Dual Banach algebras: representations and injectivity

We study representations of Banach algebras on reflexive Banach spaces. Algebras which admit such representations which are bounded below seem to be a good generalisation of Arens regular Banach algebras; this class includes dual Banach algebras as defined by Runde, but also all group algebras, and all discrete (weakly cancellative) semigroup algebras. Such algebras also behave in a similar way to C$^*$- and W$^*$-algebras; we show that interpolation space techniques can be used in the place of GNS type arguments. We define a notion of injectivity for dual Banach algebras, and show that this is equivalent to Connes-amenability. We conclude by looking at the problem of defining a well-behaved tensor product for dual Banach algebras.Comment: 40 pages; Update corrects some mathematics, and merges two sections to make for easier readin

Connes-amenability of bidual and weighted semigroup algebras

We investigate the notion of Connes-amenability for dual Banach algebras, as introduced by Runde, for bidual algebras and weighted semigroup algebras. We provide some simplifications to the notion of a $\sigma WC$-virtual diagonal, as introduced by Runde, especially in the case of the bidual of an Arens regular Banach algebra. We apply these results to discrete, weighted, weakly cancellative semigroup algebras, showing that these behave in the same way as C$^*$-algebras with regards Connes-amenability of the bidual algebra. We also show that for each one of these cancellative semigroup algebras $l^1(S,\omega)$, we have that $l^1(S,\omega)$ is Connes-amenable (with respect to the canonical predual $c_0(S)$) if and only if $l^1(S,\omega)$ is amenable, which is in turn equivalent to $S$ being an amenable group. This latter point was first shown by Gr{\"o}nb\ae k, but we provide a unified proof. Finally, we consider the homological notion of injectivity, and show that here, weighted semigroup algebras do not behave like C$^*$-algebras.Comment: 25 page

A bicommutant theorem for dual Banach algebras

A dual Banach algebra is a Banach algebra which is a dual space, with the multiplication being separately weak$^*$-continuous. We show that given a unital dual Banach algebra \mc A, we can find a reflexive Banach space $E$, and an isometric, weak$^*$-weak$^*$-continuous homomorphism \pi:\mc A\to\mc B(E) such that \pi(\mc A) equals its own bicommutant.Comment: 6 page

Multipliers of locally compact quantum groups via Hilbert C∗^*-modules

A result of Gilbert shows that every completely bounded multiplier $f$ of the Fourier algebra $A(G)$ arises from a pair of bounded continuous maps $\alpha,\beta:G \rightarrow K$, where $K$ is a Hilbert space, and $f(s^{-1}t) = (\beta(t)|\alpha(s))$ for all $s,t\in G$. We recast this in terms of adjointable operators acting between certain Hilbert C$^*$-modules, and show that an analogous construction works for completely bounded left multipliers of a locally compact quantum group. We find various ways to deal with right multipliers: one of these involves looking at the opposite quantum group, and this leads to a proof that the (unbounded) antipode acts on the space of completely bounded multipliers, in a way which interacts naturally with our representation result. The dual of the universal quantum group (in the sense of Kustermans) can be identified with a subalgebra of the completely bounded multipliers, and we show how this fits into our framework. Finally, this motivates a certain way to deal with two-sided multipliers.Comment: 24 pages; many typos corrected; some rewritin

Multipliers, Self-Induced and Dual Banach Algebras

In the first part of the paper, we present a short survey of the theory of multipliers, or double centralisers, of Banach algebras and completely contractive Banach algebras. Our approach is very algebraic: this is a deliberate attempt to separate essentially algebraic arguments from topological arguments. We concentrate upon the problem of how to extend module actions, and homomorphisms, from algebras to multiplier algebras. We then consider the special cases when we have a bounded approximate identity, and when our algebra is self-induced. In the second part of the paper, we mainly concentrate upon dual Banach algebras. We provide a simple criterion for when a multiplier algebra is a dual Banach algebra. This is applied to show that the multiplier algebra of the convolution algebra of a locally compact quantum group is always a dual Banach algebra. We also study this problem within the framework of abstract Pontryagin duality, and show that we construct the same weak$^*$-topology. We explore the notion of a Hopf convolution algebra, and show that in many cases, the use of the extended Haagerup tensor product can be replaced by a multiplier algebra.Comment: Numerous typos corrected, and a proof rectified (v3 was an incorrect upload of a previous version