Nonseparable approximate equivalence

Hadwin, Donald W.

doi:10.1090/S0002-9947-1981-0613792-6

Nonseparable approximate equivalence
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by Donald W. Hadwin

Trans. Amer. Math. Soc. 266 (1981), 203-231

DOI: https://doi.org/10.1090/S0002-9947-1981-0613792-6

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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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