Contractive projections and operator spaces

Parallel to the study of finite dimensional Banach spaces, there is a growing interest in the corresponding local theory of operator spaces. We define a family of Hilbertian operator spaces H_n^k,0< k < n+1, generalizing the row and column Hilbert spaces R_n,C_n and show that an atomic subspace X of B(H) which is the range of a contractive projection on B(H) is isometrically completely contractive to a direct sum of the H_n^k and Cartan factors of types 1 to 4. In particular, for finite dimensional X, this answers a question posed by Oikhberg and Rosenthal. Explicit in the proof is a classification up to complete isometry of atomic w*-closed JW*-triples without an infinite dimensional rank 1 w^*-closed idealComment: 40 pages, latex, the paper was submitted in October of 2000 and an announcement with the same title appeared in C. R. Acad. Sci. Paris 331 (2000), 873-87

Operator space characterizations of C*-algebras and ternary rings

We prove that an operator space is completely isometric to a ternary ring of operators if and only if the open unit balls of all of its matrix spaces are bounded symmetric domains. From this we obtain an operator space characterization of C*-algebras.Comment: 20 pages, latex, submitted in November 200

State spaces of JB*-triples

An atomic decomposition is proved for Banach spaces which satisfy some affine geometric axioms compatible with notions from the quantum mechanical measuring process. This is then applied to yield, under appropriate assumptions, geometric characterizations, up to isometry, of the unit ball of the dual space of a JB*-triple, and up to complete isometry, of one-sided ideals in C*-algebras.Comment: 28 page

Cohomology of Jordan triples via Lie algebras

We develop a cohomology theory for Jordan triples, including the infinite dimensional ones, by means of the cohomology of TKK Lie algebras. This enables us to apply Lie cohomological results to the setting of Jordan triples. Some preliminary results for von Neumann algebras are obtained.Comment: 27 pages, to appear in: Topics in Functional Analysis and Algebra, Proceedings of a special session of the USA-Uzbekistan Conference on Analysis and Mathematical Physics, CSU Fullerton, May 20-23, 2014, Contemporary Mathematics, 201