The following observation provides the technical tool which is the key to studying BGC. Theorem 1.4. Let B, Bit and BGC be as in Definition 1.3, and let x E G. Then x E BGC if and only if CG(x) is an open subgroup of G. In view of Theorem 1.4 the subscript B in BGC is redundant and will be omitted following the proof of this theorem. Furthermore, the superscript in Gc may be read as referring to (norm) continuity or to (open) centralizers. Proof. Suppose first that CG(x) is open in G. Then the function g ~*Bir(gxg~x) is constant when restricted to the open subgroup CG(x) of G, so in particular it is norm continuous at the identity. Since Bit is a norm-bounded representation of G (1.1 (i)), it follows that the function in question is continuous on all of G. Conversely, suppose that CG(x) is not an open subgroup of G. Then any neighborhood V of e must contain some v E V CG(x). We shall show that Bit(gxg~x) — Btt(x) is bounded away from zero for all such g, hence x & BGC. As [g, x] =£ e there is a neighborhood W of e such that W n W[g, x] = 0. Let 0 t^ tp E B such that tp = 0 of f W and X(tp) > 8 tp. Then Bit([g, x]) is supported on W[ g, x]. Thus Bit(gxg-x) Bit(x) = (Bit([g,x]) l)Bir(x) > Bir([g,x]) I/C > B([g,x])tp-tp/Ctp >X(m)/C||m|| >8/C. □ Corollary 1.5. Let G be a connected group. Then Gc = Z(G). Proof. For any group G one has Z(G) c Gc trivially. Conversely, suppose that x E Gc. Then CG(x) is an open subgroup of G. But G is connected, so CG(x) = G. Thus x E Z(G). □ Corollary 1.6. Suppose that Gc = G. Then G0 c Z(G). The remainder of this note is devoted to the study of Gc. In §2 we consider the case when G = Gc and we obtain complete information when G is compactly generated. §3 is concerned with other cases. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use CONTINUITY OF GROUP REPRESENTATIONS 101 2. In this section we characterize the compactly generated groups for which G = Gc and we obtain various equivalent formulations of this property. Lemma 2.1. Suppose that G = Gc. Let A and B be compact sets of G. Then [A,B] ={[x,y]:xEA,y E B) is a finite set. Proof. Let w, x, y, z E G with z E CG(x) n CG(y) (2.2) and w E CG(x) n CG(y) n CG(z). (2.3) Then [zx, vvy] = z;oty;c~lz~|y~1w~' = xyzwz~xw~xx~~x by (2.2) and (2.3)" @default.
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• W2004564780 title "Open centralizers and the continuity of group representations" @default.
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