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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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