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Continuous functions on compact groups

Author: David P. Blecher
Journal: Proc. Amer. Math. Soc. 125 (1997), 1177-1185
MSC (1991): Primary 43A77, 43A25; Secondary 46L05, 43A15
MathSciNet review: 1363447
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Abstract: We show that every scalar valued continuous function $f(g)$ on a compact group $G$ may be written as $f(g) = \langle \Phi (g) \xi , \eta \rangle $ for all $g \in G$, where $\xi , \eta $ are vectors in a separable Hilbert space $\mathcal {H}$, and $\Phi (g)$ is a strongly continuous unitary valued function on $G$ which is a product of unitary representations and antirepresentations of $G$ on $ \mathcal {H}$. This product is countable, but always converges uniformly on $G$. Moreover the supremum norm of $f$ is matched by $\Vert \xi \Vert \Vert \eta \Vert $. This may be viewed as a `Fourier product representation' for $f$, and complements a result of Eymard for the Fourier algebra. For `Fourier polynomials' we show that the Hilbert space may be taken to be finite dimensional, and the product finite, which is more or less obvious except in that we are able to match the correct norm. The main ingredients of the proof are the Peter-Weyl theory, Tannaka's duality theorem, and a method developed with Paulsen using a characterization of operator algebras due to the author, Ruan and Sinclair. We also give the analogues of these formulae for compact quantum groups.

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Additional Information

David P. Blecher
Affiliation: Department of Mathematics, University of Houston, Houston, Texas 77204-3476

Received by editor(s): May 25, 1995
Received by editor(s) in revised form: October 26, 1995
Additional Notes: Supported by a grant from the NSF
Communicated by: J. Marshall Ash
Article copyright: © Copyright 1997 American Mathematical Society

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