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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Nonseparable approximate equivalence


Author: Donald W. Hadwin
Journal: Trans. Amer. Math. Soc. 266 (1981), 203-231
MSC: Primary 46L05
DOI: https://doi.org/10.1090/S0002-9947-1981-0613792-6
MathSciNet review: 613792
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Abstract: This paper extends Voiculescu's theorem on approximate equivalence to the case of nonseparable representations of nonseparable $ {C^ \ast }$-algebras. The main result states that two representations $ f$ and $ g$ are approximately equivalent if and only if $ {\text{rank}}f(x) = {\text{rank}}g(x)$ for every $ x$. For representations of separable $ {C^ \ast }$-algebras a multiplicity theory is developed that characterizes approximate equivalence. Thus for a separable $ {C^ \ast }$-algebra, the space of representations modulo approximate equivalence can be identified with a class of cardinal-valued functions on the primitive ideal space of the algebra. Nonseparable extensions of Voiculescu's reflexivity theorem for subalgebras of the Calkin algebra are also obtained.


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DOI: https://doi.org/10.1090/S0002-9947-1981-0613792-6
Article copyright: © Copyright 1981 American Mathematical Society