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Multipliers, self-induced and dual Banach algebras. (English) Zbl 1214.43004
The first part of the paper is devoted to a short survey of the theory of multipliers of Banach algebras and completely contractive Banach algebras. The author concentrates upon the problem of how to extend module actions and homomorphisms from algebras to multiplier algebras. Then the special cases are considered when a bounded approximate identity exists and when the considered algebra is self-induced. In the second part of the paper the author mainly considers dual Banach algebras and provides a simple criterion for a multiplier algebra to be 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. The author also studies this problem in the framework of abstract Pontryagin duality. He explores the notion of Hopf convolution algebra and shows that in many cases the use of the extended Haagerup tensor product can be replaced by a multiplier algebra.

MSC:
43A20 \(L^1\)-algebras on groups, semigroups, etc.
43A30 Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.
46H05 General theory of topological algebras
46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX)
46L07 Operator spaces and completely bounded maps
46L89 Other “noncommutative” mathematics based on \(C^*\)-algebra theory
43A22 Homomorphisms and multipliers of function spaces on groups, semigroups, etc.
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
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