Inner Ideals and Facial Structure of the Quasi-State Space of a JB-Algebra

In this paper we relate the geometric structure of the quasi-state space A*1, + of a JB-algebra A to the algebraic structure of A. We prove that each non-empty w*-closed face F of A*1, + coincides with the set of elements of A* that attain their norms at a closed projection p in A**, uniquely determined by F. When A is unital, we establish an order anti- isomorphism between the complete lattice of w*-closed faces of A*1, + and the complete lattice of norm-closed inner ideals of A. These results are known for the special case when A is the self-adjoint part of a C*-algebra. The proofs use associative techniques and the Gelfand–Naimark representation. In the general case such techniques are not available, and it is necessary to introduce new techniques including the use of a new inequality of Cauchy–Schwarz type for elements of A*.