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CHARACTERISATION OF THE FOURIER TRANSFORM ON COMPACT GROUPS

Published online by Cambridge University Press:  13 November 2015

N. SHRAVAN KUMAR*
Affiliation:
Department of Mathematics, Indian Institute of Technology Delhi, Delhi-110016, India email [email protected]
S. SIVANANTHAN
Affiliation:
Department of Mathematics, Indian Institute of Technology Delhi, Delhi-110016, India email [email protected]
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Abstract

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Let $G$ be a compact group. The aim of this note is to show that the only continuous *-homomorphism from $L^{1}(G)$ to $\ell ^{\infty }\text{-}\bigoplus _{[{\it\pi}]\in {\hat{G}}}{\mathcal{B}}_{2}({\mathcal{H}}_{{\it\pi}})$ that transforms a convolution product into a pointwise product is, essentially, a Fourier transform. A similar result is also deduced for maps from $L^{2}(G)$ to $\ell ^{2}\text{-}\bigoplus _{[{\it\pi}]\in {\hat{G}}}{\mathcal{B}}_{2}({\mathcal{H}}_{{\it\pi}})$.

Type
Research Article
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

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