Single elements and module isomorphisms of some operator algebra modules

In this paper, we first introduce the concept of single elements in a module. A systematic study of single elements in the Alg-module is initiated, where is a completely distributive subspace lattice on a Hilbert space . Furthermore, as an application of single elements, we study module isomorphisms between norm closed Alg-modules, where is a nest, and obtain the following result: Suppose that are norm closed Alg-modules and that is a module isomorphism. Then and there exists a non-zero complex number such that .

     

Algebraic isomorphisms and -subspace lattices

The class of -lattices was originally defined in the second author's thesis and subsequently by Longstaff, Nation, and Panaia. A subspace lattice on a Banach space which is also a -lattice is called a -subspace lattice, abbreviated JSL. It is demonstrated that every single element of has rank at most one. It is also shown that has the strong finite rank decomposability property. Let and be subspace lattices that are also JSL's on the Banach spaces and , respectively. The two properties just referred to, when combined, show that every algebraic isomorphism between and preserves rank. Finally we prove that every algebraic isomorphism between and is quasi-spatial.

     

Segal-Bargmann transforms of one-mode interacting Fock spaces associated with Gaussian and Poisson measures

Let and denote the Gaussian and Poisson measures on , respectively. We show that there exists a unique measure on such that under the Segal-Bargmann transform the space is isomorphic to the space of analytic -functions on with respect to . We also introduce the Segal-Bargmann transform for the Poisson measure and prove the corresponding result. As a consequence, when and have the same variance, and are isomorphic to the same space under the - and -transforms, respectively. However, we show that the multiplication operators by on and on act quite differently on .

     

The -indices of Tsirelson type spaces

If and are countable ordinals such that , denote by the completion of with respect to the implicitly defined norm

where the supremum is taken over all finite subsets of such that and . It is shown that the Bourgain -index of is . In particular, if \alpha =\omega^{\alpha_{1}}\cdot m_{1}+\dots+\omega^{\alpha_{n}}\cdot m_{n}$"> in Cantor normal form and is not a limit ordinal, then there exists a Banach space whose -index is .

     

(APD)-property of -algebras by extensions of -algebras with (APD)

A unital -algebra is said to have the (APD)-property if every nonzero element in has the approximate polar decomposition. Let be a closed ideal of . Suppose that and have (APD). In this paper, we give a necessary and sufficient condition that makes have (APD). Furthermore, we show that if and or is a simple purely infinite -algebra, then has (APD).

     

The -invariant and Gorensteinness of graded rings associated to filtrations of ideals in regular local rings

Let be a regular local ring and let be a filtration of ideals in such that is a Noetherian ring with . Let and let be the -invariant of . Then the theorem says that is a principal ideal and for all if and only if is a Gorenstein ring and . Hence , if is a Gorenstein ring, but the ideal is not principal.

     

Norming algebras and automatic complete boundedness of isomorphisms of operator algebras

We combine the notion of norming algebra introduced by Pop, Sinclair and Smith with a result of Pisier to show that if and are operator algebras, then any bounded epimorphism of onto is completely bounded provided that contains a norming -subalgebra. We use this result to give some insights into Kadison's Similarity Problem: we show that every faithful bounded homomorphism of a -algebra on a Hilbert space has completely bounded inverse, and show that a bounded representation of a -algebra is similar to a -representation precisely when the image operator algebra -norms itself. We give two applications to isometric isomorphisms of certain operator algebras. The first is an extension of a result of Davidson and Power on isometric isomorphisms of CSL algebras. Secondly, we show that an isometric isomorphism between subalgebras of -diagonals () satisfying extends uniquely to a -isomorphism of the -algebras generated by and ; this generalizes results of Muhly-Qiu-Solel and Donsig-Pitts.

     

Hyperbolic unit groups

In this paper we study the groups whose integral group rings have hyperbolic unit groups . We classify completely the torsion subgroups of and the polycyclic-by-finite subgroups of the group . Finally, we classify the groups for which the boundary of has dimension zero.

     

Degenerating families of rank two bundles

We construct families of rank two bundles on , in characteristic two, where for , is a sum of line bundles, and is non-split. We construct families of rank two bundles on , in characteristic , where for , is a sum of line bundles, and is non-split.

     

A proof of Weinberg's conjecture on lattice-ordered matrix algebras

Let be a subfield of the field of real numbers and let () be the matrix algebra over . It is shown that if is a lattice-ordered algebra over in which the identity matrix 1 is positive, then is isomorphic to the lattice-ordered algebra with the usual lattice order. In particular, Weinberg's conjecture is true.

     

Enriched Reedy categories

We define the notion of an enriched Reedy category and show that if is a -Reedy category for some symmetric monoidal model category and is a -model category, the category of -functors and -natural transformations from to is again a model category.

     

Minimal sufficiency of order statistics in convex models

Let be a convex and dominated statistical model on the measurable space , with minimal sufficient, and let . Then , the -algebra of all permutation invariant sets belonging to the -fold product -algebra , is shown to be minimal sufficient for the corresponding model for independent observations, .

The main technical tool provided and used is a functional analogue of a theorem of Grzegorek (1982) concerning generators of .

     

The structure of quantum spheres

We show that the C*-algebra of a quantum sphere , 1$">, consists of continuous fields of operators in a C*-algebra , which contains the algebra of compact operators with , such that is a constant function of , where is the quotient map and is the unit circle.

     

Spaces of type and the Hankel convolution

In this paper we introduce new function spaces that are denoted by , -1/2$"> and and that are spaces of type where the Hankel convolution and the Hankel transformation are defined. The spaces will play the same role in the Hankel setting that the spaces play in the theory of Fourier transformation.     

Isolated invariant curves of a foliation

We bound the equisingularity type of the set of isolated separatrices of a holomorphic foliation of in terms of the Milnor number of . This result gives a bound for the degree of an algebraic invariant curve of a foliation of in terms of the degree of , provided that all the branches of are isolated separatrices.

     

Games and general distributive laws in Boolean algebras

The games and are played by two players in -complete and max -complete Boolean algebras, respectively. For cardinals such that or , the -distributive law holds in a Boolean algebra iff Player 1 does not have a winning strategy in . Furthermore, for all cardinals , the -distributive law holds in iff Player 1 does not have a winning strategy in . More generally, for cardinals such that , the -distributive law holds in iff Player 1 does not have a winning strategy in . For regular and , implies the existence of a Suslin algebra in which is undetermined.

     

Some finiteness conditions on the set of overrings of an integral domain

Let be an integral domain with quotient field and integral closure . An overring of is a subring of containing , and denotes the set of overrings of . We consider primarily two finiteness conditions on : (FO), which states that is finite, and (FC), the condition that each chain of distinct elements of is finite. (FO) is strictly stronger than (FC), but if , each of (FO) and (FC) is equivalent to the condition that is a Prüfer domain with finite prime spectrum. In general satisfies (FC) iff satisfies (FC) and all chains of subrings of containing have finite length. The corresponding statement for (FO) is also valid.

     

Menger curvature and regularity of fractals

We show that if is an -regular set in for which the triple integral of the Menger curvature is finite and if , then almost all of can be covered with countably many curves. We give an example to show that this is false for .

     

The Russo-Dye Theorem in a weakly closed -module

Suppose that is admissible. It is shown that the convex hull of unitary elements of a weakly closed -module contains the whole unit ball of if and only if and for any 0$">, 0$">.

     

Vector measure Banach spaces containing a complemented copy of

Let a Banach space and a -algebra of subsets of a set . We say that a vector measure Banach space has the bounded Vitaly-Hahn-Sacks Property if it satisfies the following condition: Every vector measure , for which there exists a bounded sequence in verifying for all , must belong to . Among other results, we prove that, if is a vector measure Banach space with the bounded V-H-S Property and containing a complemented copy of , then contains a copy of .