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 $\mathcal A$ with $\sigma$-finite representation and values in a finite dimensional algebra is multiplicative $($even $\mathcal B$-morphism$)$ on some maximal abelian subalgebra $\mathcal B$ of $\mathcal A$.

Keywords: Pure states on JB and $C^*$ algebras, $C^*$ extreme completely positive maps, restriction property

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