Ross Stokke

(Department of Mathematics, University of Winnipeg)

Homomorphisms Of Fourier--Stieltjes Algebras

Date Friday, March 27, 2020

The Fourier and Fourier--Stieltjes algebras \(A(G)\) and \(B(G)\) of a locally compact group \(G\) are among the principal objects of study in abstract harmonic analysis. An old problem, solved in the abelian case by Paul Cohen in 1960, asks for a description of all homomorphisms \(\varphi: A(G) \rightarrow B(H)\). While the general problem remains open, M. Ilie and N. Spronk have described all completely positive, completely contractive and completely bounded homomorphisms \(\varphi: A(G) \rightarrow B(H)\) when \(G\) is amenable; H.L. Pham described the positive and contractive homomorphisms with no conditions imposed on \(G\).

A related problem asks for a description of the homomorphisms \(\varphi: B(G) \rightarrow B(H)\). After introducing the relevant terminology, we will exhibit a large collection of completely positive, completely contractive and completely bounded homomorphisms \(\varphi: B(G) \rightarrow B(H)\) and will establish converse statements in some situations. For example, we will characterize all completely positive, completely contractive and completely bounded homomorphisms \(\varphi: B(G) \rightarrow B(H)\) when \(G\) is a Euclidean- or \(p\)-adic-motion group. In these cases, our description of the completely positive/completely contractive homomorphisms is quite different from the Fourier algebra situation.