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Weak approximation by positive maps on \(C^*\)-algebras. (English) Zbl 0593.41020
Let A and B be \(C^*\)-algebras with units \(1_ A\) and \(1_ B\) respectively. The A \({}^*\)-linear map \(\phi\) :A\(\to B\) is said to be positive if for every \(a\in A\), there is some \(b\in B\) such that \(\phi (a^*a)=b^*b.\) For \(a_ 1\), \(a_ 2\in A\) we write \(a_ 1\leq a_ 2\) if there is some \(a\in A\) such that \(a_ 2-a_ 1=a^*a.\) We put \(P(A,B)_ 1=\{\phi: A\to B:\phi \quad is\quad positive,\quad \phi (1_ A)\leq \quad 1_ B\}.\) If \(\phi (a)^*\phi (a)\leq \phi (a^*a)\) for all \(a\in A\), then \(\phi\) is called Schwartz map. A \(J^*\)-subalgebra (resp. \(C^*\)-subalgebra) of A is a \({}^*\)-subspace which is closed under the Jordan product \(a_ 1\circ a_ 2=(a_ 1a_ 2+a_ 2a_ 1)/2\) (resp. the usual product \(a_ 1a_ 2)\). Let \(\phi_ 0,\phi_ 1,...,\phi_ n\), be a sequence in \(P(A,B)_ 1\) and put \(C=\{a\in A:\phi_ n(a)\to \phi_ 0(a),\phi_ n(a^*\circ a)\to \phi_ 0(a^*\circ a)=\phi_ 0(a)^*\phi_ 0(\) a)\(\}\). In this note the author gives sufficient conditions that \(A=C\) in terms of extreme points of \(P(A,B)_ 1\), when \(B=\beta (H)\) (the \(C^*\)-algebra of bounded linear operators on Hilbert space H). Moreover the author gives several related corollaries.
Reviewer: J.C.Rho
41A36 Approximation by positive operators
46L05 General theory of \(C^*\)-algebras
Full Text: EuDML
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