On amenable and compact groups

In order to summarize the contents of this paper it is necessary to use some of the definitions in [5]. These definitions are repeated below at the beginning of Section 1. Although this paper adds to the material in [5] we have attempted to reduce the number of references to [o] to a minimum. An approximate diagonal in a Banach algebra 31 is a bounded net {ma} in 31 a for all a in 31 in the norm topologies where 31? 31 is the usual projective tensor product [2] and -k is the continuous linear map 31?31->31 defined by ir(a?b) =ab. The net {m 31) ** with aM = Ma and ?r** (M) a = a for all a ? 31. In the algebraic cohomology of algebras the existence of an element m of A A with ma = am and ir(m) the identity of A is equivalent to the statement that H1(A, X) =0 for all 31 modules X almost never holds?in fact, in the commutative case for all A bimodules X. Among Banach algebras the statement jEP^S^X) =0 (see [3; Theorem 1] and [5; Proposition 8.1]) we can always find a Banach 31 module ? with S^1(3I,36)=t^O unless 31 zz?n. The existence of an approxi? mate diagonal is not unusual and is equivalent to the statement !M* (31,36*) = 0 for all Banach 31 modules X; thatis 31 is an emenable Banach algebra. The element M plays the same role in the theory of amenable tlgebras that the invariant mean does for amenable groups. The definitions used from [5] and the theorem connecting amenability with approximate diagonals are given in ? 1. In ? 2 we show that if 31 has a virtual diagonal with Ma? 31? St for all a in 3T then 913 (31, 3E) ? .0 for all Banach 31 modules 3?. Algebras satisfying this condition are grdup algebras of compact topological groups and the algebra c0 of convergent sequences of complex numbers. Although we do not prove it the result also applies to any annihilator B* algebra with only finite dimensional minimal ideals. In ? 3 we show that if 31 is a semi simple commutative amenable Banach algebra then there is a Banach 31 module X with 3/2(3I,3E) ^0, unless 31 is finite

It is an open problem whether an infinite-dimensional amenable Banach algebra exists whose underlying Banach space is reflexive. We give sufficient conditions for a reflexive, amenable Banach algebra to be finite-dimensional (and thus a finite direct sum of full matrix algebras). If $$ {\frak A} $$ is a reflexive, amenable Banach algebra such that for each maximal left ideal L of $$ {\frak A} $$ (i) the quotient $$ {\frak A}/L $$ has the approximation property and (ii) the canonical map from $$ {\frak A} \check{\otimes} L^\perp $ to $({\frak A} / L) \check{\otimes} L^\perp $$ is open, then $$ {\frak A} $$ is finite-dimensional. As an application, we show that, if $${\frak A}$$ is an amenable Banach algebra whose underlying Banach space is an Lp-space with $$ p\in (1,\infty) $$ such that for each maximal left ideal L the quotient $$ {\frak A}/L $$ has the approximation property, then $$ {\frak A} $$ is finite-dimensional.

We study growth of 1-cocycles of locally compact groups, with values in unitary representations. Discussing the existence of 1-cocycles with linear growth, we obtain the following alternative for a class of amenable groups G containing polycyclic groups and connected amenable Lie groups: either G has no quasi-isometric embedding into Hilbert space, or G admits a proper cocompact action on some Euclidean space. On the other hand, noting that almost coboundaries (i.e. 1-cocycles approximable by bounded 1-cocycles) have sublinear growth, we discuss the converse, which turns out to hold for amenable groups with "controlled" Folner sequences; for general amenable groups we prove the weaker result that 1-cocycles with sufficiently small growth are almost coboundaries. Besides, we show that there exist, on a-T-menable groups, proper cocycles with arbitrary small growth.

This article concerns the relationship between the spectral and the norm properties of certain subalgebras of the convolution measure algebra M(G) (Borel regular finite measures) on a locally compact amenable group G.

Let A be a Banach algebra and I be a closed ideal of A. We say that A is amenable relative to I, if A/I is an amenable Banach algebra. We study the relative amenability of Banach algebras and investigate the relative amenability of triangular Banach algebras and Banach algebras associated to locally compact groups. We generalize some of the previous known results by applying the concept of relative amenability of Banach algebras, especially, we present a generalization of Johnson?s theorem in the concept of relative amenability.

In this paper, we introduce new notions of $I$-biflatness and $I$-biprojectivity, for a Banach algebra $A$, where $I$ is a closed ideal of $A$. We show that $M(G)$ is $L^{1}(G)$-biprojective ($I$-biflat) if and only if $G$ is a compact group (an amenable group), respectively. Also, we show that, for a non-zero ideal $I$, if the Fourier algebra $A(G)$ is $I$-biprojective, then $G$ is a discrete group. Some examples are given to show the differences between these new notions and the classical ones.

On the non-commutative Neveu decomposition and ergodic theorems for amenable group action

Let G be a compact group whose local weight b(G) has uncountable cofinality. Let H be an amenable locally compact group and A(G × H) be the Fourier algebra of G × H. We prove that the group von Neumann algebra VN(G × H) = A(G × H)* has the weak uniform A(G × H)** factorization property of level b(G). As a corollary we show that A(G × H) is strongly Arens irregular, and the topological centre of UC 2(G × H)* is equal to the Fourier–Stieltjes algebra B(G × H).

A. M. Sinclair has proved that if is a semisimple Banach algebra then every continuous Jordan derivation from into is a derivation ([12, theorem 3·3]; ‘Jordan derivation’ is denned in Section 6 below). If is a Banach -bimodule one can consider Jordan derivations from into and ask whether Sinclair's theorem is still true. More recent work in this area appears in [1]. Simple examples show that it cannot hold for all modules and all semisimple algebras. However, for more restricted classes of algebras, including C*-algebras one does get a positive result and we develop two approaches. The first depends on symmetric amenability, a development of the theory of amenable Banach algebras which we present here for the first time in Sections 2, 3 and 4. A Banach algebra is symmetrically amenable if it has an approximate diagonal consisting of symmetric tensors. Most, but not all, amenable Banach algebras are symmetrically amenable and one can prove results for symmetric amenability similar to those in [8] for amenability. However, unlike amenability, symmetric amenability does not seem to have a concise homological characterisation. One of our results [Theorem 6·2] is that if is symmetrically amenable then every continuous Jordan derivation into an -bimodule is a derivation. Special techniques enable this result to be extended to other algebras, for example all C*-algebras. This approach to Jordan derivations appears in Section 6.

Embedding Nilpotent Finite Dimensional Banach Algebras into Amenable Banach Algebras

Let $G$ be a locally compact amenable group. The goal of this paper is to investigate the problem of surjective zero products preserving maps from the Fourier algebra of $G$ into a completely contractive Banach algebra. We show that if $B$ is a completely contractive Banach algebra which is faithful and factors weakly, then every surjective completely bounded linear map from $A(G)$ into $B$ which preserves zero products is a weighted homomorphism. Moreover an equivalent condition is given for such a map to be a homomorphism. In particular, this result implies that if $B$ is a commutative C$^*$-algebra or a matrix space and $T:A(G)\rightarrow B$ is a continuous surjective linear map which preserves zero products, then $T$ is a weighted homomorphism and there is an equivalent condition for $T$ to be a homomorphism.

We prove that for actions of a discrete countable residuallyfinite amenable group on a compact metric space with specification property, periodic measures are dense in theset of invariant measures. We also prove that certain expansiveactions of a countable discrete group by automorphisms of compact abelian groups have specification property.

Building on our previous work, we study the non-relative homology of quantum group convolution algebras. Our main result establishes the equivalence of amenability of a locally compact quantum group and 1-injectivity of as an operator -module. In particular, a locally compact group G is amenable if and only if its group von Neumann algebra VN(G) is 1-injective as an operator module over the Fourier algebra A(G). As an application, we provide a decomposability result for completely bounded -module maps on , and give a simpliûed proof that amenable discrete quantum groups have co-amenable compact duals, which avoids the use of modular theory and the Powers-Størmer inequality, suggesting that our homological techniques may yield a new approach to the open problem of duality between amenability and co-amenability.

Let G be a noncompact nondiscrete σ \sigma -compact locally compact metric group. A Baire category argument gives measurable sets { A γ : γ ∈ Γ } \{ {A_\gamma }:\gamma \in \Gamma \} of finite measure with card ( Γ ) = c (\Gamma ) = c which are independent on the open sets. One approximates { A γ : γ ∈ Γ } \{ {A_\gamma }:\gamma \in \Gamma \} by arrays of continuous bounded functions with compact support and then scatters these arrays to construct functions { f γ : γ ∈ Γ } \{ {f_\gamma }:\gamma \in \Gamma \} in CB ( G ) {\text {CB}}(G) with a certain independence property. If G is also amenable as a discrete group, the existence of these independent functions shows that on CB ( G ) {\text {CB}}(G) there are 2 c {2^c} mutually singular elements of LIM each of which is singular to TLIM.

Orthogonal ℓ1-sets and extreme non-Arens regularity of preduals of von Neumann algebras