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

Management into design education: a case study

The design project set in a studio learning environment remains central to much of the under- graduate curriculum activity for the aspiring architect. Yet much recent discussion has identified the need to look beyond this design curriculum horizon and to extend into studies in management. The Burton Report, among others, has encouraged diversity in architectural education. A degree course in Architectural Design and Management has been developed at Northumbria University as a direct response to this encouragement. However, challenges continue as students tend to see supporting studies such as management as peripheral, or even irrelevant and professional accreditation authorities seek evidence of performance via an academic portfolio only. Subsequently, management and professional studies in the programme have been developed as the process within which design happens and which allows a direct link into the studio programme. So, as for the professional practitioner, student design activity happens in teams, and has deadlines, studio design programmes have group projects and deadlines and these are structured with learning outcomes such as teamwork, timekeeping, and reflective learning

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

Isometries between quantum convolution algebras

Given locally compact quantum groups \G_1 and \G_2, we show that if the convolution algebras L^1(\G_1) and L^1(\G_2) are isometrically isomorphic as algebras, then \G_1 is isomorphic either to \G_2 or the commutant \G_2'. Furthermore, given an isometric algebra isomorphism \theta:L^1(\G_2) \rightarrow L^1(\G_1), the adjoint is a *-isomorphism between L^\infty(\G_1) and either L^\infty(\G_2) or its commutant, composed with a twist given by a member of the intrinsic group of L^\infty(\G_2). This extends known results for Kac algebras (although our proofs are somewhat different) which in turn generalised classical results of Wendel and Walter. We show that the same result holds for isometric algebra homomorphisms between quantum measure algebras (either reduced or universal). We make some remarks about the intrinsic groups of the enveloping von Neumann algebras of C$^*$-algebraic quantum groups.Comment: 23 pages, typos corrected, references adde