MULTIPLICATIVITY OF EXTREMAL POSITIVE MAPS ON ABELIAN PARTS OF OPERATOR ALGEBRAS

Abstract. It is shown that finitely many mutually orthogonal pure states on a JB algebra with σ-finite covers restrict simultaneously to pure (i.e. multiplicative) states on some maximal associative JB subalgebra. This re-sult does not hold for any infinite system of orthogonal pure states; a coun-terexample is constructed on any infinite dimensional, separable, irreducible C∗-algebra with non-commutative quotient by the compact operators. Nev-ertheless, 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 A with σ-finite rep-resentation and values in a finite dimensional algebra is multiplicative (even B-morphism) on some maximal abelian subalgebra B of A