Dual Banach Algebra Research Articles

ESSENTIAL AMENABILITY OF DUAL BANACH ALGEBRAS

We show that an essentially amenable Banach algebra need not have an approximate identity. This answers a question posed by Ghahramani and Loy [‘Generalized notions of amenability’, J. Funct. Anal. 208 (2004), 229–260]. Essentially Connes-amenable dual Banach algebras are introduced and studied.

Fourier Spaces and Completely Isometric Representations of Arens Product Algebras

AbstractMotivated by the definition of a semigroup compactication of a locally compact group and a large collection of examples, we introduce the notion of an (operator) homogeneous left dual Banach algebra (HLDBA) over a (completely contractive) Banach algebra $A$ . We prove a Gelfand-type representation theorem showing that every HLDBA over A has a concrete realization as an (operator) homogeneous left Arens product algebra: the dual of a subspace of $A^{\ast }$ with a compatible (matrix) norm and a type of left Arens product $\Box$ . Examples include all left Arens product algebras over $A$ , but also, when $A$ is the group algebra of a locally compact group, the dual of its Fourier algebra. Beginning with any (completely) contractive (operator) $A$ -module action $Q$ on a space $X$ , we introduce the (operator) Fourier space $({\mathcal{F}}_{Q}(A^{\ast }),\Vert \cdot \Vert _{Q})$ and prove that $({\mathcal{F}}_{Q}(A^{\ast })^{\ast },\Box )$ is the unique (operator) HLDBA over $A$ for which there is a weak $^{\ast }$ -continuous completely isometric representation as completely bounded operators on $X^{\ast }$ extending the dual module representation. Applying our theory to several examples of (completely contractive) Banach algebras $A$ and module operations, we provide new characterizations of familiar HLDBAs over A and we recover, and often extend, some (completely) isometric representation theorems concerning these HLDBAs.

Operator Connes-amenability of completely bounded multiplier Banach algebras

For a completely contractive Banach algebra $B$, we find conditions under which the completely bounded multiplier algebra $\mathcal{M}_{cb}(B)$ is a dual Banach algebra and the operator amenability of $B$ is equivalent to the operator Connes-amenability of $\mathcal{M}_{cb}(B)$. We also show that, in this case, these are equivalent to the existence of a normal virtual operator diagonal.

A generalization on character Connes-amenability

In the current paper, we introduce the concepts of left ?-approximate Connes-amenability and left character approximate Connes-amenability of a dual Banach algebra A that ? is a ?*-continuous homomorphism fromAto C. We also characterize left ?-approximate Connes-amenability ofAin terms of certain derivations and study some hereditary properties for such Banach algebras. Some examples show that these new notions are different from approximate Connes-amenability and left character Connesamenability for dual Banach algebras.

Twisted Orlicz algebras, II

AbstractLet G be a locally compact group, let be a 2‐cocycle, and let () be a complementary pair of strictly increasing continuous Young functions. We continue our investigation in [14] of the algebraic properties of the Orlicz space with respect to the twisted convolution ⊛ coming from Ω. We show that the twisted Orlicz algebra posses a bounded approximate identity if and only if it is unital if and only if G is discrete. On the other hand, under suitable condition on Ω, becomes an Arens regular, dual Banach algebra. We also look into certain cohomological properties of , namely amenability and Connes‐amenability, and show that they rarely happen. We apply our methods to compactly generated group of polynomial growth and demonstrate that our results could be applied to variety of cases.

Pointwise amenability for dual Banach algebras

We shall develop two notions of pointwise amenability, namely pointwise Connes amenability and pointwise $w^*$-approximate Connes amenability, for dual Banach algebras which take the $w^*$-topology into account. We shall study these concepts for the Banach sequence algebras $\ell^1(\omega)$ and the weighted semigroup algebras $ \ell^{1}(\mathbb{N}_{\wedge},\omega)$. For a weight $\omega$ on a discrete semigroup $S$, we shall investigate pointwise amenability/Connes amenability of $\ell^1(S,\omega)$ in terms of diagonals.

Dual Banach algebras associated to the coset space of a locally compact group

For H a closed group of a locally compact group G, B. Forrest ([4]) has defined the Fourier and Fourier Stieltjes algebras associated to the coset space G/H, A(G/H) and B(G/H) respectively. He proved that when H is compact, it is possible to extend many classical results to this new setting. We continue this investigation with the study of the dual space of A(G/H), showing that it can be identified with a particular type of w⁎-closed ideal of VN(G) determined by H. We obtain a characterization of all w⁎-closed left ideals of VN(G) that are of this particular form. We introduce and study the analogs of some of the classical subspaces of VN(G), UBC(Gˆ), W(Gˆ) and AP(Gˆ) in the new setting. We obtain results similar to those of D.E. Ramirez ([2]), E. Granirer ([6]) and A.T. Lau ([9]) obtained for these spaces in the group setting.

English

We study the notion of character Connes amenability of dual Banach algebras and show that if A is an Arens regular Banach algebra, then A** is character Connes amenable if and only if A is character amenable, which will resolve positively Runde’s problem for this concept of amenability. We then characterize character Connes amenability of various dual Banach algebras related to locally compact groups. We also investigate character Connes amenability of Lau product and module extension of Banach algebras. These help us to give examples of dual Banach algebras which are not Connes amenable.

Ideal Connes-Amenability of Dual Banach Algebras

Following Runde, we define the concept of ideal Connes-amenability for dual Banach algebras. For an Arens regular dual Banach algebra $${\mathcal {A}}$$ , we prove that the ideal Connes-amenability of $$\mathcal {A^{**}}$$ , the second dual of $${\mathcal {A}}$$ necessities ideal Connes-amenability of $${{\mathcal {A}}}$$ . As a typical example, we show that von Neumann algebras are always ideally Connes-amenable. For a locally compact group G, the Fourier–Stieltjes algebra of G is ideally Connes-amenable, but not ideally amenable.

Character Connes-amenability of dual Banach algebras

The concept of left character Connes-amenability for a dual Banach algebra $${\mathcal {A}}$$ is introduced. We obtain a cohomological characterization of left character Connes-amenability as well as the relation between left $$\varphi $$ -Connes-amenability and existence of left $$\varphi $$ -normal virtual diagonals for a $$\omega ^{*}$$ -continuous character $$\varphi $$ . We prove that left character amenability of $${\mathcal {A}}$$ is equivalent to left character Connes-amenability of $${\mathcal {A}}^{**}$$ when $${\mathcal {A}}$$ is Arens regular. Moreover for a locally compact group G, we show that M(G) is left character Connes-amenable. In addition by means of some examples we show that for the new notion, the corresponding class of dual Banach algebras is larger than Connes-amenable dual Banach algebras.

Derivations on generalized semidirect products of Banach algebras

Let A and B be Banach algebras, let θ:A→B be a continuous Banach algebra homomorphism, and let I be a closed ideal in B. Then the l1-direct sum of A and I with a special product becomes a Banach algebra, denoted by A⋈θI, which we call the generalized semidirect product of A and I. In this article, among other things, we first characterize derivations on A⋈θI and then we compute the first cohomology group of A⋈θI when the first cohomology groups of A with coefficients in A and I are trivial. As an application we characterize the first cohomology group of second duals of dual Banach algebras. Then we provide a nice characterization of the first cohomology group of A⋈idA. Furthermore, we calculate the first cohomology group of some concrete Banach algebras related to locally compact groups.

Extension of Derivations, and Connes- Amenability of the Enveloping Dual Banach Algebra

If $D:A \to X$ is a derivation from a Banach algebra to a contractive, Banach $A$-bimodule, then one can equip $X^{**}$ with an $A^{**}$-bimodule structure, such that the second transpose $D^{**}: A^{**} \to X^{**}$ is again a derivation. We prove an analogous extension result, where $A^{**}$ is replaced by $\mathsf{F}(A)$, the enveloping dual Banach algebra of $A$, and $X^{**}$ by an appropriate kind of universal, enveloping, normal dual bimodule of $X$. Using this, we obtain some new characterizations of Connes-amenability of $\mathsf{F}(A)$. In particular we show that $\mathsf{F}(A)$ is Connes-amenable if and only if $A$ admits a so-called $\operatorname{WAP}$-virtual diagonal. We show that when $A=L^1(G)$, existence of a $\operatorname{WAP}$-virtual diagonal is equivalent to the existence of a virtual diagonal in the usual sense. Our approach does not involve invariant means for $G$.

Module Connes amenability of hypergroup measure algebras

Abstract We define the concept of module Connes amenability for dual Banach algebras which are also Banach modules with a compatible action. We distinguish a closed subhypergroup K0 of a locally compact measured hypergroup K, and show that, under different actions, amenability of K, M.K0/-module Connes amenability of M.K/, and existence of a normal M.K0/-module virtual diagonal are related.

Erratum to “A Connes-amenable, dual Banach algebra need not have a normal, virtual diagonal”

In Trans. Amer. Math. Soc., vol. 358 (2006), pp. 391–402, we claimed that, for an amenable, non-compact [ S I N ] [\mathrm {SIN}] -group G G , the dual Banach algebra W A P ( G ) ∗ \mathcal {WAP}(G)^\ast is Connes-amenable, but lacks a normal virtual diagonal. The proof presented contains a gap. In this erratum, we indicate how the faulty proof can be repaired.

Connes amenability-like properties

We introduce and study the notions of $w^*$-approximate Connes amenability and pseudo-Connes amenability for dual Banach algebras. We prove that the dual Banach sequence algebra $\ell ^1$ is not $w^*$-approximately Connes amenable. We show that in general

Approximate injectivity of dual Banach algebras

We introduce and study the notion of w∗-approximate Connes-amenability and w∗-approximate injectivity for dual Banach algebras. It is shown that w∗-approximate Connes-amenability and w∗-approximate injectivity are the same properties. 2010 Mathematics subject classification: Primary 99X99; Secondary 99Y99..

Shift invariant preduals of ℓ 1(ℤ)

The Banach space $\ell_1(\Z)$ admits many non-isomorphic preduals, for example, $C(K)$ for any compact countable space $K$, along with many more exotic Banach spaces. In this paper, we impose an extra condition: the predual must make the bilateral shift on $\ell_1(\Z)$ weak$^*$-continuous. This is equivalent to making the natural convolution multiplication on $\ell_1(\Z)$ separately weak*-continuous and so turning $\ell_1(\Z)$ into a dual Banach algebra. We call such preduals \emph{shift-invariant}. It is known that the only shift-invariant predual arising from the standard duality between $C_0(K)$ (for countable locally compact $K$) and $\ell_1(\Z)$ is $c_0(\Z)$. We provide an explicit construction of an uncountable family of distinct preduals which do make the bilateral shift weak$^*$-continuous. Using Szlenk index arguments, we show that merely as Banach spaces, these are all isomorphic to $c_0$. We then build some theory to study such preduals, showing that they arise from certain semigroup compactifications of $\Z$. This allows us to produce a large number of other examples, including non-isometric preduals, and preduals which are not Banach space isomorphic to $c_0$.

Weak$^*$-continuous derivations in dual Banach algebras

Let $\mathcal{A}$ be a dual Banach algebra. We investigate the first weak$^*$-continuous cohomology group of $\mathcal{A}$ with coefficients in $\mathcal{A}$. Hence, we obtain conditions on ${\mathcal{A}}$ for which \[ H^1_{w^*}(\mathcal{A}, \mathcal{A})=\lbrace 0\rbrace \,. \]

Projections on weak*-closed subspace of dual Banach algebras

Our first purpose in this paper is to provide necessary conditions for a weak*-closed translation invariant subspace in the semigroup algebra of a locally compact topological foundation semigroup to be completely complemented. We give conditions when a weak*-closed left translation invariant subspace in Ma(S)* of a compact cancellative foundation semigroup S is the range of a weak*-weak* continuous projection on Ma(S)* commuting with translations. Let G be a locally compact group and A be a Banach G-module. Our second purpose in this paper is to study some projections on A* and B(A*) which commutes with translations and convolution.

Completely bounded multipliers over locally compact quantum groups

In this paper, we consider several interesting multiplier algebras associated with a locally compact quantum groupG. Firstly, we study the completely bounded right multiplier algebra M r cb (L1(G)). We show that M r cb(L1(G)) is a dual Banach algebra with a natural operator predual Q r (L1(G)), and the completely isometric representation of M r cb(L1(G)) on B(L2(G)), studied recently by Junge, Neufang and Ruan, is actually weak*-weak* continuous. Secondly, we study the left uniformly continuous space LUC(G) and its Banach algebra dual LUC(G) ⁄ . We prove that LUC(G) is a unital C*-subalgebra of L1(G) if the quantum groupG is semi-regular. We show the connection between LUC(G)⁄ and the quan- tum measure algebra M(G), as well as their representations on L1(G) and B(L2(G)). Finally, we study the right uniformly complete qotient space UCQ r (L1(G)) and its Banach algebra dual UCQ r (L1(G))⁄. For co-amenable quanum groupsG, we obtain the weak*-homeomorphic completely isometric algebra iso- morphism M r cb (L1(G)) » M(G) and the completely isometric isomorphism UCQ r (L1(G)) » LUC(G).