Abstract
We undertake a detailed study of the sets of multiplicity in a second countable locally compact group G and their operator versions. We establish a symbolic calculus for normal completely bounded maps from the space B(L2(G)) of bounded linear operators on L2 (G) into the von Neumann algebra VN(G) of G and use it to show that a closed subset E subset of G is a set of multiplicity if and only if the set E* = {(s,t) is an element of G x G : ts(1) is an element of E} is a set of operator multiplicity. Analogous results are established for M1sets and M0sets. We show that the property of being a set of multiplicity is preserved under various operations, including taking direct products, and establish an Inverse Image Theorem for such sets. We characterise the sets of finite width that are also sets of operator multiplicity, and show that every compact operator supported on a set of finite width can be approximated by sums of rank one operators supported on the same set. We show that, if G satisfies a mild approximation condition, pointwise multiplication by a given measurable function psi : G > C defines a closable multiplier on the reduced C*algebra G(r)*(G) of G if and only if Schur multiplication by the function N(psi): G x G > C, given by N(psi)(s, t) = psi(ts(1)), is a closable operator when viewed as a densely defined linear map on the space of compact operators on L2(G). Similar results are obtained for multipliers on VN(C).
Original language  English 

Pages (fromto)  14541508 
Number of pages  55 
Journal  Journal of Functional Analysis 
Volume  268 
Issue number  6 
Early online date  06 Dec 2014 
DOIs  
Publication status  Published  15 Mar 2015 
Keywords
 math.OA
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Ivan Todorov
Person: Academic