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

DOI:
https://doi.org/10.1090/S0002-9939-97-03652-6

MathSciNet review:
1363447

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Abstract | References | Similar Articles | Additional Information

Abstract: We show that every scalar valued continuous function on a compact group may be written as for all , where are vectors in a separable Hilbert space , and is a strongly continuous unitary valued function on which is a product of unitary representations and antirepresentations of on . This product is countable, but always converges uniformly on . Moreover the supremum norm of is matched by . This may be viewed as a `Fourier product representation' for , 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.

**1.**D. P. Blecher,*The standard dual of an operator space*, Pacific J. Math.**153**(1992), 15-30. MR**93d:47083****2.**-,*Factorizations in universal operator spaces and algebras*, To appear,*Rocky Mountain Math. J.*).**3.**D. Blecher and V. Paulsen,*Explicit construction of universal operator algebras and an application to polynomial factorization*, Proc. Amer. Math. Soc.**112**(1991), 839-850. MR**91j:46093****4.**D. Blecher, Z-j. Ruan, and A. Sinclair,*A characterization of operator algebras*, J. Functional Anal.**89**(1990), 188-201. MR**91b:47098****5.**E. Christensen, E. Effros, and A. Sinclair,*Completely bounded multilinear maps and algebraic cohomology*, Inv. Math.**90**(1987), 279-296. MR**89k:46084****6.**P. Eymard,*L'algebre de Fourier d'un groupe localement compact*, Bull. Soc. Math. France**92**(1964), 181-236. MR**37:4208****7.**J. M. G. Fell and R. S. Doran,*Representations of *-algebras, locally compact groups, and Banach *-algebraic bundles*, Vol. 126 in Pure and Appl. Math. Series, Academic Press, 1988. MR**90c:46002****8.**V. Paulsen,*Completely bounded maps and dilations*, Pitman Research Notes in Math., Longman, London, 1986. MR**88h:46111****9.**Z-j. Ruan,*Amenability of Hopf von Neumann algebras and Kac algebras*, J. Functional Anal.**139**(1996), 466-499.**10.**A. Van Daele,*The Haar measure on a compact quantum group*, Proc. Amer. Math. Soc.**123**(1995), 3125-3128. MR**95m:46097****11.**S. Wang,*Ph. D Thesis*, Berkeley (1993).**12.**S. L. Woronowicz,*Compact matrix pseudogroups*, Comm. Math. Phys.**111**(1987), 613-665. MR**88m:46079****13.**-,*Compact quantum groups*, Preprint (1993).**14.**K. Yosida,*Functional Analysis*, 3rd Edition, Springer-Verlag, 1971. MR**39:741****15.**R. C. Blei,*Projectively bounded Frechet measures*, Trans. A.M.S. (to appear).**16.**G. Zhao and B. M. Schreiber,*Algebras of multilinear forms on groups*, Contemp. Math.**189**(1995), 497-511.**17.**G. Zhao,*Completely bounded multilinear forms on groups*, preprint (1995).

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Additional Information

**David P. Blecher**

Affiliation:
Department of Mathematics, University of Houston, Houston, Texas 77204-3476

Email:
dblecher@math.uh.edu

DOI:
https://doi.org/10.1090/S0002-9939-97-03652-6

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