Topological inner invariant means

2003 ◽  
Vol 40 (3) ◽  
pp. 293-299
Author(s):  
R. Memarbashi ◽  
A. Riazi
Keyword(s):  
Compact Group ◽  
Locally Compact Group ◽  
Locally Compact ◽  
Invariant Means ◽  
Inner Amenability

For a locally compact group G, we investigate topological inner invariant means on L8(G) and its subspaces. In particular, we characterize strict inner amenability of L1(G) to study the relation between this notion and strict inner amenability of G.

On the amenability of a class of Banach algebras with application to measure algebra

2019 ◽  
Vol 69 (5) ◽  
pp. 1177-1184
Author(s):  
Mohammad Reza Ghanei ◽  
Mehdi Nemati
Keyword(s):  
Compact Group ◽  
Invariant Subspace ◽  
Banach Algebras ◽  
Locally Compact Group ◽  
Approximate Identity ◽  
Invariant Mean ◽  
Measure Algebra ◽  
Locally Compact ◽  
Invariant Means ◽  
Inner Amenability

Abstract Let 𝓛 be a Lau algebra and X be a topologically invariant subspace of 𝓛* containing UC(𝓛). We prove that if 𝓛 has a bounded approximate identity, then strict inner amenability of 𝓛 is equivalent to the existence of a strictly inner invariant mean on X. We also show that when 𝓛 is inner amenable the cardinality of the set of topologically left invariant means on 𝓛* is equal to the cardinality of the set of topologically left invariant means on RUC(𝓛). Applying this result, we prove that if 𝓛 is inner amenable and 〈𝓛2〉 = 𝓛, then the essential left amenability of 𝓛 is equivalent to the left amenability of 𝓛. Finally, for a locally compact group G, we consider the measure algebra M(G) to study strict inner amenability of M(G) and its relation with inner amenability of G.

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.

Uniformly Continuous Functionals and M-Weakly Amenable Groups

2013 ◽  
Vol 65 (5) ◽  
pp. 1005-1019 ◽  
Author(s):  
Brian Forrest ◽  
Tianxuan Miao
Keyword(s):  
Compact Group ◽  
Amenable Group ◽  
Locally Compact Group ◽  
Approximate Identity ◽  
Fourier Algebra ◽  
Locally Compact ◽  
Amenable Groups ◽  
Invariant Means ◽  
Continuous Functionals ◽  
Uniformly Continuous

AbstractLet G be a locally compact group. Let AM(G) (A0(G))denote the closure of A(G), the Fourier algebra of G in the space of bounded (completely bounded) multipliers of A(G). We call a locally compact group M-weakly amenable if AM(G) has a bounded approximate identity. We will show that when G is M-weakly amenable, the algebras AM(G) and A0(G) have properties that are characteristic of the Fourier algebra of an amenable group. Along the way we show that the sets of topologically invariant means associated with these algebras have the same cardinality as those of the Fourier algebra.

A BIJECTION OF INVARIANT MEANS ON AN AMENABLE GROUP WITH THOSE ON A LATTICE SUBGROUP

2021 ◽  
pp. 1-6
Author(s):  
JOHN HOPFENSPERGER
Keyword(s):  
Compact Group ◽  
Lie Groups ◽  
Amenable Group ◽  
Locally Compact Group ◽  
Locally Compact ◽  
Discrete Subgroups ◽  
Invariant Means ◽  
Lattice Subgroup

Abstract Suppose G is an amenable locally compact group with lattice subgroup $\Gamma $ . Grosvenor [‘A relation between invariant means on Lie groups and invariant means on their discrete subgroups’, Trans. Amer. Math. Soc.288(2) (1985), 813–825] showed that there is a natural affine injection $\iota : {\text {LIM}}(\Gamma )\to {\text {TLIM}}(G)$ and that $\iota $ is a surjection essentially in the case $G={\mathbb R}^d$ , $\Gamma ={\mathbb Z}^d$ . In the present paper it is shown that $\iota $ is a surjection if and only if $G/\Gamma $ is compact.

Invariant Means on a Class of von Neumann Algebras Related to Ultraspherical Hypergroups II

2017 ◽  
Vol 60 (2) ◽  
pp. 402-410
Author(s):  
N. Shravan Kumar
Keyword(s):  
Compact Group ◽  
Von Neumann Algebras ◽  
Locally Compact Group ◽  
Fourier Algebra ◽  
Locally Compact ◽  
Von Neumann ◽  
Invariant Means

AbstractLet K be an ultraspherical hypergroup associated with a locally compact group G and a spherical projector π and let VN(K) denote the dual of the Fourier algebra A(K) corresponding to K. In this note, we show that the set of invariant means on VN(K) is singleton if and only if K is discrete. Here K need not be second countable. We also study invariant means on the dual of the Fourier algebra A0(K), the closure of A(K) in the cb-multiplier norm. Finally, we consider generalized translations and generalized invariant means.

The Von Neumann Algebra VN(G) of a Locally Compact Group and Quotients of Its Subspaces

1997 ◽  
Vol 49 (6) ◽  
pp. 1117-1138
Author(s):  
Zhiguo Hu
Keyword(s):  
Compact Group ◽  
Von Neumann Algebra ◽  
Quotient Space ◽  
Linear Mapping ◽  
Locally Compact Group ◽  
Continuous Linear ◽  
Locally Compact ◽  
Von Neumann ◽  
Quotient Spaces ◽  
Invariant Means

AbstractLet VN(G) be the von Neumann algebra of a locally compact group G. We denote by μ the initial ordinal with |μ| equal to the smallest cardinality of an open basis at the unit of G and X = ﹛α ; α < μ﹜.We show that if G is nondiscrete then there exist an isometric *-isomorphism of l∞(X) into VN(G) and a positive linear mapping π of VN(G) onto l∞(X) such that π o = idl∞(X) and and π have certain additional properties. Let UCB((Ĝ)) be the C*–algebra generated by operators in VN(G) with compact support and F(Ĝ) the space of all T∈ VN(G) such that all topologically invariant means on VN(G) attain the same value at T. The construction of the mapping π leads to the conclusion that the quotient space UCB((Ĝ))/F((Ĝ)) ∪UCB((Ĝ)) has l∞(X) as a continuous linear image if G is nondiscrete. When G is further assumed to be non-metrizable, it is shown that UCB((Ĝ))/F((Ĝ)) ∪UCB((Ĝ)) contains a linear isomorphic copy of l∞(X). Similar results are also obtained for other quotient spaces.

Inner amenability of locally compact groups and their algebras

2007 ◽  
Vol 44 (2) ◽  
pp. 265-274
Author(s):  
M. Lashkhrizadeh Bami ◽  
B. Mohammadzadeh
Keyword(s):  
Compact Group ◽  
Open Problem ◽  
Locally Compact Group ◽  
Locally Compact Groups ◽  
Invariant Mean ◽  
Compact Groups ◽  
Locally Compact ◽  
Inner Amenability

It is shown that there is a locally compact group G which is not inner amenable, but L∞ ( G ) has a topological inner invariant mean which is not a mixed identity. This resolves negatively an open problem raised by Nasr-Isfahani. Motivated by this problem, we also give a characterization of strict inner amenability for certain locally compact groups.

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