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

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.

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Uniformly Continuous Functionals and M-Weakly Amenable Groups

Canadian Journal of Mathematics ◽

10.4153/cjm-2013-019-x ◽

2013 ◽

Vol 65 (5) ◽

pp. 1005-1019 ◽

Cited By ~ 2

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.

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Weak Containment by Restrictions of Induced Representations

International Mathematics Research Notices ◽

10.1093/imrn/rny063 ◽

2018 ◽

Vol 2020 (7) ◽

pp. 2034-2053

Author(s):

Matthew Wiersma

Keyword(s):

Compact Group ◽

Closed Subgroup ◽

Discrete Group ◽

Amenable Group ◽

Locally Compact Group ◽

Approximate Identity ◽

Locally Compact ◽

Induced Representations ◽

C Algebras ◽

Weak Containment

Abstract A QSIN group is a locally compact group G whose group algebra $\mathrm L^{1}(G)$ admits a quasi-central bounded approximate identity. Examples of QSIN groups include every amenable group and every discrete group. It is shown that if G is a QSIN group, H is a closed subgroup of G, and $\pi \!: H\to \mathcal B(\mathcal{H})$ is a unitary representation of H, then $\pi$ is weakly contained in $\Big (\mathrm{Ind}_{H}^{G}\pi \Big )|_{H}$. This provides a powerful tool in studying the C*-algebras of QSIN groups. In particular, it is shown that if G is a QSIN group which contains a copy of $\mathbb{F}_{2}$ as a closed subgroup, then $\mathrm C^{\ast }(G)$ is not locally reflexive and $\mathrm C^{\ast }_{r}(G)$ does not admit the local lifting property. Further applications are drawn to the “(weak) extendability” of Fourier spaces $\mathrm A_{\pi }$ and Fourier–Stieltjes spaces $\mathrm B_{\pi }$.

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

Canadian Journal of Mathematics ◽

10.4153/cjm-2009-020-0 ◽

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**.

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Norm One Idempotent cb-Multipliers with Applications to the Fourier Algebra in the cb-Multiplier Norm

Canadian Mathematical Bulletin ◽

10.4153/cmb-2011-098-0 ◽

2011 ◽

Vol 54 (4) ◽

pp. 654-662 ◽

Cited By ~ 3

Author(s):

Brian E. Forrest ◽

Volker Runde

Keyword(s):

Compact Group ◽

Open Subgroup ◽

Locally Compact Group ◽

Approximate Identity ◽

Indicator Function ◽

Fourier Algebra ◽

Locally Compact

AbstractFor a locally compact group G, let A(G) be its Fourier algebra, let McbA(G) denote the completely bounded multipliers of A(G), and let AMcb (G) stand for the closure of A(G) in McbA(G). We characterize the norm one idempotents in McbA(G): the indicator function of a set E ⊂ G is a norm one idempotent in McbA(G) if and only if E is a coset of an open subgroup of G. As applications, we describe the closed ideals of AMcb (G) with an approximate identity bounded by 1, and we characterize those G for which AMcb (G) is 1-amenable in the sense of B. E. Johnson. (We can even slightly relax the norm bounds.)

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Completely bounded bimodule maps and spectral synthesis

International Journal of Mathematics ◽

10.1142/s0129167x17500677 ◽

2017 ◽

Vol 28 (10) ◽

pp. 1750067 ◽

Cited By ~ 2

Author(s):

M. Alaghmandan ◽

I. G. Todorov ◽

L. Turowska

Keyword(s):

Tensor Product ◽

Compact Group ◽

Closed Subset ◽

Spectral Synthesis ◽

Locally Compact Group ◽

Product Formula ◽

Fourier Algebra ◽

Locally Compact ◽

Multiplication Algebra

We initiate the study of the completely bounded multipliers of the Haagerup tensor product [Formula: see text] of two copies of the Fourier algebra [Formula: see text] of a locally compact group [Formula: see text]. If [Formula: see text] is a closed subset of [Formula: see text] we let [Formula: see text] and show that if [Formula: see text] is a set of spectral synthesis for [Formula: see text] then [Formula: see text] is a set of local spectral synthesis for [Formula: see text]. Conversely, we prove that if [Formula: see text] is a set of spectral synthesis for [Formula: see text] and [Formula: see text] is a Moore group then [Formula: see text] is a set of spectral synthesis for [Formula: see text]. Using the natural identification of the space of all completely bounded weak* continuous [Formula: see text]-bimodule maps with the dual of [Formula: see text], we show that, in the case [Formula: see text] is weakly amenable, such a map leaves the multiplication algebra of [Formula: see text] invariant if and only if its support is contained in the antidiagonal of [Formula: see text].

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

Canadian Journal of Mathematics ◽

10.4153/cjm-1994-046-x ◽

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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Some properties of locally compact groups

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700004389 ◽

1967 ◽

Vol 7 (4) ◽

pp. 433-454 ◽

Cited By ~ 10

Author(s):

Neil W. Rickert

Keyword(s):

Compact Group ◽

Lie Groups ◽

Locally Compact Group ◽

Factor Group ◽

Locally Compact Groups ◽

Compact Groups ◽

Locally Compact ◽

Finite Dimensional ◽

Arcwise Connected ◽

New Formulation

In this paper a number of questions about locally compact groups are studied. The structure of finite dimensional connected locally compact groups is investigated, and a fairly simple representation of such groups is obtained. Using this it is proved that finite dimensional arcwise connected locally compact groups are Lie groups, and that in general arcwise connected locally compact groups are locally connected. Semi-simple locally compact groups are then investigated, and it is shown that under suitable restrictions these satisfy many of the properties of semi-simple Lie groups. For example, a factor group of a semi-simple locally compact group is semi-simple. A result of Zassenhaus, Auslander and Wang is reformulated, and in this new formulation it is shown to be true under more general conditions. This fact is used in the study of (C)-groups in the sense of K. Iwasawa.

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On One-parameter Subgroups in Finite Dimensional Locally Compact Group with No Small Subgroups

Nagoya Mathematical Journal ◽

10.1017/s0027763000023047 ◽

1952 ◽

Vol 4 ◽

pp. 89-96

Author(s):

Masatake Kuranishi

Keyword(s):

Topological Group ◽

Compact Group ◽

Lie Group ◽

Locally Compact Group ◽

Locally Compact ◽

Compact Topological Group ◽

Finite Dimensional ◽

Locally Compact Topological Group

Let G be a locally compact topological group and let U be a neighborhood of the identity in G. A curve g(λ) (|λ| ≦ 1) in G, which satisfies the conditions, g(s)g(t) = g(s + t) (|s|, |f|, |s + t| ≦ l),is called a one-parameter subgroup of G. If there exists a neighborhood U1 of the identity in G such that for every element x of U1 there exists a unique one-parameter subgroup g(λ) which is contained in U and g(1) =x, we shall call, for the sake of simplicity, that U has the property (S). It is well known that the neighborhoods of the identity in a Lie group have the property (S). More generally it is proved that if G is finite dimensional, locally connected, and is without small subgroups, G has the same property. In this note, these theorems will be generalized to the case when G is unite dimensional and without small subgroups.

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