For a locally compact group, G, its Fourier and Fourier–Stieltjes algebras A(G) and B(G) are Banach algebras of continuous functions on G that uniquely determine G as a topological group; when G is abelian, A(G) and B(G) can be identified via the Fourier–Stieltjes transform with the group and measure convolution algebras on the dual group of G. An old problem, solved in the abelian case by Paul Cohen in 1960, asks for a description of all homomorphisms from A(G) into B(H). For non-abelian groups, M. Ilie, N. Spronk, M. Daws and H.L. Pham have, among others, made significant contributions to this problem. The difficulty of the problem of describing homomorphisms from A into B(H) where A is some other closed translation-invariant subalgebra of B(G) is significantly impacted by the complexity of the Gelfand spectrum of A. While the Gelfand spectrum of A(G) is just G and the spectrum of B(G) is often inaccessible, the spine of B(G), A’(G), is a subalgebra of B(G) containing A(G) whose spectrum is of intermediate complexity between the spectra of A(G) and B(G). The spine algebra was introduced by J. Inoue and J. Taylor for abelian groups and by M. Ilie and N. Spronk for nonabelian locally compact groups.

For any upper semilattice D of locally precompact topologies on G, we will define an associated generalized spine subalgebra AD’(G) of B(G); when D is the set of all locally precompact topologies, we obtain the full spine algebra A’(G). We will discuss properties of generalized spine algebras and identify their spectra as certain semilattices of topological groups. Using almost periodic compactifications, we will introduce a collection of examples of generalized spine algebras over whose spectra we exhibit fine control. Notions of compatible fusions of homomorphisms and affine maps will be introduced and used to characterize all completely positive, completely contractive and, when G is amenable, completely bounded homomorphisms from a generalized spine algebra AD’(G) to a Fourier–Stieltjes algebra B(H). These results are new, even when AD’(G) is the full spine algebra A’(G) and even when G and H are abelian. Examples illustrating the scope of these theorems will be discussed.

This is joint work with Nico Spronk and Aasaimani Thamizhazhagan.