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

Abstract

We classify, up to a linear-topological isomorphism, all separableLp-spaces, 1≤p<∞, associated with von Neumann algebras of type I. In particular, anyLp-space associated with an infinite-dimensional atomic von Neumann algebra is isomorphic tolp, or toCp, or to\(Sp = (\sum {_{n = 1}^\infty C_p^n )_{l_p } } \). Further, anyLp-space,p∈[1,∞),p∈2 associated with an infinite-dimensional von Neumann algebraM of type I is isomorphic to one of the following nine Banach spaces: lp, Lp, SP, Cp, Sp ⊕ Lp, Lp(Sp), Cp ⊕ Lp, Lp(Cp), Cp ⊕ Lp(Sp). In the casep=1 all the spaces in this list are pairwise non-isomorphic.