Linear-topological classification of separable L-p-spaces associated with von Neumann algebras of type I

We classify, up to a linear-topological isomorphism, all separable L-p-spac es, 1 less than or equal to p < infinity, associated with von Neumann algeb ras of type I. In particular, any L-p-space associated with an infinite-dim ensional atomic von Neumann algebra is isomorphic to l(p), or to C-p, or to S-p = (Sigma(n=1)(infinity) C-p(n))l(p). Further, any L-p-space, p epsilon [1, infinity), p not equal 2 associated with an infinite-dimensional von N eumann algebra M of type I is isomorphic to one of the following nine Banac h spaces: l(p), L-p, S-p, C-p, S-p circle plus L-p, L-p(S-p), C-p circle plus L-p, L- p(C-p), C-p circle plus L-p(S-p). In the case p = 1 all the spaces in this list are pairwise non-isomorphic.