A new proof of the Tikuisis–White–Winter theorem

Abstract

AbstractA trace on a {\mathrm{C}^{*}}-algebra is amenable (resp. quasidiagonal) if it admits a net of completely positive, contractive maps into matrix algebras which approximately preserve the trace and are approximately multiplicative in the 2-norm (resp. operator norm). Using that the double commutant of a nuclear {\mathrm{C}^{*}}-algebra is hyperfinite, it is easy to see that traces on nuclear {\mathrm{C}^{*}}-algebras are amenable. A recent result of Tikuisis, White, and Winter shows that faithful traces on separable, nuclear {\mathrm{C}^{*}}-algebras in the UCT class are quasidiagonal. We give a new proof of this result using the extension theory of {\mathrm{C}^{*}}-algebras and, in particular, using a version of the Weyl–von Neumann Theorem due to Elliott and Kucerovsky.