The separable representations of

Author:
Doug Pickrell

Journal:
Proc. Amer. Math. Soc. **102** (1988), 416-420

MSC:
Primary 22E65,; Secondary 47B99,47D10

DOI:
https://doi.org/10.1090/S0002-9939-1988-0921009-X

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.

**[1]**Richard V. Kadison,*Infinite unitary groups*, Trans. Amer. Math. Soc.**72**(1952), 386–399. MR**0048455**, https://doi.org/10.1090/S0002-9947-1952-0048455-3**[2]**A. A. Kirillov,*Representations of the infinite-dimensional unitary group*, Dokl. Akad. Nauk. SSSR**212**(1973), 288–290 (Russian). MR**0340487****[3]**G. I. Ol′šanskiĭ,*Unitary representations of the infinite-dimensional classical groups 𝑈(𝑝,∞), 𝑆𝑂₀(𝑝,∞), 𝑆𝑝(𝑝,∞), and of the corresponding motion groups*, Funktsional. Anal. i Prilozhen.**12**(1978), no. 3, 32–44, 96 (Russian). MR**509382****[4]**A. N. Pressley and G. Segal,*Loop groups*, Oxford Univ. Press, 1987.**[5]**G. Segal,*Faddeev's anomaly in Gauss's law*, preprint.**[6]**Şerban Strătilă and Dan Voiculescu,*Representations of AF-algebras and of the group 𝑈(∞)*, Lecture Notes in Mathematics, Vol. 486, Springer-Verlag, Berlin-New York, 1975. MR**0458188****[7]**Harold Widom,*Asymptotic behavior of block Toeplitz matrices and determinants. II*, Advances in Math.**21**(1976), no. 1, 1–29. MR**0409512**, https://doi.org/10.1016/0001-8708(76)90113-4**[8]**Robert P. Boyer,*Representation theory of the Hilbert-Lie group 𝑈(ℌ)₂*, Duke Math. J.**47**(1980), no. 2, 325–344. MR**575900****[9]**D. Pickrell,*Measures on infinite Grassmann manifolds*. II (in preparation).**[10]**I. David Berg,*An extension of the Weyl-von Neumann theorem to normal operators*, Trans. Amer. Math. Soc.**160**(1971), 365–371. MR**0283610**, https://doi.org/10.1090/S0002-9947-1971-0283610-0**[11]**I. E. Segal,*The structure of a class of representations of the unitary group on a Hilbert space*, Proc. Amer. Math. Soc.**8**(1957), 197–203. MR**0084122**, https://doi.org/10.1090/S0002-9939-1957-0084122-8

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1988-0921009-X

Keywords:
Unitary group,
separable representation,
Calkin algebra,
spectrum

Article copyright:
© Copyright 1988
American Mathematical Society