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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:
Characterizing the algebra \(A\) inside its multiplier algebras;
Representation theory for multipliers over locally compact quantum groups (completely isometric representations of \(L_1(G)\) and \(A(G)\));
Characterizing \(A\) inside its bidual \(A^{**}\) in terms of multipliers;
Multipliers and topological centres.
The authors discuss the relation of their results to other works and at the end provide some applications.

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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