# Ross Stokke

## 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.