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Zbl 0593.41020
Limaye, B.V.; Namboodiri, M.N.N.
Weak approximation by positive maps on $C\sp*$-algebras. (English)
[J] Math. Slovaca 36, 91-99 (1986). ISSN 0139-9918; ISSN 1337-2211/e

Let A and B be $C\sp*$-algebras with units $1\sb A$ and $1\sb B$ respectively. The A ${}\sp*$-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 $\phi (a\sp*a)=b\sp*b.$ For $a\sb 1$, $a\sb 2\in A$ we write $a\sb 1\le a\sb 2$ if there is some $a\in A$ such that $a\sb 2-a\sb 1=a\sp*a.$ We put $P(A,B)\sb 1=\{\phi: A\to B:\phi \quad is\quad positive,\quad \phi (1\sb A)\le \quad 1\sb B\}.$ If $\phi (a)\sp*\phi (a)\le \phi (a\sp*a)$ for all $a\in A$, then $\phi$ is called Schwartz map. A $J\sp*$-subalgebra (resp. $C\sp*$-subalgebra) of A is a ${}\sp*$-subspace which is closed under the Jordan product $a\sb 1\circ a\sb 2=(a\sb 1a\sb 2+a\sb 2a\sb 1)/2$ (resp. the usual product $a\sb 1a\sb 2)$. Let $\phi\sb 0,\phi\sb 1,...,\phi\sb n$, be a sequence in $P(A,B)\sb 1$ and put $C=\{a\in A:\phi\sb n(a)\to \phi\sb 0(a),\phi\sb n(a\sp*\circ a)\to \phi\sb 0(a\sp*\circ a)=\phi\sb 0(a)\sp*\phi\sb 0($ a)$\}$. In this note the author gives sufficient conditions that $A=C$ in terms of extreme points of $P(A,B)\sb 1$, when $B=\beta (H)$ (the $C\sp*$-algebra of bounded linear operators on Hilbert space H). Moreover the author gives several related corollaries.
[J.C.Rho]

MSC 2000:: *41A36 Approximation by positive operators
46L05 General theory of C*-algebras

Keywords: extreme point; positive map; spectrum; weak approximation; $C\sp*$- algebras; Jordan product; Hilbert space

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