Are covering (enveloping) morphisms minimal?

We prove that for certain classes of modules such that direct sums of -covers ( -envelopes) are -covers ( -envelopes), -covering ( -enveloping) homomorphisms are always right (left) minimal. As a particular case we see that over noetherian rings, essential monomorphisms are left minimal. The same type of results are given when direct products of -covers are -covers. Finally we prove that over commutative noetherian rings, any direct product of flat covers of modules of finite length is a flat cover.     

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.

     

(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).

     

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 .

     

Some exact sequences for Toeplitz algebras of spherical isometries

A family of commuting bounded operators on a Hilbert space is said to be a spherical isometry if in the weak operator topology. We show that every commuting family of spherical isometries is jointly subnormal, which means that it has a commuting normal extension on some Hilbert space Suppose now that the normal extension is minimal. Then we show that every bounded operator in the commutant of has a unique norm preserving extension to an operator in the commutant of Moreover, if is the commutator ideal in then is *-isomorphic to We also show that the commutant of the minimal normal extension is completely isometric, via the compression mapping, to the space of Toeplitz-type operators associated to We apply these results to construct exact sequences for Toeplitz algebras on generalized Hardy spaces associated to strictly pseudoconvex domains.

     

A dichotomy theorem for subsets of the power set of the natural numbers

The following dichotomy is established for any pair , of hereditary families of finite subsets of : Given , an infinite subset of , there exists an infinite subset of so that either , or , where denotes the set of all finite subsets of .

     

On Burch's inequality and a reduction system of a filtration

Let be a multiplicative filtration of a local ring such that the Rees algebra is Noetherian. We recall Burch's inequality for and give an upper bound of the a-invariant of the associated graded ring using a reduction system of . Applying those results, we study the symbolic Rees algebra of certain ideals of dimension .

     

The Dirichlet-Jordan test and multidimensional extensions

If is a foliation of an open set by smooth -dimensional surfaces, we define a class of functions , supported in , that are, roughly speaking, smooth along and of bounded variation transverse to . We investigate geometrical conditions on that imply results on pointwise Fourier inversion for these functions. We also note similar results for functions on spheres, on compact 2-dimensional manifolds, and on the 3-dimensional torus. These results are multidimensional analogues of the classical Dirichlet-Jordan test of pointwise convergence of Fourier series in one variable.

     

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.

     

Metrically generated theories

Many examples are known of natural functors describing the transition from categories of generalized metric spaces to the ``metrizable" objects in some given topological construct . If preserves initial morphisms and if is initially dense in , then we say that is -metrically generated. Our main theorem proves that is -metrically generated if and only if can be isomorphically described as a concretely coreflective subconstruct of a model category with objects sets structured by collections of generalized metrics in and natural morphisms. This theorem allows for a unifying treatment of many well-known and varied theories. Moreover, via suitable comparison functors, the various relationships between these theories are studied.

     

Minimal rank and reflexivity of operator spaces

Let be an -dimensional space of linear operators between the linear spaces and over an algebraically closed field . Improving results of Larson, Ding, and Li and Pan we show the following.

Theorem. Let be a basis of . Assume that every nonzero operator in has rank larger than . Then a linear operator belongs to if and only if for every , is a linear combination of .

     

A non-Hausdorff model for the complement of a complexified hyperplane arrangement

Given a hyperplane arrangement in a real vector space , we introduce a real algebraic prevariety , and exhibit the complement of in the complexification of as the total space of an affine bundle over with fibers modeled on the dual vector space .

     

On an approximate automorphism on a -algebra

It is shown that for an approximate algebra homomorphism on a Banach -algebra , there exists a unique algebra -homomorphism near the approximate algebra homomorphism. This is applied to show that for an approximate automorphism on a unital -algebra , there exists a unique automorphism near the approximate automorphism. In fact, we show that the approximate automorphism is an automorphism.

     

Sobolev spaces and the Cayley transform

The generalised Cayley transform  from an Iwasawa -group into the corresponding real unit sphere  induces isomorphisms between suitable Sobolev spaces and . We study the differential of  , and we obtain a criterion for a function to be in  .

     

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 .

     

Projective boundedness and convolution of Fréchet measures

Fréchet measures of order ( -measures) are the measure-theoretic analogues of bounded -linear forms on products of spaces. In an LCA setting, convolution of -measures is always defined, while there exist -measures whose convolution cannot be defined. In a three-dimensional setting, we demonstrate the existence of an -measure which cannot be convolved with arbitrary -measures.

     

An order characterization of commutativity for -algebras

In this paper, we investigate the problem of when a -algebra is commutative through operator-monotonic increasing functions. The principal result is that the function is operator-monotonic increasing on a -algebra if and only if is commutative. Therefore, -algebra is commutative if and only if in for all positive elements in .

     

Simple real rank zero algebras with locally Hausdorff spectrum

Let be a unital, simple, separable -algebra with real rank zero, stable rank one, and weakly unperforated ordered group. Suppose, also, that can be locally approximated by type I algebras with Hausdorff spectrum and bounded irreducible representations (the bound being dependent on the local approximating algebra). Then is tracially approximately finite dimensional (i.e., has tracial rank zero).

Hence, is an -algebra with bounded dimension growth and is determined by -theoretic invariants.

The above result also gives the first proof for the locally case.

     

A note on invertibility preservers on Banach algebras

Let be be semisimple Banach algebras and let be a unital bijective linear operator that preserves invertibility. If the socle of is an essential ideal of , then is a Jordan isomorphism.