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

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 TA. 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 = H1H2 ⊕ … where dim Hi < ∞ (i = 1, 2,…) such that each T∈π(A) can be written T = D + K where D= D1D2 ⊕ …, with DiL(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
Copyright
Copyright © Cambridge Philosophical Society 1991

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References

REFERENCES

[1]Anderson, J.. A C*-algebra A for which Ext(A) is not a group. Ann. of Math. 107 (1978), 455458.CrossRefGoogle Scholar
[2]Choi, M.-D. and Effros, E. G.. Injectivity and operator spaces. J. Fund. Anal. 24 (1977), 156209.CrossRefGoogle Scholar
[3]Powers, R. T.. Simplicity of the C*-algebra associated with the free group on two generators. Duke Math. J. 42 (1975), 151156.CrossRefGoogle Scholar
[4]Wassermann, S.. On tensor products of certain group C*-algebras. J. Fund. Anal. 23 (1976), 239254.CrossRefGoogle Scholar
[5]Wassermann, S.. C*-algebras associated with groups with Kazhdan's property T (preprint).Google Scholar