Inner invariant means and the regular conjugation representation of L1(G)

1988 ◽  
pp. 283-287
Author(s):  
Chuan Kuan Yuan
Keyword(s):  
Invariant Means ◽  
Conjugation Representation

Locally compact groups whose conjugation representations satisfy a Kazhdan type property or have countable support

1994 ◽  
Vol 116 (1) ◽  
pp. 79-97 ◽  
Author(s):  
Eberhard Kaniuth ◽  
Annette Markfort
Keyword(s):  
Modular Function ◽  
Equivalence Classes ◽  
Irreducible Representations ◽  
Compact Groups ◽  
Locally Compact ◽  
Countable Support ◽  
Type Property ◽  
Invariant Means ◽  
Image Position ◽  
Conjugation Representation

For a locally compact group G with left Haar measure and modular function δ the conjugation representation γG of G on L2(G) is defined byf ∈ L2(G), x, y ∈ G. γG has been investigated recently (see [19, 20, 21, 24, 32, 35]). For semi-simple Lie groups, a related representation has been studied in [25]. γG is of interest not least because of its connection to questions on inner invariant means on L∞(G). In what follows suppγG denotes the support of γG in the dual space Ĝ, that is the closed subset of all equivalence classes of irreducible representations which are weakly contained in γG. The purpose of this paper is to establish relations between properties such as a variant of Kazhdan's property and discreteness or countability of supp γG and the structure of G.

scholarly journals Remark on invariant means

1967 ◽  
Vol 18 (1) ◽  
pp. 120-120 ◽  
Author(s):  
Robert Kaufman
Keyword(s):  
Invariant Means

The Extensions of an Invariant Mean and the Set LIM ∽ TLIM

1994 ◽  
Vol 46 (4) ◽  
pp. 808-817
Author(s):  
Tianxuan Miao
Keyword(s):  
Compact Group ◽  
Discrete Group ◽  
Locally Compact Group ◽  
Invariant Mean ◽  
Locally Compact ◽  
Invariant Means

AbstractLet with . If G is a nondiscrete locally compact group which is amenable as a discrete group and m ∈ LIM(CB(G)), then we can embed into the set of all extensions of m to left invariant means on L∞(G) which are mutually singular to every element of TLIM(L∞(G)), where LIM(S) and TLIM(S) are the sets of left invariant means and topologically left invariant means on S with S = CB(G) or L∞(G). It follows that the cardinalities of LIM(L∞(G)) ̴ TLIM(L∞(G)) and LIM(L∞(G)) are equal. Note that which contains is a very big set. We also embed into the set of all left invariant means on CB(G) which are mutually singular to every element of TLIM(CB(G)) for G = G1 ⨯ G2, where G1 is nondiscrete, non–compact, σ–compact and amenable as a discrete group and G2 is any amenable locally compact group. The extension of any left invariant mean on UCB(G) to CB(G) is discussed. We also provide an answer to a problem raised by Rosenblatt.

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