Open Access
April 2007 A class of Banach spaces
Wassim Nasserddine
Proc. Japan Acad. Ser. A Math. Sci. 83(4): 56-59 (April 2007). DOI: 10.3792/pjaa.83.56


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.


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Wassim Nasserddine. "A class of Banach spaces." Proc. Japan Acad. Ser. A Math. Sci. 83 (4) 56 - 59, April 2007.


Published: April 2007
First available in Project Euclid: 30 April 2007

zbMATH: 1131.43005
MathSciNet: MR2326203
Digital Object Identifier: 10.3792/pjaa.83.56

Primary: 43A30 , ‎46E15
Secondary: 47B37

Keywords: ‎Banach spaces , Beurling-Domar weight , Fourier transform and cotransform on nonabelian groups , uncertainty principle

Rights: Copyright © 2007 The Japan Academy

Vol.83 • No. 4 • April 2007
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