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.

     

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.

     

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 .

     

Dynamical systems method (DSM) for unbounded operators

Let be an unbounded linear operator in a real Hilbert space , a generator of a semigroup, and let be a nonlinear map. The DSM (dynamical systems method) for solving equation consists of solving the Cauchy problem , , where is a suitable operator, and proving that i) 0$">, ii) , and iii) .

Conditions on and are given which allow one to choose such that i), ii), and iii) hold.

     

On differentiability properties of typical continuous functions and Haar null sets

Let () be the set of all continuous functions on which have a derivative ( , respectively) at least at one point . B. R. Hunt (1994) proved that is Haar null (in Christensen's sense) in .

In the present article it is proved that neither nor its complement is Haar null in . Moreover, the same assertion holds if we consider the approximate derivative (or the ``strong' preponderant derivative) instead of the ordinary derivative; these results are proved using a new result on typical (in the sense of category) continuous functions, which is of interest in its own right.

     

The minimal spanning tree and the upper box dimension

We show that the -weight of an MST over points in a metric space with upper box dimension has a bound independent of if d$"> and does not have one if .

     

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.

     

Uniformly complete quotient space

The uniformly complete quotient space of a locally compact group is introduced. It is shown that the operator space dual is a completely contractive Banach algebra, which contains the completely bounded Fourier multiplier algebra as a completely contractively complemented Banach subalgebra. A natural completely isometric representation of on is studied and some equivalent amenability conditions associated with are proved.

     

Eigenvalues of scaling operators and a characterization of -splines

A finitely supported sequence that sums to defines a scaling operator on functions a transition operator on sequences and a unique compactly supported scaling function that satisfies normalized with It is shown that the eigenvalues of on the space of compactly supported square-integrable functions are a subset of the nonzero eigenvalues of the transition operator on the space of finitely supported sequences, and that the two sets of eigenvalues are equal if and only if the corresponding scaling function is a uniform -spline.

     

Multiplicative bijections of

Let be a compact Hausdorff space which satisfies the first axiom of countability, let and let , be the set of all continuous functions from to If , ,is a bijective multiplicative map, then there exist a homeomorphism and a continuous map such that for all and for all

     

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.

     

Hardy's inequality on exterior domains

Let be a smooth exterior domain in and . We prove that when , Hardy's inequality is valid on .

     

The duality theory of a finite dimensional discrete quantum group

Suppose that is a finite dimensional discrete quantum group and is a Hilbert space. This paper shows that if there exists an action of on so that is a modular algebra and the inner product on is -invariant, then there is a unique C*-representation of on supplemented by the The commutant of in is exactly the -invariant subalgebra of . As an application, a new proof of the classical Schur-Weyl duality theory of type A is given.

     

Two characterizations of pure injective modules

Let be a commutative ring with identity and an -module. It is shown that if is pure injective, then is isomorphic to a direct summand of the direct product of a family of finitely embedded modules. As a result, it follows that if is Noetherian, then is pure injective if and only if is isomorphic to a direct summand of the direct product of a family of Artinian modules. Moreover, it is proved that is pure injective if and only if there is a family of -algebras which are finitely presented as -modules, such that is isomorphic to a direct summand of a module of the form , where for each , is an injective -module.

     

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.

     

Degree of a holomorphic map between unit balls from to

Let be a rational proper holomorphic map between the unit ball in and the unit ball in Write

where and are holomorphic polynomials, with Recall that the degree of is defined by

   deg

In this paper, we give a bound estimate for the degree of improving the bound given by Forstneric (1989).

     

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.

     

Automatic continuity of -derivations on -algebras

Let be a -algebra acting on a Hilbert space , let be a linear mapping and let be a -derivation. Generalizing the celebrated theorem of Sakai, we prove that if is a continuous -mapping, then is automatically continuous. In addition, we show the converse is true in the sense that if is a continuous --derivation, then there exists a continuous linear mapping such that is a --derivation. The continuity of the so-called - -derivations is also discussed.

     

The Nevanlinna counting functions for Rudin's orthogonal functions

and denote the Hardy spaces on the open unit disc . Let be a function in and . If is an inner function and , then is orthogonal in . W.Rudin asked if the converse is true and C. Sundberg and C. Bishop showed that the converse is not true. Therefore there exists a function such that is not an inner function and is orthogonal in . In this paper, the following is shown: is orthogonal in if and only if there exists a unique probability measure on [0,1] with supp such that for nearly all in where is the Nevanlinna counting function of . If is an inner function, then is a Dirac measure at .