Cohomological Characterizations of Biprojective and Biflat Banach Algebras

Selivanov, Yu. V.

doi:10.1007/pl00010082

Cited by 15 publications

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“…Similar ideas have also been explored in the context of Banach cohomology theory (see for example [51]). We shall argue that self-induced algebras form a natural setting to consider multipliers in.…”

Section: Self-induced Banach Algebrasmentioning

confidence: 89%

“…However, multiplier algebras which appear in abstract harmonic analysis do often seem to be dual spaces. Our result allows us to show that, in particular, M (L 1 (G)) (and its completely bounded counterpart) are dual Banach algebras, for a locally compact quantum group G. Our ideas are influenced by [51].…”

Section: When Multiplier Algebras Are Dualmentioning

confidence: 99%

“…If products are not linearly dense in A, then, following [51], one could instead consider the unitisation of A. In our applications, this will not be needed.…”

Section: Thus By the Preceding Paragraph It Follows That (Lmentioning

confidence: 99%

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Multipliers, self-induced and dual Banach algebras

Daws¹

2010

Dissertationes Math.

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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.Acknowledgements. I wish to thank the referees for their careful and timely reading of this manuscript, and for comments which have improved the exposition.2010 Mathematics Subject Classification: Primary 43A20, 43A30, 46H05, 46H25, 46L07, 46L89; Secondary 16T05, 43A22, 81R50. Key words and phrases: multiplier, double centraliser, Fourier algebra, locally compact quantum group, dual Banach algebra, Hopf convolution algebra. Received 12.2.2010; revised version 29.4.2010.[4] IntroductionMultipliers are a useful way of embedding a non-unital algebra into a unital algebra: a problem which occurs often in algebraic analysis. The theory has reached maturity when applied to C * -algebras (see, for example, [59, Chapter 2]) where it is best studied in the context of Hilbert C * -modules, [33]. Indeed, one can also study "unbounded operators" for C * -algebras, [61], which form a vital tool in the study of quantum groups. For Banach algebras with a bounded approximate identity, much of the theory carries over (see [24, Section 1.d] or [8, Theorem 2.9.49]) although we remark that there seems to be no parallel to the unbounded theory.This paper starts with a survey of multipliers; we start with some generality, working with multipliers of modules, and not just algebras. This material is surely well-known to experts, but we are not aware of any particularly definitive source. For example, in [41], Ng uses similar ideas (but for C * -algebras, working in the category of operator modules) motived by the study of cohomology theories for Hopf operator algebras (that is, loosely speaking, quantum groups). However, most of the proofs are left in an unpublished manuscript. The particular aspects of the theory which we develop are somewhat motivated by Ng's presentation.We quickly turn to discussing Banach algebras, but we shall not (as is usually the case) require a bounded app...

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Section: Self-induced Banach Algebrasmentioning

confidence: 89%

Section: When Multiplier Algebras Are Dualmentioning

confidence: 99%

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Multipliers, self-induced and dual Banach algebras

Daws¹

2010

Dissertationes Math.

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“…(iii) The algebra K( 2⊗ 2 ) of all compact operators on the Banach space 2⊗ 2 provides an example of a non-amenable, biflat semisimple Banach algebra with a left-bounded approximate identity [43]. …”

Section: Example 46 (Biflat Banach Algebras)mentioning

confidence: 99%

Cyclic Cohomology of Projective Limits of Topological Algebras

Lykova

2006

Proceedings of the Edinburgh Mathematical Society

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We present methods for the computation of the Hochschild and cyclic continuous cohomology and homology of some locally convex topological algebras. Let (Aα, T α,β ) (Λ, ) be a reduced projective system of complete Hausdorff locally convex algebras with jointly continuous multiplications, and let A be the projective limit algebra A = lim← α Aα. We prove that, for the continuous cyclic cohomology HC * and continuous periodic cohomology HP * of A and Aα, α ∈ Λ, for all n 0, HC n (A) = lim→ α HC n (Aα), the inductive limit of HC n (Aα), and, for k = 0, 1, HP k (A) = lim→ α HP k (Aα). For a projective limit algebra A = lim← m Am of a countable reduced projective system (Am, T m, ) N of Fréchet algebras, we also establish relations between the cyclic-type continuous homology of A and Am, m ∈ N. For example, we show the exactness of the following short sequence for all n 0:We present a class of Fréchet algebras A for which the continuous periodic cohomology HP k (A), k = 0, 1, is isomorphic to the continuous cyclic cohomology HC 2 +k (A) starting from some integer . We apply the above results to calculate the continuous cyclic-type homology and cohomology of some Fréchet locally m-convex algebras.

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“…Ya. Helemskii [34,36] and have been studied by many authors (see, e.g., [34,35,46,78,80,81,37,82,53,69,83,87,88,89,91,8,63,102,64,65,67,9,76]). The original motivation to study such algebras was the vanishing of their cohomology groups 14 , H n (A, X), with coefficients in arbitrary topological A-bimodules X for all n ≥ 3 (see, e.g., [42,Theorem 2.4.21]).…”

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confidence: 99%

Structure theory of homologically trivial and annihilator locally C^*-algebras

Pirkovskii

Selivanov

2010

Banach Center Publ.

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We study the structure of certain classes of homologically trivial locally C * -algebras. These include algebras with projective irreducible Hermitian A-modules, biprojective algebras, and superbiprojective algebras. We prove that, if A is a locally C * -algebra, then all irreducible Hermitian A-modules are projective if and only if A is a direct topological sum of elementary C * -algebras. This is also equivalent to A being an annihilator (dual, complemented, left quasi-complemented, or topologically modular annihilator) topological algebra. We characterize all annihilator σ-C * -algebras and describe the structure of biprojective locally C * -algebras. Also, we present an example of a biprojective locally C * -algebra that is not topologically isomorphic to a Cartesian product of biprojective C * -algebras. Finally, we show that every superbiprojective locally C * -algebra is topologically * -isomorphic to a Cartesian product of full matrix algebras.

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Cohomological Characterizations of Biprojective and Biflat Banach Algebras

Cited by 15 publications

References 9 publications

Multipliers, self-induced and dual Banach algebras

Multipliers, self-induced and dual Banach algebras

Cyclic Cohomology of Projective Limits of Topological Algebras

Structure theory of homologically trivial and annihilator locally C^*-algebras

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Cohomological Characterizations of Biprojective and Biflat Banach Algebras

Cited by 15 publications

References 9 publications

Multipliers, self-induced and dual Banach algebras

Multipliers, self-induced and dual Banach algebras

Cyclic Cohomology of Projective Limits of Topological Algebras

Structure theory of homologically trivial and annihilator locally C*-algebras

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Structure theory of homologically trivial and annihilator locally C^*-algebras