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Multipliers on a new class of Banach algebras, locally compact quantum groups, and topological centres. (English) Zbl 1192.43002
Let \(A\) be a Banach algebra and let \(LM(A)\) and \(RM(A)\) be the Banach algebras of left and right multipliers on \(A\). The authors undertake a thorough study of these algebras for some prominent classical Banach algebras, such as the weighted convolution algebra \(L_1(D,w)\) and the Fourier algebra \(A(G)\) of a locally compact group \(G\). Their results can be assembled into four major topics:
(1)
Characterizing the algebra \(A\) inside its multiplier algebras;
(2)
Representation theory for multipliers over locally compact quantum groups (completely isometric representations of \(L_1(G)\) and \(A(G)\));
(3)
Characterizing \(A\) inside its bidual \(A^{**}\) in terms of multipliers;
(4)
Multipliers and topological centres.
The authors discuss the relation of their results to other works and at the end provide some applications.

MSC:
43A10 Measure algebras on groups, semigroups, etc.
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
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