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Commutants of unitaries in UHF algebras and functorial properties of exactness. (English) Zbl 0796.46043
It is shown that the maximal \(C^*\)-algebra tensor product \(N\otimes^{\max} A\) of a von Neumann algebra \(N\) and a unital separable \(C^*\)-algebra \(A\) with lifting property has a faithful *-representation \(d\) on a Hilbert space such that \(n\in N\mapsto d(n\otimes 1)\) is normal. In particular, there is one and only one \(C^*\)-norm on the algebraic tensor product \({\mathcal L} (H)\odot C^*(F)\) of the algebra \({\mathcal L}(H)\) of operators on a Hilbert space and the universal enveloping \(C^*\)- algebra \(C^*(F)\) of a free group \(F\).
This leads to a new proof of the following: Separable exact \(C^*\)- algebras are quotient algebras of \(C^*\)-subalgebras of the CAR-algebra. Here we improve this result on exact \(C^*\)-algebras as follows:
For every unital separable exact \(C^*\)-algebra \(A\) and every uniformly hyperfinite \(C^*\)-algebra \(B\), there exists a unitary \(U\) in \(B\) such that \(A\) is isomorphic to a quotient of the relative commutant \(\{U\}^ c= \{U\}'\cap B\) of \(U\) in \(B\) by a closed approximately finite dimensional ideal \(J\) of \(\{U\}^ c\).
Then we study functorial properties of exact \(C^*\)-algebras. Among others, we obtain that amalgamated reduced free products of finite- dimensional \(C^*\)-algebras are exact.

46L05 General theory of \(C^*\)-algebras
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