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Injectivity and decomposition of completely bounded maps. (English) Zbl 0591.46050
Operator algebras and their connections with topology and ergodic theory, Proc. Conf., Buşteni/Rom. 1983, Lect. Notes Math. 1132, 170-222 (1985).
[For the entire collection see Zbl 0562.00005.]
Let CB(A,B) (resp. CP(A,B)) be the set of all linear completely bounded (resp. completely positive) maps from a \(C^*\)-algebra A into a \(C^*\)- algebra B. Let span CP(A,B) be the linear span of CP(A,B). The author proves in a deep sense the striking converse of G. Wittstock’s result that if B is injective, \(CB(A,B)=span CP(A,B)\) for every \(C^*\)- algebra A [J. Funct. Anal. 40, 127-150 (1981; Zbl 0495.46005)].
For \(T\in span CP(A,B)\) denote \(\| T\|_{dec}\) the infinimum of those \(\lambda\geq 0\), for which there exists \(S_ 1,S_ 2\in CP(A,B)\) such that \(\| S_ i\| \leq \lambda\) and the map of A to \(B\otimes M_ 2\), \[ x\to \left( \begin{matrix} S_ 1(x)\\ T(x)\end{matrix} \begin{matrix} T(x^*)^*\\ S_ 2(x)\end{matrix} \right) \] is completely positive. Then the main result (4\(\Rightarrow 1\) of Theorem 2.1) says that a von Neumann algebra N is injective if there exists a constant \(c>0\) such that for every linear map T from \(\ell_ n^{\infty}\) to N, \(\| T\|_{dec}\leq c\| T\|_{cb}\). For a non-injective von Neumann algebra N the article shows that (Theorem 2.6) for every infinite dimensional \(C^*\)-algebra A, there exists a completely bounded map T of A to N which does not belong to span(A,N).
Reviewer: J.Tomiyama

46L05 General theory of \(C^*\)-algebras
46L10 General theory of von Neumann algebras