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.

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.

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.

Decomposability of von Neumann Algebras and the Mazur Property of Higher Level

2006 ◽  
Vol 58 (4) ◽  
pp. 768-795 ◽  
Author(s):  
Zhiguo Hu ◽  
Matthias Neufang
Keyword(s):  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Group Structure ◽  
Cardinal Number ◽  
Banach Algebras ◽  
Locally Compact Group ◽  
Measure Algebra ◽  
Locally Compact ◽  
Von Neumann ◽  
Topological Centre

AbstractThe decomposability number of a von Neumann algebra ℳ (denoted by dec(ℳ)) is the greatest cardinality of a family of pairwise orthogonal non-zero projections in ℳ. In this paper, we explore the close connection between dec(ℳ) and the cardinal level of the Mazur property for the predual ℳ* of ℳ, the study of which was initiated by the second author. Here, our main focus is on those von Neumann algebras whose preduals constitute such important Banach algebras on a locally compact group G as the group algebra L1(G), the Fourier algebra A(G), the measure algebra M(G), the algebra LUC(G)*, etc. We show that for any of these von Neumann algebras, say ℳ, the cardinal number dec(ℳ) and a certain cardinal level of the Mazur property of ℳ* are completely encoded in the underlying group structure. In fact, they can be expressed precisely by two dual cardinal invariants of G: the compact covering number κ(G) of G and the least cardinality ᙭(G) of an open basis at the identity of G. We also present an application of the Mazur property of higher level to the topological centre problem for the Banach algebra A(G)**.

scholarly journals Johnson pseudo-Connes amenability of dual Banach algebras

Filomat ◽  
2021 ◽  
Vol 35 (2) ◽  
pp. 551-559
Author(s):  
Amir Sahami ◽  
Seyedeh Shariati ◽  
Abdolrasoul Pourabbas
Keyword(s):  
Compact Group ◽  
Finite Index ◽  
Banach Algebras ◽  
Locally Compact Group ◽  
Locally Compact ◽  
Hereditary Properties ◽  
Connes Amenability

We introduce the notion of Johnson pseudo-Connes amenability for dual Banach algebras. We study the relation between this new notion with the various notions of Connes amenability like Connes amenability, approximate Connes amenability and pseudo Connes amenability. We also investigate some hereditary properties of this new notion. We prove that for a locally compact group G,M(G) is Johnson pseudo-Connes amenable if and only if G is amenable. Also we show that for every non-empty set I,MI(C) under this new notion is forced to have a finite index. Finally, we provide some examples of certain dual Banach algebras and we study their Johnson pseudo-Connes amenability.

scholarly journals Operator Amenability of the Fourier Algebra in the cb-Multiplier Norm

2007 ◽  
Vol 59 (5) ◽  
pp. 966-980 ◽  
Author(s):  
Brian E. Forrest ◽  
Volker Runde ◽  
Nico Spronk
Keyword(s):  
Compact Group ◽  
Free Group ◽  
Discrete Group ◽  
Amenable Group ◽  
Locally Compact Group ◽  
Approximate Identity ◽  
Fourier Algebra ◽  
Locally Compact ◽  
Residually Finite ◽  
Finite Dimensional

AbstractLet G be a locally compact group, and let Acb(G) denote the closure of A(G), the Fourier algebra of G, in the space of completely boundedmultipliers of A(G). If G is a weakly amenable, discrete group such that C*(G) is residually finite-dimensional, we show that Acb(G) is operator amenable. In particular, Acb() is operator amenable even though , the free group in two generators, is not an amenable group. Moreover, we show that if G is a discrete group such that Acb(G) is operator amenable, a closed ideal of A(G) is weakly completely complemented in A(G) if and only if it has an approximate identity bounded in the cb-multiplier norm.

Unit Elements in the Double Dual of a Subalgebra of the Fourier Algebra A(G)

2009 ◽  
Vol 61 (2) ◽  
pp. 382-394
Author(s):  
Tianxuan Miao
Keyword(s):  
Compact Group ◽  
Banach Algebra ◽  
Locally Compact Group ◽  
Approximate Identity ◽  
Fourier Algebra ◽  
Locally Compact ◽  
Arens Product ◽  
The Right ◽  
Topological Centers ◽  
The Relationship

Abstract. Let 𝒜 be a Banach algebra with a bounded right approximate identity and let ℬ be a closed ideal of 𝒜. We study the relationship between the right identities of the double duals ℬ** and 𝒜** under the Arens product. We show that every right identity of ℬ** can be extended to a right identity of 𝒜** in some sense. As a consequence, we answer a question of Lau and Ülger, showing that for the Fourier algebra A(G) of a locally compact group G, an element ϕ ∈ A(G)** is in A(G) if and only if A(G)ϕ ⊆ A(G) and Eϕ = ϕ for all right identities E of A(G)**. We also prove some results about the topological centers of ℬ** and 𝒜**.

scholarly journals ON HOMOLOGICAL PROPERTIES FOR SOME MODULES OF UNIFORMLY CONTINUOUS FUNCTIONS OVER CONVOLUTION ALGEBRAS

2011 ◽  
Vol 84 (2) ◽  
pp. 177-185
Author(s):  
RASOUL NASR-ISFAHANI ◽  
SIMA SOLTANI RENANI
Keyword(s):  
Compact Group ◽  
Group Algebra ◽  
Dual Space ◽  
Continuous Functions ◽  
Locally Compact Group ◽  
Measure Algebra ◽  
Locally Compact ◽  
Convolution Algebras ◽  
Uniformly Continuous

AbstractFor a locally compact group G, let LUC(G) denote the space of all left uniformly continuous functions on G. Here, we investigate projectivity, injectivity and flatness of LUC(G) and its dual space LUC(G)* as Banach left modules over the group algebra as well as the measure algebra of G.

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