Tensor products of group algebras

Abstract:

Let C be a commutative Banach algebra. A commutative Banach algebra A is a Banach C-algebra if A is a Banach C-module and $c \cdot (aa’) = (c \cdot a)a’$ for all $c \in C,a,a’ \in A$. If ${A_1}, \cdots ,{A_n}$ are commutative Banach C-algebras, then the C-tensor product ${A_1}{ \otimes _C} \cdots { \otimes _C}{A_n} \equiv D$ is defined and is a commutative Banach C-algebra. The maximal ideal space ${\mathfrak {M}_D}$ of D is identified with a closed subset of ${\mathfrak {M}_{{A_1}}} \times \cdots \times {\mathfrak {M}_{{A_n}}}$ in a natural fashion, yielding a generalization of the Gelbaum-Tomiyama characterization of the maximal ideal space of ${A_1}{ \otimes _\gamma } \cdots { \otimes _\gamma }{A_n}$. If $C = {L^1}(K)$ and ${A_i} = {L^1}({G_i})$, for LCA groups K and ${G_i},i = 1, \cdots ,n$, then the ${L^1}(K)$-tensor product D of ${L^1}({G_1}), \cdots ,{L^1}({G_n})$ is uniquely written in the form $D = N \oplus {D_e}$, where N and ${D_e}$ are closed ideals in D, ${L^1}(K) \cdot N = \{ 0\}$, and ${D_e}$ is the essential part of D, i.e. ${D_e} = {L^1}(K) \cdot D$. Moreover, if ${D_e} \ne \{ 0\}$, then ${D_e}$ is isometrically ${L^1}(K)$-isomorphic to ${L^1}({G_1}{ \otimes _K} \cdots { \otimes _K}{G_n})$, where ${G_1}, \cdots ,{G_n}$ is a K-tensor product of ${G_1}, \cdots ,{G_n}$ with respect to naturally induced actions of K on ${G_1}, \cdots ,{G_n}$. The above theorems are a significant generalization of the work of Gelbaum and Natzitz in characterizing tensor products of group algebras, since here the algebra actions are arbitrary. The Cohen theory of homomorphisms of group algebras is required to characterize the algebra actions between group algebras. Finally, the space of multipliers ${\operatorname {Hom}_{{L^1}(K)}}({L^1}(G),{L^\infty }(H))$ is characterized for all instances of algebra actions of ${L^1}(K)$ on ${L^1}(G)$ and ${L^1}(H)$, generalizing the known result when $K = G = H$ and the module action is given by convolution.