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Lifting problems and local reflexivity for \(C^ *\)-algebras. (English) Zbl 0613.46047
In [Ann. Math., II. Ser. 104, 585-609 (1976; Zbl 0361.46067)], M.-D. Choi and E. G. Effros proved that if A is an operator system (a unital linear self-adjoint subspace of operators on a Hilbert space), B is a unital \(C^*\)-algebra and J is a closed two-sided ideal in B, then one obstruction to the existence of a completely positive lifting of a completely positive map \(\phi\) :A\(\to B/J\) is the irregular behavior of minimal tensor products. Specifically, the obstruction is caused by the fact that the kernel of the map \(A\otimes_{\min}B\to A\otimes_{\min}(B/J)\) may not equal \(A\otimes_{\min}J\). In the paper under review, the authors prove that the above obstruction is the only one, by proving the following theorem:
Suppose that \(J\) is a nuclear ideal in a unital \(C^*\)-algebra \(B\). Then the following are equivalent:
(1) One can always lift a completely positive unital map \(\phi\) :A\(\to B/J\) to a completely positive unital map \(\psi:A\to B\) for any separable unital \(C^*\)-algebra \(A\).
(2) The kernel of \(B\otimes_{\min}C\to (B/J)\otimes_{\min}C\) is \(J\otimes_{\min}C\) for all unital \(C^*\)-algebras \(C\).
A \(C^*\)-algebra \(A\) is said to be approximately injective if given finite-dimensional operator systems \(E_ 1\subseteq E_ 2\subseteq {\mathcal B}({\mathcal H})\), any completely positive map \(\phi_ 1:E_ 1\to A\) has completely positive approximate extensions \(\phi_ 2:E_ 2\to A\), meaning that given \(\epsilon >0\), there exists a completely positive \(\phi_ 2:E_ 2\to A\) such that \(\| \phi_ 2|_{E_ 1}- \phi_ 1\| <\epsilon\). The theorem above is actually proven under the weaker hypothesis that J is approximately injective, although it is unknown whether any hypothesis on J is necessay.
The authors then consider the property of approximate injectivity, prove that it is weaker than nuclearity or (trivially) injectivity, and that it is preserved under direct limits.
Finally, the authors consider a condition on a \(C^*\)-algebra B that implies condition (2) of the main theorem, for all ideals J of B. The condition is a variant of one considered by R. J. Archbold and C. J. K. Batty in [J. London Math., II. Ser. 22, 127-138 (1980; Zbl 0437.46049)], and is also a \(C^*\)-analog of local reflexivity. Let B be a unital \(C^*\)-algebra. Then (local reflexivity) for any finite- dimensional operator system E, any completely positive unital map \({\bar \phi}: E\to B^{**}\) can be approximated by completely positive unital maps \(\phi: E\to B\) in the point-weak* topology if and only if, for any \(C^*\)-algebra C, the algebraic tensor product \(B^{**}\otimes C\), under a natural imbedding into \((B\otimes_{\min}C)^{**}\), inherits the minimal norm (Theorem 5.1). These conditions then imply (Proposition 5.3) that for any ideal J in B, kernel (B\(\otimes_{\min}C\to (B/J)\otimes_{\min}C)=J\otimes_{\min}C\), and in fact a partial converse is proven also (Proposition 5.5).
Reviewer: E.Gootman

MSC:
46L05 General theory of \(C^*\)-algebras
46A22 Theorems of Hahn-Banach type; extension and lifting of functionals and operators
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