Universal state space embeddability of Jordan-Banach algebras

Abstract

We study extensions of states between projection structures of JB algebras and generalized orthomodular posets. It is shown that projection orthoposet of a JB algebra A admits the universal extension property if and only if the Gleason theorem is valid for A. As a consequence we get that any positive Stone algebra-valued measure on projection lattice of a quotient of a JBW algebra without type I2 direct summand extends to a positive measure on an arbitrary larger generalized orthomodular lattice. The classical extension theorem of Horn and Tarski [5] says that any probability measure on a Boolean algebra B extends to a probability measure on any larger Boolean algebra B' containing B. The first measure theoretic generalization of this result was given by P. Ptak [10] who showed that B' can be replaced by any quantum logic L with reasonably rich state space. We say in this case that Boolean algebras have the universal state extension property. In further development it has been proved in [6] that the universal state extension property for projection structures of unital C*-algebras is equivalent with the existence of non-commutative integral (so called Gleason property). As a consequence we get that projection lattices of von Neumann algebras without type I2 direct summand enjoy the universal state extension property. The aim of this note is to extend the above stated results for (not necessarily unital) Jordan Banach algebras and corresponding orthomodular structures not containing a largest element. This generalization is based on extension technique using determinacy of pure states on Jordan algebras. Moreover, general extension theorem for complete vector space valued measures on projections will be proved. First we recall a few notions and fix the notation. (Our standard references for Jordan algebras, ordered vector spaces, and quantum logics are [7, 1, 11], correspondingly.) * By a Jordan algebra we shall mean a real vector space A equipped with the product (i.e. bilinear form) written as (a, b) -? a o b and satisfying the following properties for all a, b E A: a o b = b o a, a o (bo a 2) = (aob) oa 2. A is said to be urnital if it admits a unit with respect to the Jordan product. In addition, if A is a Banach algebra with respect to the product o and if the norm I I satisfies the following conditions for all a,b E A: IIa211 < Ila2 + b211, Received by the editors May 1, 1997. 1991 Mathematics Subject Classification. Primary 46L70, 46L50, 28B15, 81P10.