Homomorphisms of convolution algebras
\textit{P. J.~Cohen} [``On homomorphisms of group algebras'', Am. J. Math. 82, 213--226 (1960; Zbl 0099.25601)] characterised all homomorphisms \(\phi: L^1(G)\to M(H)\) between the group and measure algebras of locally compact abelian groups \(G\) and \(H\). There has been a considerable effort to generalise this result to all locally compact groups. One may approach the question from two directions, dual to each other. First, we may simply leave out the condition that \(G\) and \(H\) are abelian. Secondly, we may consider homomorphisms \(\phi: A(G) \to B(H)\), where \(A(G)\) is the Fourier algebra of \(G\) and \(B(H)\) the Fourier-Stieltjes algebra of \(H\) (and again \(G\) and \(H\) are not necessarily abelian). As Cohen employed the Fourier-Stieltjes transform in his solution, perhaps the latter approach is more amenable. In this setting, \textit{M.~Ilie} and \textit{N.~Spronk} [``Completely bounded homomorphisms of the Fourier algebras'', J. Funct. Anal. 225, No.~2, 480--499 (2005; Zbl 1077.43004)] characterised all completely bounded homomorphisms \(\phi: A(G) \to B(H)\) for amenable \(G\) by corresponding these homomorphisms to continuous piece-wise affine maps \(\alpha: H\supseteq Y\to G\); this covers Cohen's theorem as well as earlier extensions by \textit{B.~Host} [``Le théorème des idempotents dans B(G)'', Bull. Soc. Math. Fr. 114, 215--223 (1986; Zbl 0606.43002)] and \textit{M.~Ilie} [``On Fourier algebra homomorphisms'', J. Funct. Anal. 213, No.~1, 88--110 (2004; Zbl 1052.43006)]. Recently, \textit{H. L.~Pham} [``Contractive homomorphisms of the Fourier algebras'', Bull. Lond. Math. Soc. 42, No.~5, 937--947 (2010; Zbl 1248.43003)] showed that every contractive homomorphism \(\phi: A(G) \to B(H)\) has a factorisation \(\phi = \ell_s\circ \iota\circ j_\theta \circ \ell_t\), where \(\ell_t\) is the left translation on \(A(G)\) by \(t\in G\), \(j_\theta: A(G)\to B(H_0)\) is the homomorphism \(u\mapsto u\circ \theta\) defined by a continuous homomorphism or anti-homomorphism \(\theta : H_0\to G\) from an open subgroup \(H_0\) of \(H\) to \(G\), \(\iota : B(H_0)\to B(H)\) is the natural inclusion (extension of functions by \(0\)), and \(\ell_s\) is the left translation on \(B(H)\) by \(s\in H\). This factorisation is of a similar form as the factorisation of contractive homomorphisms \(\phi: L^1(G)\to M(H)\) in the case of abelian groups due to \textit{I.~Glicksberg} [``Homomorphisms of certain algebras of measures'', Pac. J. Math. 10, 167--191 (1960; Zbl 0104.33805)]. Continuing the work of Glicksberg, \textit{F. P.~Greenleaf} [``Norm decreasing homomorphisms of group algebras'', Pac. J. Math. 15, 1187--1219 (1965; Zbl 0136.11402)] characterised all contractive homomorphisms \(\phi: L^1(G)\to M(H)\), where \(G\) and \(H\) are not necessarily abelian. However, as noted by \textit{J. E.~Kerlin} and \textit{W. D.~Pepe} [``Norm decreasing homomorphisms between group algebras'', Pac. J. Math. 57, 445--451 (1975; Zbl 0309.43014)], the characterisation due to Greenleaf is less tractable than in the abelian case; they then went on to give necessary and sufficient conditions to certain factorisations of contractive homomorphisms more in line with the abelian case. This is where the paper under review comes in. Answering a question of Kerlin and Pepe, the author gives a counterexample that shows that the types of factorisations used by Kerlin and Pepe are no longer valid in the general case. The main result, however, is a different factorisation of contractive homomorphisms \(\phi: L^1(G)\to M(H)\) into elementary parts. Along the way, the author also derives, and extends, one of the main results of J. E.~Kerlin and W. D.~Pepe [loc. cit.]. Many of the techniques and results are improvements over those of Greenleaf. In particular, Greenleaf's characterisation of contractive subgroups of \(M(H)\) is improved by completing the description of their topological structures. The main result yields also several corollaries. These include a new characterisation of contractive epimorphisms \(\phi: L^1(G)\to L^1(H)\) as well as considerations on larger algebras such as \(\text{LUC}(G)^*\) and \(\text{WAP}(G)^*\) (the duals of the C*-algebras of left uniformly continuous functions and weakly almost periodic functions, respectively).
scientific article; zbMATH DE number 5978501
scientific article