Functional Analysis

In 1979, B.Bjornestal obtained local estimate for a modulus of uniform continuity of metric projection operator on closed subspace in uniformly convex and uniformly smooth Banach space B . In the present paper we give the global version of this result for the projection operator on an arbitrary closed convex set in B .

The conjecture about relation between infinite-dimensional Grassmannian and string theory is based on the fact that moduli spaces of algebraic curves are embedded into Grassmannian via Krichever construction. We describe a multidimensional analog of Krichever construction, that can be used in the attempt to relate Grassmannian to membranes.

We consider a homogeneous space X=(X,d,m) of dimension ν≥1 and a local regular Dirichlet form in L^{2}(X,m). We prove that if a Poincaré inequality holds on every pseudo-ball B(x,R) of X, with local characteristic constant c_{0}(x) and c_{1}(r), then a Green's function estimate from above and below is obtained. A Saint-Venant-like principle is recovered in terms of the Energy's decay.

The Bisognano-Wichmann property on the geometric behavior of the modular group of the von Neumann algebras of local observables associated to wedge regions in Quantum Field Theory is shown to provide an intrinsic sufficient criterion for the existence of a covariant action of the (universal covering of) the Poincaré group. In particular this gives, together with our previous results, an intrinsic characterization of positive-energy conformal pre-cosheaves of von Neumann algebras. To this end we adapt to our use Moore theory of central extensions of locally compact groups by polish groups, selecting and making an analysis of a wider class of extensions with natural measurable properties and showing henceforth that the universal covering of the Poincaré group has only trivial central extensions (vanishing of the first and second order cohomology) within our class.

We consider a homogeneous space X=(X,d,m) of dimension ν≥1 and a local regular Dirichlet form in L 2 (X,m). We prove that if a Poincaré inequality holds on every pseudo-ball B(x,R) of X , then an Harnack's inequality can be proved on the same ball with local characteristic constant c 0 and c 1

The connection between polynomial solutions of finite-difference equations and finite-dimensional representations of the s l 2 -algebra is established (the talk given at the Wigner Symposium, Guadalajara, Mexico, August 1995, to be published in the Proceedings)

A study of Hilbert C ∗ -bimodules over commutative C ∗ -algebras is carried out and used to establish a sufficient condition for two quantum Heisenberg manifolds to be isomorphic.

The overview contains 450 references of books, chapters of monographs, papers, preprints and Ph.~D.~thesises which are concerned with the theory and/or various applications of Hilbert C*-modules. To show a way through this amount of literature a four pages guide is added clustering sources around major research problems and research fields, and giving information on the historical background. Two smaller separate parts list references treating Hilbert modules over Hilbert*-algebras and Hilbert modules over (non-self-adjoint) operator algebras. Any additions, corrections and forthcoming information are welcome.

The aim of the present paper is to describe self-duality and C*- reflexivity of Hilbert {\bf A}-modules M over monotone complete C*-algebras {\bf A} by the completeness of the unit ball of M with respect to two types of convergence being defined, and by a structural criterion. The derived results generalize earlier results of {\sc H.~Widom} [Duke Math.~J.~23, 309-324, MR 17 \# 1228] and {\sc W.~L.~Paschke} [Trans. Amer.~Math.~Soc.~182, 443-468, MR 50 \# 8087, Canadian J.~Math.~26, 1272-1280, MR 57 \# 10433]. For Hilbert C*-modules over commutative AW*-algebras the equivalence of the self-duality property and of the Kaplansky-Hilbert property is reproved, (cf. {\sc M.~Ozawa} [J.~Math.~Soc.~Japan 36, 589-609, MR 85m:46068] ). Especially, one derives that for a C*-algebra {\bf A} the {\bf A}-valued inner pro\-duct of every Hilbert {\bf A}-module M can be continued to an {\bf A}-valued inner product on it's {\bf A}-dual Banach {\bf A}-module M ' turning M ' to a self-dual Hilbert {\bf A}-module if and only if {\bf A} is monotone complete (or, equivalently, additively complete) generalizing a result of {\sc M.~Hamana} [Internat.~J.~Math.~3(1992), 185-204]. A classification of countably generated self-dual Hilbert {\bf A}-modules over monotone complete C*-algebras {\bf A} is established. The set of all bounded module operators End A (M) on self-dual Hilbert {\bf A}-modules M over monotone complete C*-algebras {\bf A} is proved again to be a monotone complete

The two reference lists contain 54/22 references of papers and preprints concerned with the theory and/or various applications of Hilbert modules over Hilbert ∗ -algebras and over (non-self-adjoint) operator algebras. They are far from being complete, but they give additional information about two research fields which are closely related to the theory of Hilbert C*-modules, i.e. they are complements to the reference guide about this circle of problems. Any additions, corrections and forthcoming information are welcome.