We consider the generalized Segal-Bargmann transform C t for a compact group K; introduced in Hall (J. Funct. Anal. 122 (1994) 103). Let K C denote the complexification of K: We give a necessary-and-sufficient pointwise growth condition for a holomorphic function on K C to be in the image under C t of C N ðKÞ: We also characterize the image under C t of Sobolev spaces on K: The proofs make use of a holomorphic version of the Sobolev embedding theorem.

Abstract. This paper contains the first written exposition of some ideas (announced in a previous survey) on an approach to quantum gravity based on Tomita-Takesaki modular theory and A. Connes non-commutative geometry aiming at the reconstruction of spectral geometries from an operational formalism of states and categories of observables in a covariant theory. Care has been taken to provide a coverage of the relevant background on modular theory, its applications in non-commutative geometry and physics and to the detailed discussion of the main foundational issues raised by the proposal.

Abstract. The purpose of this short note is to outline the current status of some recent research programs aiming at a categorification of parts of A.Connes non-commutative geometry and to provide an outlook on some possible future developments in categorical non-commutative geometry.

In the setting of C*-categories, we provide a definition of spectrum of a commutative full C*-category as a one-dimensional unital Fell bundle over a suitable groupoid (equivalence relation) and prove a categorical Gel'fand duality theorem generalizing the usual Gel'fand duality between the categories of commutative unital C*-algebras and compact Hausdorff spaces. Although many of the individual ingredients that appear along the way are well known, the somehow unconventional way we "glue" them together seems to shed some new light on the subject.

We derive an iterative procedure for solving a generalized Sylvester matrix equation $AXB+CXD = E$ A X B + C X D = E , where $A,B,C,D,E$ A , B , C , D , E are conforming rectangular matrices. Our algorithm is based on gradients and hierarchical identification principle. We convert the matrix iteration process to a first-order linear difference vector equation with matrix coefficient. The Banach contraction principle reveals that the sequence of approximated solutions converges to the exact solution for any initial matrix if and only if the convergence factor belongs to an open interval. The contraction principle also gives the convergence rate and the error analysis, governed by the spectral radius of the associated iteration matrix. We obtain the fastest convergence factor so that the spectral radius of the iteration matrix is minimized. In particular, we obtain iterative algorithms for the matrix equation $AXB=C$ A X B = C , the Sylvester equation, and the Kalman–Yakubovich equation. We give numerical experiments of the proposed algorithm to illustrate its applicability, effectiveness, and efficiency.