The Trace Conjecture – A Counterexample

In the paper Geometric K-Theory for Lie Groups and Foliations, Baum and Connes conjecture in a remark following Corollary 2 of their famous Isomorphism conjecture that for a finitely generated group with torsion, the trace map tr: K₀(C^*)R maps K₀(C^*) onto the additive subgroup of Q generated by all rational numbers of the form 1/n where n is the order of a finite subgroup of . We construct a counterexample to this conjecture.     

Subalgebras that are cyclic as submodules

Let R be an associative, commutative, unital ring. By a R-algebra we mean a unital R-module A together with a R-module homomorphism : _R ⁿ AA (n2). We raise the question whether such an algebra possesses either an idempotent or a nilpotent element. In section 1 an affirmative answer is obtained in case R=k is an algebraically closed field and dim_kA<, as well as in case R=, dimS<, and n0(2). Section 2 deals with the case of reduced rings R and R-algebras which are finitely generated and projective as R-modules. In section 3 we show that the generic algebra over an integral domain D fails to have nilpotent elements in any integral domain extending its base ring D_n,m, and thus acquires an idempotent element in some integral domain extending D_n,m.Partially supported by National Science Foundation Grant GP-38229.     

Verallgemeinerung eines Satzes von F. Riesz

Let X be a compact Hausdorff space, C(X) the class of all continuous functions f: XR, considered as an ordered vector space over R with respect to its canonical order; furthermore let F be an order complete ordered vector space over R. A linear operator A: C(X)F is called bounded, if it transforms each bounded set of C(X) into a bounded set of F.The purpose of this paper is to represent such a bounded linear operator A: C(X)F as an integral Af= d with respect to some content , defined on the algebraa(X) which is generated by the open sets UX.This representation is unique, if is required to be regular (p. 11).     

Reguläre Differentialformen

The main object of this paper is the module _A of regular differential forms on a. finitely generated algebra or a local analytic algebra A over a field k, which was defined in [9]._A is a finitely generated module over the De Rham-algebra _A with an exterior differentiation d. In the reduced, pure dimensional, characteristic zero case A can be described as a submodule of the module Q(A)_AA of meromorphic differential forms, and it has the image of _A as submodule. Now in this case one can compare the De Rnam-cohomology of _A with the cohomology of _A, which is of interest in studying analytic or algebraic singularities.     

Cofibrations of algebras

Summary The chain complex of a twisted free product A_*t, FK, is chain homotopy equivalent to a differential graded algebra, which is identified to be a confibration of algebras as defined by Quillen. Under certain connectivity conditions we obtain a long exact sequence connecting the homologies of A, K, and A_*t FK. In particular we derive a long exact sequence connecting the homologies of Y, X and (Y U_g CX) (, C, are the loop, the cone and the suspension constructions respectively). A chain complex equivalent to the chain complex of the Milnor free group FX is recognized, from which results a theorem of Bott and Samelson that H(X) is freely generated as a graded algebra by H(X).     

A conditional dilatation equation

Summary The following theorem holds true. Theorem. Let X be a normed real vector space of dimension 3 and let k > 0 be a fixed real number. Suppose that f: X X and g: X × X are functions satisfying x – y = k f(x) – f(y) = g(x, y)(x – y) for all x, y X. Then there exist elements and t X such that f(x) = x + t for all x X and such that g(x, y) = for all x, y X with x – y = k.     

A condition for the absence of the singular continuous spectrum in the Friedrichs model. II

One considers a self-adjoint operator H for which one has a unitary group U such that the operator H UHU ^–1 is analytic with respect to . Under certain additional restrictions on H, one proves the absence of the singular continuous spectrum of H. In this connection one admits such a behavior of the essential spectrum of H for Im 0 which excludes the application of the method of analytic dilatations. In our analysis, analogies with the method of the inverse scattering problem play an important role.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 127, pp. 3–6, 1983.     

Über die endliche Lösbarkeit des Plateau-Problems in Riemannschen Mannigfaltigkeiten

We prove finiteness results of Plateau's problem for boundary curves in a normal chart of radius R < /2 in an analytic Riemannian manifold. It is shown that for analytic there exist only finitely many minimal surfaces of the type of the disc bounded by , which represent an absolute minimum of area, and if boundary branch points are excluded, there exist only finitely many relative minima in the normal chart. If lies on the boundary of a strict convex manifold, there cannot exist any boundary branch points.

---

---

Die vorliegende Arbeit entstand als Dissertation am Mathematischen] Institut der Universität Saarbrücken.

---

Herrn Prof. Dr. F. Tomi möchte ich für die Bereitstellung des Themas und für seine Betreuung sehr herzlich danken.     

K1 of Twisted Rings of Polynomials

We prove that for an arbitrary endomorphism of a ring R the group K1(R[t]) splits into the direct sum of K1(R) and Ñil (r;). Moreover, for any such R and Ñil (R; ) is isomorphic to Ñil (R ; ) for some ring R with : R R – an isomorphism.     

Mutational equations in metric spaces

This paper summarizes an extension of differential calculus to a mutational calculus for maps from one metric space to another. The simple idea is to replace half-lines allowing to define difference quotients of maps and their various limits in the case of vector space by transitions with which we can also define differential quotients of a map. Their various limits are called mutations of a map. Many results of differential calculus and set-valued analysis, including the Inverse Function Theorem, do not really rely on the linear structure and can be adapted to the nonlinear case of metric spaces and exploited. Furthermore, the concept of differential equation can be extended tomutational equation governing the evolution in metric spaces. Basic Theorems as the Nagumo Theorem, the Cauchy-Lipschitz Theorem, the Center Manifold Theorem and the second Lyapunov Method hold true for mutational equations.This work was motivated by evolution equations of tubes in visual servoing on one hand, mathematical morphology on the other, when the metric spaces are power spaces. This paper begins by listing some consequences of general theorems concerning mutational equations for tubes.     

Central limit theorems for Walsh series with general gaps

. [1]. 3 - .     

Generalized localization and convergence tests for double trigonometric Fourier series of functions fromL p,p>1

. E , f(x)L _p (T ^N ),P1,f(x)=0 E (E— ^N =[-, ]^N) E , . , .

---

---

In closing the author thanks V. A. Il'in and . A. Alimov for their constant attention paid to the present work.     

Orlicz spaces for a family of measures

. .     

On contiguity of probability measures corresponding to semimartingales

, (ⁿ), - (P ⁿ ), P ⁿ (A ⁿ )>0P ⁿ (A ⁿ )0,n. [15] - , . , P ⁿ P ⁿ T ⁿ T ⁿ .     

Finitely generated profinitely dense free groups in higher rank semi-simple groups

We prove that if is an arithmetic subgroup of a non-compact linear semi-simple groupG such that the associated simply connected algebraic group over has the so-called congruence subgroup property, then contains a finitely generated profinitely dense free subgroup. As a corollary we obtain af·g·p·d·f subgroup of SL _n () (n 3. More generally, we prove that if is an irreducible arithmetic non-cocompact lattice in a higher rank group, then containsf·g·p·d·f groups.Partially supported by GIF grant No. G-454-213.06/95     

Additive functions

1<q<2 L:= _n=1 1/q ⁿ=1/q–1. [0,1] _n()=1, A _n:= _i=1 ^n–1 _i(x)/q^i+1/n x _n(x)=0, _n>. , = _{n=1 n}(x)/qⁿ. F: [0,L]R , F(x)= _{n=1 n}(x)a_n, _n=1 ¦a _n¦<. [0,L]. q(1,2), . , q(1, 2), . .     

Transferring estimates for zonal convolution operators

_p - , . Ad- .     

Жизнь и деятельность яноша Бояи

: . BOLYAI.     

Sharpening of Stečkin's theorem to strong approximation with exponentp

— , B _n , B p1., , p=1 , - . , , , p.     

A generalization of second variation

f p- , l p . p=1 . . p - , f -.