The generalized Berg theorem and BDF-theorem

Let be a separable simple -algebra with finitely many extreme traces. We give a necessary and sufficient condition for an essentially normal element , i.e., is normal ( is the quotient map), having the form for some normal element and We also show that a normal element can be quasi-diagonalized if and only if the Fredholm index for all In the case that is a simple -algebra of real rank zero, with stable rank one and with continuous scale, and has countable rank, we show that a normal element with zero Fredholm index can be written as

where is an (increasing) approximate identity for consisting of projections, is a bounded sequence of numbers and with for any given

     

Duality of restriction and induction for -coactions

Consider a coaction of a locally compact group on a - algebra , and a closed normal subgroup of . We prove, following results of Echterhoff for abelian , that Mansfield's imprimitivity between and implements equivalences between Mansfield induction of representations from to and restriction of representations from to , and between restriction of representations from to and Green induction of representations from to . This allows us to deduce properties of Mansfield induction from the known theory of ordinary crossed products.

     

Wavelet transform and orthogonal decomposition of

Let be the Weyl-Poincaré group and be the Iwasawa decomposition of with . Then the ``affine Weyl-Poincaré group' can be realized as the complex tube domain or the classical Cartan domain . The square-integrable representations of and give the admissible wavelets and wavelet transforms. An orthogonal basis of the set of admissible wavelets associated to is constructed, and it gives an orthogonal decomposition of space on (or the Cartan domain
) with every component being the range of wavelet transforms of functions in with .

     

Partial regularity of solutions to a class of degenerate systems

We consider the system , in coupled with suitable initial-boundary conditions, where is a bounded domain in with smooth boundary and is a continuous and positive function of . Our main result is that under some conditions on there exists a relatively open subset of such that is locally Hölder continuous on , the interior of is empty, and is essentially bounded on .

     

Twin trees and -gons

We define a natural generalization of generalized -gons to the case of -graphs (where is a totally ordered abelian group and ). We term these objects -gons. We then show that twin trees as defined by Ronan and Tits can be viewed as -gons, where is ordered lexicographically. This allows us to then generalize twin trees to the case of -trees. Finally, we give a free construction of -gons in the cases where is discrete and has a subgroup of index that does not contain the minimal element of .

     

Structural properties of universal minimal dynamical systems for discrete semigroups

We show that for a discrete semigroup there exists a uniquely determined complete Boolean algebra - the algebra of clopen subsets of . is the phase space of the universal minimal dynamical system for and it is an extremally disconnected compact Hausdorff space. We deal with this connection of semigroups and complete Boolean algebras focusing on structural properties of these algebras. We show that is either atomic or atomless; that is weakly homogenous provided has a minimal left ideal; and that for countable semigroups is semi-Cohen. We also present a class of what we call group-like semigroups that includes commutative semigroups, inverse semigroups, and right groups. The group reflection of a group-like semigroup can be constructed via universal minimal dynamical system for and, moreover, and are the same.

     

Bodies with similar projections

Aleksandrov's projection theorem characterizes centrally symmetric convex bodies by the measures of their orthogonal projections on lower dimensional subspaces. A general result proved here concerning the mixed volumes of projections of a collection of convex bodies has the following corollary. If is a convex body in whose projections on -dimensional subspaces have the same -dimensional volume as the projections of a centrally symmetric convex body , then the Quermassintegrals satisfy , for , with equality, for any , if and only if is a translate of . The case where is centrally symmetric gives Aleksandrov's projection theorem.

     

Small cancellation groups and translation numbers

In this paper we prove that C(4)-T(4)-P, C(3)-T(6)-P and C(6)-P small cancellation groups are translation discrete in the strongest possible sense and that in these groups for any and any there is an algorithm deciding whether or not the equation has a solution. There is also an algorithm for calculating for each the maximum such that is an -th power of some element. We also note that these groups cannot contain isomorphic copies of the group of -adic fractions and so in particular of the group of rational numbers. Besides we show that for and groups all translation numbers are rational and have bounded denominators.

     

The homotopy groups of the at the prime 3

In this paper, we try to compute the homotopy groups of the -localized Toda-Smith spectrum at the prime 3 by using the Adams-Novikov spectral sequence, and have almost done so. This computation involves non-trivial differentials and of the Adams-Novikov spectral sequence, different from the case . We also determine the homotopy groups of some -localized finite spectra relating to . We further show some of the non-trivial differentials on elements relating so-called -elements in the Adams-Novikov spectral sequence for .

     

Congruences between Modular Forms, Cyclic Isogenies of Modular Elliptic Curves, and Integrality of -Functions

Let be a congruence subgroup of type and of level . We study congruences between weight 2 normalized newforms and Eisenstein series on modulo a prime above a rational prime . Assume that , is a common eigenfunction for all Hecke operators and is ordinary at . We show that the abelian variety associated to and the cuspidal subgroup associated to intersect non-trivially in their -torsion points. Let be a modular elliptic curve over with good ordinary reduction at . We apply the above result to show that an isogeny of degree divisible by from the optimal curve in the -isogeny class of elliptic curves containing to extends to an étale morphism of Néron models over if . We use this to show that -adic distributions associated to the -adic -functions of are -valued.

     

Isomorphism of lattices of recursively enumerable sets

Let , and for , let be the lattice of subsets of which are recursively enumerable relative to the ``oracle' . Let be , where is the ideal of finite subsets of . It is established that for any , is effectively isomorphic to if and only if , where is the Turing jump of . A consequence is that if , then . A second consequence is that can be effectively embedded into preserving least and greatest elements if and only if .

     

Only single twists on unknots can produce composite knots

Let be a knot in the -sphere , and a disc in meeting transversely more than once in the interior. For non-triviality we assume that over all isotopy of . Let () be a knot obtained from by cutting and -twisting along the disc (or equivalently, performing -Dehn surgery on ). Then we prove the following: (1) If is a trivial knot and is a composite knot, then ; (2) if is a composite knot without locally knotted arc in and is also a composite knot, then . We exhibit some examples which demonstrate that both results are sharp. Independently Chaim Goodman-Strauss has obtained similar results in a quite different method.

     

Linear isometries between subspaces of continuous functions

We say that a linear subspace of is strongly separating if given any pair of distinct points of the locally compact space , then there exists such that . In this paper we prove that a linear isometry of onto such a subspace of induces a homeomorphism between two certain singular subspaces of the Shilov boundaries of and , sending the Choquet boundary of onto the Choquet boundary of . We also provide an example which shows that the above result is no longer true if we do not assume to be strongly separating. Furthermore we obtain the following multiplicative representation of : for all and all , where is a unimodular scalar-valued continuous function on . These results contain and extend some others by Amir and Arbel, Holszty\'{n}ski, Myers and Novinger. Some applications to isometries involving commutative Banach algebras without unit are announced.

     

Kaehler structures on

Let be a compact connected semi-simple Lie group, let , and let be an Iwasawa decomposition. To a given -invariant Kaehler structure on , there corresponds a pre-quantum line bundle on . Following a suggestion of A.S. Schwarz, in a joint paper with V. Guillemin, we studied its holomorphic sections as a -representation space. We defined a -invariant -structure on , and let denote the space of square-integrable holomorphic sections. Then is a unitary -representation space, but not all unitary irreducible -representations occur as subrepresentations of . This paper serves as a continuation of that work, by generalizing the space considered. Let be a Borel subgroup containing , with commutator subgroup . Instead of working with , we consider , for all parabolic subgroups containing . We carry out a similar construction, and recover in the unitary irreducible -representations previously missing. As a result, we use these holomorphic sections to construct a model for : a unitary -representation in which every irreducible -representation occurs with multiplicity one.

     

Boundary value maps, Szegö maps and intertwining operators

We consider one series of unitarizable representations, the cohomological induced modules with dominant regular infinitesimal character. The minimal -type of determines a homogeneous vector bundle . The derived functor modules can be realized on the solution space of a first order differential operator on . Barchini, Knapp and Zierau gave an explicit integral map from the derived functor module, realized in the Langlands classification, into the space of smooth sections of the vector bundle . In this paper we study the asymptotic behavior of elements in the image of . We obtain a factorization of the standard intertwining opeartors into the composition of the Szegö map and a passage to boundary values.

     

Cohomological construction of quantized universal enveloping algebras

Given an associative algebra and the category of its finite dimensional modules, additional structures on the algebra induce corresponding ones on the category . Thus, the structure of a rigid quasi-tensor (braided monoidal) category on is induced by an algebra homomorphism (comultiplication), coassociative up to conjugation by (associativity constraint) and cocommutative up to conjugation by (commutativity constraint), together with an antiautomorphism (antipode) of satisfying the compatibility conditions. A morphism of quasi-tensor structures is given by an element with suitable induced actions on , and . Drinfeld defined such a structure on for any semisimple Lie algebra with the usual comultiplication and antipode but nontrivial and , and proved that the corresponding quasi-tensor category is isomomorphic to the category of representations of the Drinfeld-Jimbo (DJ) quantum universal enveloping algebra (QUE), .

In the paper we give a direct cohomological construction of the which reduces to the trivial associativity constraint, without any assumption on the prior existence of a strictly coassociative QUE. Thus we get a new approach to the DJ quantization. We prove that can be chosen to satisfy some additional invariance conditions under (anti)automorphisms of , in particular, gives an isomorphism of rigid quasi-tensor categories. Moreover, we prove that for pure imaginary values of the deformation parameter, the elements , and can be chosen to be formal unitary operators on the second and third tensor powers of the regular representation of the Lie group associated to with depending only on even powers of the deformation parameter. In addition, we consider some extra properties of these elements and give their interpretation in terms of additional structures on the relevant categories.

     

Symmetric powers of complete modules over a two-dimensional regular local ring

Let be a two-dimensional regular local ring with infinite residue field. For a finitely generated, torsion-free -module , write for the th symmetric power of , mod torsion. We study the modules , , when is complete (i.e., integrally closed). In particular, we show that , for any minimal reduction and that the ring is Cohen-Macaulay.

     

Approximation by harmonic functions

For a compact set we construct a restoring covering for the space of real-valued functions on which can be uniformly approximated by harmonic functions. Functions from restricted to an element of this covering possess some analytic properties. In particular, every nonnegative function , equal to 0 on an open non-void set, is equal to 0 on . Moreover, when , the algebra of complex-valued functions on which can be uniformly approximated by holomorphic functions is analytic. These theorems allow us to prove that if a compact set has a nontrivial Jensen measure, then contains a nontrivial compact set with analytic algebra .