# Journal of Operator Theory

Volume 48, Issue 2, Fall 2002 pp. 369-383.

Multiplicativity of extremal positive maps on abelian parts of operator algebras**Authors**: Jan Hamhalter

**Author institution:**Czech Technical University, Faculty of Electrical Engineering, Department of Mathematics, Technicka 2, 166 27 Prague 6, Czech Republic

**Summary:**It is shown that finitely many mutually orthogonal pure states on a JB algebra with $\sigma$-finite covers restrict simultaneously to pure (i.e. multiplicative) states on some maximal associative JB subalgebra. This result does not hold for any infinite system of orthogonal pure states; a counterexample is constructed on any infinite dimensional, separable, irreducible \C algebra with non-commutative quotient by the compact operators. Nevertheless, under some natural additional conditions the restriction property does hold for all systems of orthogonal pure states. Finally, it is shown that any \C extreme completely positive map on a \C algebra $\p A$ with $\sigma$-finite representation and values in a finite dimensional algebra is multiplicative (even $\p B$-morphism) on some maximal abelian subalgebra $\p B$ of $\p A$.

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