\(W^ *\)-categories (Q1086795)

A \(W^*\)-category is the categorical counterpart of a von Neumann algebra with an abstract definition equivalent to a concrete definition in terms of operators between Hilbert spaces. We develop the elementary theory of \(W^*\)-categories including modular theory and the comparison theory of objects (equivalence and quasi-equivalence). We also characterize certain \(W^*\)-categories in terms of the \(W^*\)-category of projections in a von Neumann algebra, self-dual Hermitian modules for a von Neumann algebra or normal representations of a von Neumann algebra. This leads naturally to a discussion of the Morita equivalence of von Neumann algebras and of \(W^*\)-categories.