Bimodules Over Cartan Subalgebras

Given a Cartan subalgebra A of a non Neumann algebra M, the techniques of Feldman and Moore are used to analyze the partial isometries v in M such that v * Av is contained in A. Orthonormal bases for M consisting of such partial isometries are discussed, and convergence of the resulting generalized fourier series is shown to take place in the Bures A-topology. The Bures A-topology is shown to be equivalent to the strong topology on the unit ball of M. These ideas are applied to A-bimodules and to give a simplified and intuitive proof of the Spectral Theorem for Bimodules first proven by Muhly, Saito, and Solel.