A perturbed elementary operator and range-kernel orthogonality

Let denote the algebra of operators on a Hilbert . If and are commuting normal operators, and and are commuting quasi-nilpotents such that , then define and by , , and . It is proved that and , where is some scalar and is the quasi-nilpotent part of the operator .

     

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 .

     

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.

     

Topological entropy and AF subalgebras of graph -algebras

Let be the canonical AF subalgebra of a graph -algebra associated with a locally finite directed graph . For Brown and Voiculescu's topological entropy of the canonical completely positive map on , is known to hold for a finite graph , where is the loop entropy of Gurevic and is the block entropy of Salama. For an irreducible infinite graph , the inequality has recently been known. It is shown in this paper that

where is the graph with the direction of the edges reversed. Some irreducible infinite graphs 1)$"> with are also examined.

     

Avoiding the projective hierarchy in expansions of the real field by sequences

Some necessary conditions are given on infinitely oscillating real functions and infinite discrete sets of real numbers so that first-order expansions of the field of real numbers by such functions or sets do not define . In particular, let be such that , as for some , is o-minimal, and the expansion of by the set does not define . Then there exist 0$"> and such that as .

     

Equicompact sets of operators defined on Banach spaces

Let and be Banach spaces. We say that a set denotes the space of all compact operators from into ) is equicompact if there exists a null sequence in such that for all and all . It is easy to show that collectively compactness and equicompactness are dual concepts in the following sense: is equicompact iff is collectively compact. We study some properties of equicompact sets and, among other results, we prove: 1) a set is equicompact iff each bounded sequence in has a subsequence such that is a converging sequence uniformly for ; 2) if does not have finite cotype and is a maximal equicompact set, then, given and a finite set in , there is an operator such that for and all .

     

A note on periodic points of order preserving subhomogeneous maps

Let be the standard closed positive cone in and let be the set of integers for which there exists a continuous, order preserving, subhomogeneous map , which has a periodic point with period . It has been shown by Akian, Gaubert, Lemmens, and Nussbaum that is contained in the set consisting of those for which there exist integers and such that , , and for some . This note shows that for all .

     

Entire pluricomplex Green functions and Lelong numbers of projective currents

Let be a positive closed current of bidimension (1,1) and unit mass on the complex projective space . We prove that the set of points where has Lelong number larger than is contained in a complex line if , and for some complex line if . We also prove that in dimension 2 and if , then for some conic .

     

The real rank zero property of crossed product

Let be a unital -algebra, and let be a -dynamical system with abelian and discrete. In this paper, we introduce the continuous affine map from the trace state space of the crossed product to the -invariant trace state space of . If is of real rank zero and is connected, we have proved that is homeomorphic. Conversely, if is homeomorphic, we also get some properties and real rank zero characterization of . In particular, in that case, is of real rank zero if and only if each unitary element in with the form can be approximated by the unitary elements in with finite spectrum, where , , and if moreover is a unital inductive limit of the direct sums of non-elementary simple -algebras of real rank zero, then the above can be cancelled.

     

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.

     

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.

     

Simultaneous non-vanishing of twists

Let be a newform of even weight , level and character and let be a newform of even weight , level and character . We give a generalization of a theorem of Elliott, regarding the average values of Dirichlet -functions, in the context of twisted modular -functions associated to and . Using this result, we find a lower bound in terms of for the number of primitive Dirichlet characters modulo prime whose twisted product -functions are non-vanishing at a fixed point with .

     

Hyperelliptic curves over of every -rank without extra automorphisms

We prove that for any pair of integers such that or 0$">, there exists a (hyper)elliptic curve over of genus and -rank whose automorphism group consists of only identity and the (hyper)elliptic involution. As an application, we prove the existence of principally polarized abelian varieties over of dimension and -rank such that .

     

The maximal ideal space of with respect to the Hadamard product

It is shown that the space of all regular maximal ideals in the Banach algebra with respect to the Hadamard product is isomorphic to The multiplicative functionals are exactly the evaluations at the -th Taylor coefficient. It is a consequence that for a given function in and for a function holomorphic in a neighborhood of with and for all the function is in

     

Fiber products, Poincaré duality and -ring spectra

For a Poincaré duality space and a map , consider the homotopy fiber product . If is orientable with respect to a multiplicative cohomology theory , then, after suitably regrading, it is shown that the -homology of has the structure of a graded associative algebra. When is the diagonal map of a manifold , one recovers a result of Chas and Sullivan about the homology of the unbased loop space .

     

Separate continuity, joint continuity and the Lindelöf property

In this paper we prove a theorem more general than the following. Suppose that is Lindelöf and -favourable and is Lindelöf and Cech-complete. Then for each separately continuous function there exists a residual set in such that is jointly continuous at each point of .

     

The rank of elliptic curves with rational 2-torsion points over large fields

Let be a number field, an algebraic closure of , the absolute Galois group , the maximal abelian extension of and an elliptic curve defined over . In this paper, we prove that if all 2-torsion points of are -rational, then for each , has infinite rank, and hence has infinite rank.

     

Free modules of relative invariants and some rings of invariants that are Cohen-Macaulay

Let be a faithful representation of a finite group and a linear character. We study the module of -relative invariants. We prove a modular analogue of result of R. P. Stanley and V. Reiner in the case of nonmodular reflection groups to the effect that these modules are free on a single generator over the ring of invariants . This result is then applied to show that the ring of invariants for is Cohen-Macaulay. Since the Cohen-Macaulay property is not an issue in the nonmodular case (it is a consequence of a theorem of Eagon and Hochster), this would seem to be a new way to verify the Cohen-Macaulay property for modular rings of invariants. It is known that the Cohen-Macaulay property is inherited when passing from the ring of invariants of to that of a pointwise stabilizer of a subspace . In a similar vein, we introduce for a subspace the subgroup of elements of having as an eigenspace, and prove that Cohen-Macaulay implies is also.

     

On isomorphisms between centers of integral group rings of finite groups

For finite nilpotent groups and , and a -adapted ring (the rational integers, for example), it is shown that any isomorphism between the centers of the group rings and is monomial, i.e., maps class sums in to class sums in up to multiplication with roots of unity. As a consequence, and have identical character tables if and only if the centers of their integral group rings and are isomorphic. In the course of the proof, a new proof of the class sum correspondence is given.

     

Scalar-valued dominated polynomials on Banach spaces

It is well known that 2-homogeneous polynomials on -spaces are 2-dominated. Motivated by the fact that related coincidence results are possible only for polynomials defined on symmetrically regular spaces, we investigate the situation in several classes of symmetrically regular spaces. We prove a number of non-coincidence results which makes us suspect that there is no infinite dimensional Banach space such that every scalar-valued homogeneous polynomial on is -dominated for every .