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A separable quasidiagonal C*-algebra with a non-quasidiagonal quotient by the compact operators

Published online by Cambridge University Press:  24 October 2008

Simon Wassermann
Simon Wassermann
Affiliation:
Department of Mathematics, University of Glasgow, Glasgow G12 8QW, Scotland
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Extract

A C*-algebra A of operators on a separable Hilbert space H is said to be quasidiagonal if there is an increasing sequence E1, E2, … of finite-rank projections on H tending strongly to the identity and such that

as i → ∞ for T∈A. More generally a C*-algebra is quasidiagonal if there is a faithful *-representation π of A on a separable Hilbert space H such that π(A) is a quasidiagonal algebra of operators. When this is the case, there is a decomposition H = H1 ⊕ H2 ⊕ … where dim Hi < ∞ (i = 1, 2,…) such that each T∈π(A) can be written T = D + K where D= D1 ⊕ D2 ⊕ …, with Di∈L(Hi) (i = 1, 2,…), and K is a compact linear operator on H. As is well known (and readily seen), this is an alternative characterization of quasidiagonality.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 110 , Issue 1 , July 1991 , pp. 143 - 145
Copyright
Copyright © Cambridge Philosophical Society 1991

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References

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