The separable representations of
Author:
Doug Pickrell
Journal:
Proc. Amer. Math. Soc. 102 (1988), 416420
MSC:
Primary 22E65,; Secondary 47B99,47D10
MathSciNet review:
921009
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Abstract: In this paper we show that the separable representation theory of is completely analogous to that for , in that every separable representation is discretely decomposable and the irreducible representations all occur in the decomposition of the mixed tensor algebra of . This was previously shown to be true (for all representations, separable and nonseparable) for the normal subgroup , consisting of operators which are compact perturbations of the identity, by Kirillov and Ol'shanskii. In particular we show that all nontrivial representations of the unitary Calkin group are nonseparable. The proof exploits the analogue of the following fact about the Calkin algebra: if is a nontrivial representation of the Calkin algebra and is a normal operator on , then every point in the spectrum of is an eigenvalue.
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 R. Kadison, Infinite unitary groups, Trans. Amer. Math. Soc. 72 (1952), 386399. MR 0048455 (14:16g)
 [2]
 A. Kirillov, Representations of the infinite dimensional unitary group, Dokl. Akad. Nauk SSSR 212 (1973), 288290. MR 0340487 (49:5239)
 [3]
 G. I. Ol'shanskii, Unitary representations of the infinitedimensional classical groups and the corresponding motion groups, Funct. Anal. Appl. 12 (1978), 185195. MR 509382 (80g:22020)
 [4]
 A. N. Pressley and G. Segal, Loop groups, Oxford Univ. Press, 1987.
 [5]
 G. Segal, Faddeev's anomaly in Gauss's law, preprint.
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 S. Stratila and D. Voiculescu, Representations of AFalgebras and of the group , Lecture Notes in Math., vol. 486, SpringerVerlag, Berlin and New York, 1975. MR 0458188 (56:16391)
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 H. Widom, Asymptotic behavior of block Toeplitz matrices and determinants. II, Advances in Math. 21 (1976), 129. MR 0409512 (53:13266b)
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 R. Boyer, Representation theory of the HilbertLie group , Duke Math. J. 47 (1980), 325344. MR 575900 (81g:22024)
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 D. Pickrell, Measures on infinite Grassmann manifolds. II (in preparation).
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 I. Berg, An extension of the Weylvon Neumann theorem to normal operators, Trans. Amer. Math. Soc. 160 (1971), 365371. MR 0283610 (44:840)
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 I. Segal, The structure of a class of representations of the unitary group on a Hilbert space, Proc. Amer. Math. Soc. 8 (1957), 197203. MR 0084122 (18:812f)
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DOI:
http://dx.doi.org/10.1090/S0002993919880921009X
PII:
S 00029939(1988)0921009X
Keywords:
Unitary group,
separable representation,
Calkin algebra,
spectrum
Article copyright:
© Copyright 1988
American Mathematical Society
