## 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.

### MSC:

 46L05 General theory of $$C^*$$-algebras
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