Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



The separable representations of $ {\rm U}(H)$

Author: Doug Pickrell
Journal: Proc. Amer. Math. Soc. 102 (1988), 416-420
MSC: Primary 22E65,; Secondary 47B99,47D10
MathSciNet review: 921009
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we show that the separable representation theory of $ {\text{U(}}H)$ is completely analogous to that for $ {\text{U(}}{{\mathbf{C}}^n})$, in that every separable representation is discretely decomposable and the irreducible representations all occur in the decomposition of the mixed tensor algebra of $ H$. This was previously shown to be true (for all representations, separable and nonseparable) for the normal subgroup $ {\text{U(}}H{)_\infty }$, 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 $ \pi $ is a nontrivial representation of the Calkin algebra and $ T$ is a normal operator on $ H$, then every point in the spectrum of $ \pi (T)$ is an eigenvalue.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 22E65,, 47B99,47D10

Retrieve articles in all journals with MSC: 22E65,, 47B99,47D10

Additional Information

Keywords: Unitary group, separable representation, Calkin algebra, spectrum
Article copyright: © Copyright 1988 American Mathematical Society