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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
##### MSC:
 41A36 Approximation by positive operators 46L05 General theory of $$C^*$$-algebras
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##### References:
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