(Generalized) Spine Subalgebras of Fourier-Stieltjes algebras and their Homomorphisms
For any upper semilattice ${\cal D}$ of locally precompact topologies on a locally compact group $G$, we define an associated generalized spine subalgebra $A^*_{\cal D}(G)$ of the Fourier-Stieltjes algebra $B(G)$. We show that $A^*_{\cal D}(G)$ is a semilattice-graded $\ell^1$-direct sum of maximal copies of Fourier algebras and we identify its spectrum as a semilattice of groups. We build a collection of examples of generalized spine algebras over whose spectra we exhibit fine control. We define notions of compatible fusions of homomorphisms and affine maps, and use these definitions to characterize all completely positive, completely contractive and, when $G$ is amenable, all completely bounded homomorphisms from a generalized spine algebra $A^*_{\cal D}(G)$ to a Fourier-Stieltjes algebra $B(H)$. These results are new, even when $A^*_{\cal D}(G)$ is the full spine algebra $A^*(G)$ and even when $G$ and $H$ are abelian. We provide examples illustrating the scope of our theorems.