The Separable Representations Of U(H)

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Abstract

In this paper we show that the separable representation theory of U(H) is completely analogous to that for U(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 U(if)∞, 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 π is a nontrivial representation of the Calkin algebra and T is a normal operator on H, then every point in the spectrum of π(T) is an eigenvalue.

Original languageEnglish (US)
Pages (from-to)416-420
Number of pages5
JournalProceedings of the American Mathematical Society
Volume102
Issue number2
DOIs
StatePublished - 1988

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Algebra
Nonseparable
Mixed tensor
Tensor Algebra
Compact Perturbation
Normal Operator
Unitary group
Normal subgroup
Decomposable
Representation Theory
Irreducible Representation
Tensors
Mathematical operators
Analogue
Eigenvalue
Decomposition
Decompose
Operator

Keywords

  • Calkin algebra
  • Separable representation
  • Spectrum
  • Unitary group

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

The Separable Representations Of U(H). / Pickrell, Douglas M.

In: Proceedings of the American Mathematical Society, Vol. 102, No. 2, 1988, p. 416-420.

Research output: Contribution to journalArticle

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