Results 1  10
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19
Dual Banach algebras: representations and injectivity
, 2008
"... 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 ..."
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Cited by 10 (7 self)
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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 Connesamenability. We conclude by looking at the problem of defining a wellbehaved tensor product for dual Banach algebras.
Dual Banach algebras: Connesamenability, normal, virtual diagonals, and injectivity of the predual bimodule
, 2003
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Connesamenability of bidual and weighted semigroup algebras
"... We investigate the notion of Connesamenability, introduced by Runde in [14], for bidual algebras and weighted semigroup algebras. We provide some simplifications to the notion of a σWCvirtual diagonal, as introduced in [10], especially in the case of the bidual of an Arens regular Banach algebra. ..."
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Cited by 3 (1 self)
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We investigate the notion of Connesamenability, introduced by Runde in [14], for bidual algebras and weighted semigroup algebras. We provide some simplifications to the notion of a σWCvirtual 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 Connesamenability 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 Connesamenable (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.
A Representation Theorem for Completely Contractive Dual Banach 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 ..."
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Cited by 2 (0 self)
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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
LEFT INTROVERTED SUBSPACES OF DUALS OF BANACH ALGEBRAS AND WEAK ∗ −CONTINUOUS DERIVATIONS ON DUAL BANACH ALGEBRAS
, 2006
"... 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 du ..."
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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 Connesamenability of some dual Banach algebras. 1.
CONNES AMENABILITY OF THE SECOND DUAL OF ARENS REGULAR BANACH ALGEBRAS
, 2006
"... 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 Ban ..."
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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.
Weakly almost periodic functionals, representations, and operator spaces
, 2008
"... A theorem of Davis, Figiel, Johnson and Pe̷lczyński tells us that weaklycompact 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, sho ..."
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A theorem of Davis, Figiel, Johnson and Pe̷lczyński tells us that weaklycompact 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 weaklycompact 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.