## Tensor products of group algebras

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- by J. E. Kerlin PDF
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**175**(1973), 1-36 Request permission

## 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.

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## Additional Information

- © Copyright 1973 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**175**(1973), 1-36 - MSC: Primary 46M05; Secondary 43A20
- DOI: https://doi.org/10.1090/S0002-9947-1973-0312286-0
- MathSciNet review: 0312286