Given a real separable Hilbert space H, we denote with G(H) the geometry of closed linear subspaces of H.The strong convergence of sequences of subspaces is shown to be a L*-convergence and the weak convergence a L-convergence.The smallest L*-convergence containing the weak convergence is found, and the orthogonal image of the strong convergence, which is also a L*-convergence, is defined.
Let E be a Riesz space. By defining the spaces and of E, we prove that the center of is and show that the injectivity of the Arens homomorphism m: Z(E)” → Z(E˜) is equivalent to the equality . Finally, we also give some representation of an order continuous Banach lattice E with a weak unit and of the order dual E˜ of E in which are different from the representations appearing in the literature.
Let W be a Coxeter group and let μ be an inner product on the group algebra ℝW. We say that μ is admissible if it satisfies the axioms for a Hilbert algebra structure. Any such inner product gives rise to a von Neumann algebra containing ℝW. Using these algebras and the corresponding von Neumann dimensions we define -Betti numbers and an -Euler charactersitic for W. We show that if the Davis complex for W is a generalized homology manifold, then these Betti numbers satisfy a version of Poincaré...
We construct the category of quotients of -spaces and we show that it is Abelian. This answers a question of L. Waelbroeck from 1990.