1. S. GiNDiKrN, 'Analysis on homogeneous domains', Russian Math. Surveys 19 (1964) 1-89. 2. M. KOECHER, 'An elementary approach to bounded symmetric domains', Lecture Notes, Rice University, 1969. 3. O. Loos, 'Bounded symmetric domains and Jordan pairs', Lecture Notes, University of California, Irvine, 1977. 4. A. UNTERBERGER and H. UPMEIER, Pseudodifferential analysis on symmetric cones (CRC-P...

This note adds to the recent spate of derivations of the probabilistic apparatus of finite-dimensional quantum theory from various axiomatic packages. We offer two different axiomatic packages that lead easily to the Jordan algebraic structure of finite-dimensional quantum theory. The derivation relies on the Koecher-Vinberg Theorem, which sets up an equivalence between order-unit spaces having...

A three-dimensional polynomial algebra of order m is defined by the commutation relations [P0, P±] = ±P±, [P+, P−] = φ (P0) where φ(P0) is an m-th order polynomial in P0 with the coefficients being constants or central elements of the algebra. It is shown that two given mutually commuting polynomial algebras of orders l and m can be combined to give two distinct (l + m + 1)-th order polynomial ...