Proceedings of the Japan Academy, Series A, Mathematical Sciences

A class of Banach spaces

Wassim Nasserddine

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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.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 83, Number 4 (2007), 56-59.

Dates
First available in Project Euclid: 30 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.pja/1177941418

Digital Object Identifier
doi:10.3792/pjaa.83.56

Mathematical Reviews number (MathSciNet)
MR2326203

Zentralblatt MATH identifier
1131.43005

Subjects
Primary: 43A30: Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc. 46E15: Banach spaces of continuous, differentiable or analytic functions
Secondary: 47B37: Operators on special spaces (weighted shifts, operators on sequence spaces, etc.)

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

Citation

Nasserddine, Wassim. A class of Banach spaces. Proc. Japan Acad. Ser. A Math. Sci. 83 (2007), no. 4, 56--59. doi:10.3792/pjaa.83.56. https://projecteuclid.org/euclid.pja/1177941418


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