On stable equivalences induced by exact functors

Let and be two Artin algebras with no semisimple summands. Suppose that there is a stable equivalence between and such that is induced by exact functors. We present a nice correspondence between indecomposable modules over and . As a consequence, we have the following: (1) If is a self-injective algebra, then so is ; (2) If and are finite dimensional algebras over an algebraically closed field , and if is of finite representation type such that the Auslander-Reiten quiver of has no oriented cycles, then and are Morita equivalent.

     

Multiplication and division by inner functions in the space of Bloch functions

A subspace of the Hardy space is said to have the -property if whenever and is an inner function with . We let denote the space of Bloch functions and the little Bloch space. Anderson proved in 1979 that the space does not have the -property. However, the question of whether or not ( ) has the -property was open. We prove that for every the space does not have the -property.

We also prove that if is any infinite Blaschke product with positive zeros and is a Bloch function with , as , then the product is not a Bloch function.

     

Cuntz-Pimnser algebras, completely positive maps and Morita equivalence

Let be a completely positive map on and let be the associated GNS--correspondence. We prove a result that implies, in particular, that the Cuntz-Pimsner algebra of , , is strongly Morita equivalent to the Cuntz algebra , where is the index of .

     

Failure of Krull-Schmidt for invertible lattices over a discrete valuation ring

Let be a prime greater than , and let be the semi-direct product of a group of order by a cyclic group of order , which acts faithfully on . Let be the localization of at . We show that the Krull-Schmidt Theorem fails for the category of invertible -lattices.

     

Localization of injective modules over valuation rings

It is proved that is injective if is an injective module over a valuation ring , for each prime ideal . Moreover, if or is flat, then is injective, too. It follows that localizations of injective modules over h-local Prüfer domains are injective, too.

     

Functional calculus and -regularity of a class of Banach algebras

Suppose that is a -dynamical system such that is of polynomial growth. If is finite dimensional, we show that any element in has slow growth and that is -regular. Furthermore, if is discrete and is a ``nice representation' of , we define a new Banach -algebra which coincides with when is finite dimensional. We also show that any element in has slow growth and is -regular.

     

Witt kernels of bilinear forms for algebraic extensions in characteristic

Let be a field of characteristic and let be a purely inseparable extension of exponent . We determine the kernel of the natural restriction map between the Witt rings of bilinear forms of and , respectively. This complements a result by Laghribi who computed the kernel for the Witt groups of quadratic forms for such an extension . Based on this result, we will determine for a wide class of finite extensions which are not necessarily purely inseparable.

     

A stability property for linear groups

For suitable rings of integers , we show that the mod group cohomology for comes from when restricted to the diagonal matrices for all ranks .

     

Eventual arm and leg widths in cocharacters of P. I. Algebras

Given a p.i. algebra , we study which partitions correspond to characters with non-zero multiplicities in the cocharacter sequence of . We define the , the eventual arm width to be the maximal so that such can have parts arbitrarily large, and to be the maximum so that the conjugate could have arbitrarily large parts. Our main result is that for any , .

     

A dual graph construction for higher-rank graphs, and -theory for finite 2-graphs

Given a -graph and an element of , we define the dual -graph, . We show that when is row-finite and has no sources, the -algebras and coincide. We use this isomorphism to apply Robertson and Steger's results to calculate the -theory of when is finite and strongly connected and satisfies the aperiodicity condition.

     

Mapping spaces and homology isomorphisms

Let denote the space of pointed continuous maps from a finite cell complex to a space . Let be a generalized homology theory. We use Goodwillie calculus methods to prove that under suitable conditions on and , will send an -isomorphism in either variable to a map that is monic in homology. Interesting examples arise by letting be -theory, the finite complex be a sphere, and the map in the variable be an exotic unstable Adams map between Moore spaces.

     

Linear series over real and -adic fields

We note that the degeneration arguments given by the author in 2003 to derive a formula for the number of maps from a general curve of genus to with prescribed ramification also yields weaker results when working over the real numbers or -adic fields. Specifically, let be such a field: we see that given , , , and satisfying , there exists smooth curves of genus together with points such that all maps from to can, up to automorphism of the image, be defined over . We also note that the analagous result will follow from maps to higher-dimensional projective spaces if it is proven in the case , , and that thanks to work of Sottile, unconditional results may be obtained for special ramification conditions.

     

Extensions of orthosymmetric lattice bimorphisms

Let be an Archimedean vector lattice, let be its Dedekind completion and let be a Dedekind complete vector lattice. If is an orthosymmetric lattice bimorphism, then there exists a lattice bimorphism that not just extends but also has to be orthosymmetric. As an application, we prove the following: Let be an Archimedean -algebra. Then the multiplication in can be extended to a multiplication in , the Dedekind completion of , in such a fashion that is again a -algebra with respect to this extended multiplication. This gives a positive answer to the problem posed by C. B. Huijsmans in 1990.

     

Maps into complex space

If the dimension of is denoted by or , then a generic map satisfies , while in certain cases there is no map that satisfies .

     

for commutative rings with identity

Let , , , , be the usual operators on classes of rings: and for isomorphic and homomorphic images of rings and , , respectively for subrings, direct, and subdirect products of rings. If is a class of commutative rings with identity (and in general of any kind of algebraic structures), then the class is known to be the variety generated by the class . Although the class is in general a proper subclass of the class for many familiar varieties . Our goal is to give an example of a class of commutative rings with identity such that . As a consequence we will describe the structure of two partially ordered monoids of operators.

     

A strong hot spot theorem

A real number is said to be -normal if every -long string of digits appears in the base- expansion of with limiting frequency . We prove that is -normal if and only if it possesses no base- ``hot spot'. In other words, is -normal if and only if there is no real number such that smaller and smaller neighborhoods of are visited by the successive shifts of the base- expansion of with larger and larger frequencies, relative to the lengths of these neighborhoods.

     

On a conjecture about MRA Riesz wavelet bases

Let be a compactly supported refinable function in such that the shifts of are stable and for a -periodic trigonometric polynomial . A wavelet function can be derived from by . If is an orthogonal refinable function, then it is well known that generates an orthonormal wavelet basis in . Recently, it has been shown in the literature that if is a -spline or pseudo-spline refinable function, then always generates a Riesz wavelet basis in . It was an open problem whether can always generate a Riesz wavelet basis in for any compactly supported refinable function in with stable shifts. In this paper, we settle this problem by proving that for a family of arbitrarily smooth refinable functions with stable shifts, the derived wavelet function does not generate a Riesz wavelet basis in . Our proof is based on some necessary and sufficient conditions on the -periodic functions and in such that the wavelet function , defined by , generates a Riesz wavelet basis in .

     

Computable trees, prime models, and relative decidability

We show that for every computable tree with no dead ends and all paths computable, and every _T \emptyset$">, there is a -computable listing of the isolated paths of . It follows that for every complete decidable theory such that all the types of are computable and every _T \emptyset$">, there is a -decidable prime model of . This result extends a theorem of Csima and yields a stronger version of the theorem, due independently to Slaman and Wehner, that there is a structure with presentations of every nonzero degree but no computable presentation.

     

Remarks on spectra and multipliers for convolution operators

We prove that the spectrum of a convolution operator on a locally compact group by a self-adjoint -function is the same on and and consequently on all spaces, if and only if a Beurling algebra contains non-analytic functions on operating on into .

     

Finiteness conditions and infinite matrix rings

For a unital ring , RCFM denotes the ring of row and column finite matrices over indexed by . We give necessary and sufficient structural conditions on RCFM which are equivalent to being, respectively, Quasi-Frobenius, left artinian, and left noetherian.