Real-Multilinear Isometries on Function Algebras

Abstract

Let $A_1, \ldots , A_k$ be function algebras (or more generally, dense subspaces of uniformly closed function algebras) on locally compact Hausdorff spaces $X_1, \ldots ,X_k$, respectively, and let Y be a locally compact Hausdorff space. A k-real-linear map $T:A_1\times \cdots \times A_k\longrightarrow C_0(Y)$ is called a real-multilinear (or k-real-linear) isometry if

$$\begin{aligned} \Vert T(f_1, \ldots , f_k)\Vert =\prod _{i=1}^{k} \Vert f_i\Vert \quad ((f_1, \ldots , f_k)\in A_1\times \cdots \times A_k), \end{aligned}$$

where $\Vert \cdot \Vert$ denotes the supremum norm. In this paper we study such maps and obtain generalizations of basically all known results concerning multilinear and real-linear isometries on function algebras.