Continuous-trace groupoid -algebras. III

Suppose that is a second countable locally compact groupoid with a Haar system and with abelian isotropy. We show that the groupoid -algebra has continuous trace if and only if there is a Haar system for the isotropy groupoid and the action of the quotient groupoid is proper on the unit space of .

     

On -algebras associated with locally compact groups

Let be a locally compact group, and let denote the same group with the discrete topology. There are various associated to and We are concerned with the question of when these are isomorphic. This is intimately related to amenability. The results can be reformulated in terms of Fourier and Fourier-Stieltjes algebras and of weak containment properties of unitary representations.

     

Liouville-Green asymptotic approximation for a class of matrix differential equations and semi-discretized partial differential equations

A Liouville-Green (or WKB) asymptotic approximation theory is developed for the class of linear second-order matrix differential equations Y^″=[f(t)A+G(t)]Y on [a,+∞), where A and G(t) are matrices and f(t) is scalar. This includes the case of an “asymptotically constant” (not necessarily diagonalizable) coefficient A (when f(t)≡1). An explicit representation for a basis of the right-module of solutions is given, and precise computable bounds for the error terms are provided. The double asymptotic nature with respect to both t and some parameter entering the matrix coefficient is also shown. Several examples, some concerning semi-discretized wave and convection-diffusion equations, are given.     

-algebras generated by a subnormal operator

Using the functional calculus for a normal operator, we provide a result for generalized Toeplitz operators, analogous to the theorem of Axler and Shields on harmonic extensions of the disc algebra. Besides that result, we prove that if is an injective subnormal weighted shift, then any two nontrivial subspaces invariant under cannot be orthogonal to each other. Then we show that the -algebra generated by and the identity operator contains all the compact operators as its commutator ideal, and we give a characterization of that -algebra in terms of generalized Toeplitz operators. Motivated by these results, we further obtain their several-variable analogues, which generalize and unify Coburn's theorems for the Hardy space and the Bergman space of the unit ball.

     

The connected stable rank of the purely infinite simple -algebras

Suppose that is a unital purely infinite simple -algebra. If the class [1] of the unit 1 in has torsion, then ; if [1] is torsion-free in , then . If is a non-unital purely infinite simple -algebra, then .

     

Equivalence of norms on operator space tensor products of -algebras

The Haagerup norm on the tensor product of two -algebras and is shown to be Banach space equivalent to either the Banach space projective norm or the operator space projective norm if and only if either or is finite dimensional or and are infinite dimensional and subhomogeneous. The Banach space projective norm and the operator space projective norm are equivalent on if and only if or is subhomogeneous.

     

Refinement monoids with weak comparability and applications to regular rings and -algebras

We prove a cancellation theorem for simple refinement monoids satisfying the weak comparability condition, first introduced by K.C. O'Meara in the context of von Neumann regular rings. This result is then applied to von Neumann regular rings and -algebras of real rank zero via the monoid of isomorphism classes of finitely generated projective modules.

     

On the definition of real -algebras

We prove that, if is a real -algebra having a predual , then is the unique predual of and the product of is -continuous.

     

On nonlinear -widths

For characterization of best nonlinear approximation, DeVore,
Howard, and Micchelli have recently suggested the nonlinear -width of a subset in a normed linear space . We proved by a topological method that for and the well-known Aleksandrov -width in a Banach space the following inequalities hold: . Let be the unit ball of Besov space , of multivariate periodic functions. Then for approximation in , with some restriction on and , we established the asymptotic degree of these -widths: .

     

A -functional and the rate of convergence of some linear polynomial operators

We show that the -functional

where , is equivalent to the rate of convergence of a certain linear polynomial operator. This operator stems from a Riesz-type summability process of expansion by Legendre polynomials. We use the operator above to obtain a linear polynomial approximation operator with a rate comparable to that of the best polynomial approximation.

     

Semi-discretization of stochastic partial differential equations on by a finite-difference method

The paper concerns finite-difference scheme for the approximation of partial differential equations in , with additional stochastic noise. By replacing the space derivatives in the original stochastic partial differential equation (SPDE, for short) with difference quotients, we obtain a system of stochastic ordinary differential equations. We study the difference between the solution of the original SPDE and the solution to the corresponding equation obtained by discretizing the space variable. The need to approximate the solution in with functions of compact support requires us to introduce a scale of weighted Sobolev spaces. Employing the weighted -theory of SPDE, a sup-norm error estimate is derived and the rate of convergence is given.

     

over number fields and function fields

For the number field case we will give an upper bound on the number of the -integral points in
. The main tool here is the explicit upper bound of the number of solutions of -unit equations (Invent. Math. 102 (1990), 95--107). For the function field case we will give a bound on the height of the -integral points in . We will also give a bound for the number of ``generators" of those -integral points. The main tool here is the -unit Theorem by Brownawell and Masser (Proc. Cambridge Philos. Soc. 100 (1986), 427--434).

     

Determining the small solutions to -unit equations

In this paper we generalize the method of Wildanger for finding small solutions to unit equations to the case of -unit equations. The method uses a minor generalization of the LLL based techniques used to reduce the bounds derived from transcendence theory, followed by an enumeration strategy based on the Fincke-Pohst algorithm. The method used reduces the computing time needed from MIPS years down to minutes.

     

Probability measures in -algebras in Hilbert spaces with conjugation

Let be a real -algebra of -real bounded operators containing no central summand of type in a complex Hilbert space with conjugation . Denote by the quantum logic of all -orthogonal projections in the von Neumann algebra . Let be a probability measure. It is shown that contains a finite central summand and there exists a normal finite trace on such that , .

     

The Toda molecule equation and the -algorithm

One of the well-known convergence acceleration methods, the -algorithm is investigated from the viewpoint of the Toda molecule equation. It is shown that the error caused by the algorithm is evaluated by means of solutions for the equation. The acceleration algorithm based on the discrete Toda molecule equation is also presented.

     

Products of images

Let be the \u{C}ech-Stone remainder . We show that there exists a large class of images of such that whenever is a subset of of cardinality at most the continuum, then is again an image of . The class contains all separable compact spaces, all compact spaces of weight at most and all perfectly normal compact spaces.

     

On cardinal invariants for CCC -ideals

We show several results about cardinal invariants for -ideals of the reals. In particular we show that for every CCC -ideal on the real line .

     

The Furuta inequality in Banach -algebras

Let be real numbers with and Furuta (1987) proved that if bounded linear operators on a Hilbert space satisfy , then . This inequality is called the Furuta inequality and has many applications. In this paper, we prove that the Furuta inequality holds in a unital hermitian Banach -algebra with continuous involution.