A class of Banach spaces

Abstract

Let $G$ be a separable locally compact unimodular group of type I, $ \widehat{G}$ be its dual, $\hat{p}$ is a measurable field of, not necessary bounded, operators on $\widehat{G}$ such that $\hat{p}(\pi)$ is self-adjoint, $\hat{p}(\pi) \geq I$ for $\mu-$almost all $\pi \in \widehat{G}$, and \begin{align*} & A_{\hat{p} }(G) =\{f(x):=\int_{ \widehat{G}} Tr[\hat{f}(\pi)\pi(x)^{-1}]d\mu(\pi), \hat{f} \in L_{1}( \widehat{G} ), \|f\|_{\hat{p}} \\& \qquad =\int_{ \widehat{G} }Tr|\hat{p}(\pi)\hat{f}(\pi)|d\mu(\pi) >\infty \} \end{align*} We show that $ A_{\hat{p} }(G)$ is a Banach space endowed with the norm $\|f\|_{\hat{p}}$, and we generalize this result to the matricial group $G=G_{nm}$, $m\geq n$, of a local field.