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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.

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We investigate the notion of Connes-amenability, introduced by Runde in [14], for bidual algebras and weighted semigroup algebras. We provide some simplifications to the notion of a σWC-virtual diagonal, as introduced in [10], 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, ω), we have that l 1 (S, ω) is Connes-amenable (with respect to the canonical predual c0(S)) if and only if l 1 (S, ω) is amenable, which is in turn equivalent to S being an amenable group. This latter point was first shown by Grönbæk in [5], 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.

In this paper, we prove that every completely contractive dual Banach algebra is completely isometric to a w ∗-closed subalgebra the operator space of completely bounded linear operators on some reflexive operator space. 1

Abstract. Let X be a left introverted subspace of dual of a Banach algebra. We study Zt(X ∗), the topological center of Banach algebra X ∗. We fined the topological center of (XA) ∗ , when A has a bounded right approximate identity and A ⊆ X ∗. So we introduce a new notation of amenability for a dual Banach algebra A. A dual Banach algebra A is weakly Connesamenable if the first weak ∗ −continuous cohomology group of A with coefficients in A is zero; i.e., H1 w∗(A, A) = {o}. We study the weak Connes-amenability of some dual Banach algebras. 1.

Abstract. In this paper we study the Connes amenability of the second dual of Arens regular Banach algebras. Of course we provide a partial answer to the question posed by Volker Runde. Also we fined the necessary and sufficient conditions for the second dual of an Arens regular module extension Banach algebra to be Connes amenable when the module is reflexive.

A theorem of Davis, Figiel, Johnson and Pe̷lczyński tells us that weakly-compact operators between Banach spaces factor through reflexive Banach spaces. The machinery underlying this result is that of the real interpolation method, which has been adapted to the category of operator spaces by Xu, showing the this factorisation result also holds for completely bounded weakly-compact maps. In this note, we show that Xu’s ideas can be adapted to give an intrinsic characterisation of when a completely contractive Banach algebra arises as a closed subalgebra of the algebra of completely bounded operators on a reflexive operator space. This result was shown by Young for Banach algebras, and our characterisation is a direct analogue of Young’s, involving weakly almost periodic functionals.