K-theory of reduced C*-algebras and rapidly decreasing functions on groups

We associate to any length function L on a group a space of rapidly decreasing functions on (in the l ² sense), denoted by H _L (). When H _L () is contained in the reduced C*-algebra C _r ^* () of (), then it is a dense *-subalgebra of C _r ^* () and we prove a theorem of A. Connes which asserts that under this hypothesis H _L () has the same K-theory as C _r ^* (). We introduce another space of rapidly decreasing functions on (in the l ¹ sense), denoted by H _L ^1, (), which is always a dense *-subalgebra of the Banach algebra l ¹(), and we show that H _L ^1, () has the same K-theory as l ¹().     

On eigenvalues of perturbed quadratic matrix polynomials

Among other results, it is shown that ifC andK are arbitrary complexn×n matrices and if det( ₀ ² I₀ C+K)=0 for some ₀0 (resp. ₀=0), then the Newton diagram of the polynomialt(, ) = det(² I+(1+)C+K expanded in (–₀) and , has at least a point on or below the linex+y=b (resp. has no expanded in (–₀) and , has at least a point on or below the of 0 as an eigenvalue of ₀ ² I+₀ C+K. These are extensions of similar results deu to H. Langer, B. Najman, and K. Veseli proved for diagonable matricesC, and shed light on the eigenvalues of the perturbed quadratic matrix polynomials. Our proofs are independent and seem to be simpler     

Equivalence Bimodule Between Non-Commutative Tori

The non-commutative torus C ^*(ⁿ,) is realized as the C*-algebra of sections of a locally trivial C*-algebra bundle over S with fibres isomorphic to C ^*n/S, ₁) for a totally skew multiplier ₁ on ⁿ/S. D. Poguntke [9] proved that A is stably isomorphic to C(S) C(^*( Zⁿ/S, ₁) C(S) A M_kl( C) for a simple non-commutative torus A and an integer kl. It is well-known that a stable isomorphism of two separable C*-algebras is equivalent to the existence of equivalence bimodule between them. We construct an A-C(S) A-equivalence bimodule.     

A comparison theorem for the mean exit time from a domain in a Kähler manifold

Let M be a Kähler manifold with Ricci and antiholomorphic Ricci curvature bounded from below. Let be a domain in M with some bounds on the mean and JN-mean curvatures of its boundary . The main result of this paper is a comparison theorem between the Mean Exit Time function defined on and the Mean Exit Time from a geodesic ball of the complex projective space ⁿ () which involves a characterization of the geodesic balls among the domain . In order to achieve this, we prove a comparison theorem for the mean curvatures of hypersurfaces parallel to the boundary of , using the Index Lemma for Submanifolds.Work partially supported by a DGICYT Grant No. PS87-0115-C03-01.     

On local oscillatory integrals with variable Calderón-Zygmund kernels

Letp(1, ). In this paper, the authors investigate the uniformL ^p ( ⁿ ) in of the oscillatory singular integral operatorT defined by

The dependence of critical parameter bounds on the monotonicity of a Newton sequence

Summary A new method is proposed for the inclusion of the critical parameter ^* of some convex operator equationu=Tu (appearing e.g. in thermal explosion theory). It is based on the fact that for a fixed Newton's method starting with a suitable subsolution is not monotonically if and only if >^*. Several numerical examples arising from nonlinear boundary value problems illustrate the efficiency of the method.     

The Atiyah-Segal completion theorem for C*-algebras

We generalize the Atiyah-Segal completion theorem to C ^*-algebras as follows. Let A be a C ^*-algebra with a continuous action of the compact Lie group G. If K _* ^G (A) is finitely generated as an R(G)-module, or under other suitable restrictions, then the I(G)-adic completion K _* ^G (A) is isomorphic to RK _*([A C(EG)]^G), where RK _* is representable K-theory for - C ^*-algebras and EG is a classifying space for G. As a corollary, we show that if and are homotopic actions of G, and if K _*(C ^* (G,A,)) and K _*(C ^* (G,A,)) are finitely generated, then K _*(C ^*(G,A,))K_*(C ^* (G,A,)). We give examples to show that this isomorphism fails without the completions. However, we prove that this isomorphism does hold without the completions if the homotopy is required to be norm continuous.This work was partially supported by an NSF Graduate Fellowship and by an NSF Postdoctoral Fellowship.     

On the width of ordered sets and Boolean algebras

Thewidth (chain number) of a partial order P, < is the smallest cardinal such that ¦A¦< 1 + whenever A is an antichain (chain) in P. We prove that, if a partial order (P, <) has width and cf()=, then P contains antichains A_n (n<) such that ¦A ₀¦<¦A₁¦ <...<={¦A_n¦: n < < } and either A₀1 A₂< ... or A₀>A₁ >A₂> ... A similar structure result is obtained for partial orders with chain number if cf()=. As an application we solve a problem of van Douwen, Monk and Rubin [1] by showing that if a Boolean algebra has width , thencf() .This work has been partially supported by NATO grant No. 339/84.Presented by Bjarni Jonsson.     

Rieffel projections in certain transformation group C*-algebras

LetA be the transformation groupC ^*-algebra associated with an arbitrary orientation-preserving homeomorphism of . ThisC ^*-algebra contains an infinite family of projections, called Rieffel projections, each of which generates theK ₀-groupK ₀(A). Although these projections must beK-theoretically equivalent, it is easy to see that most are not Murray-von Neumann equivalent. The mystery of how large the matrix algebra must be to implement theK-theory equivalence, is solved by explicitly constructing the equivalence in the smallest possible algebra:A with unit adjoined.Partially supported by NSF Grant DMS 8901923.     

On the positive solutions of some nonlinear diffusion problems

We consider the nonlinear diffusion equationu _t –a(x, u _{x x} )+b(x, u)=g(x, u) with initial boundary conditions andu(t, 0)=u(t, 1)=0. Here,a, b, andg denote some real functions which are monotonically increasing with respect to the second variable. Then, the corresponding stationary problem has a positive solution if and only if(0, ^*) or(0, ^*]. The endpoint ^* can be estimated by , where ₁ u denotes the first eigenvalue of the stationary problem linearized at the pointu. The minimal positive steady state solutions are stable with respect to the nonlinear parabolic equation.

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Zusammenfassung Wir betrachten die nichtlineare Diffusionsgleichungu _t –a(x, u _x ) _x +b(x, u)=g(x, u) mit Randbedingungen undu (t, 0)=u (t, 1)=0. Dabei sinda, b, undg monoton wachsende Funktionen bzgl. des zweiten Argumentes. Das zugehörige stationäre Problem hat genau dann eine positive Lösung, falls (0, ^*) oder(0, ^*]. Der Endpunkt ^* kann durch abgeschätzt werden, wobei ₁ u den ersten Eigenwert des an der Stelleu linearisierten stationären Problems bezeichnet. Die minimale positive stationäre Lösung ist stabil bzgl. der obigen nichtlinearen parabolischen Gleichung.

     

Décroissance rapide de la distribution fλ

Let X be an open subset of ⁿ and (f₁, ...,f_p): X ^p be a holomorphic mapping. We prove that if (x⁰,0, ⁰) T^* × ^p does not belong to the characteristic variety of the _X []-module _X[]f, then there exists a conic neighborhood V × of (x⁰, ⁰) such the function is rapidely decreasing in | Im | for with Re bounded, for any (n,n)-form of class C with compact support in V. The following partial converse of this result is also established: if for all (n,n)-forms of class C with compact support in X, then .     

Description of Hyperinvariant Subspaces of a Contraction in Terms of Its Characteristic Function

If T is a completely nonunitary contraction on a Hilbert space and L is its invariant subspace corresponding to a regular factorization of its characteristic function = , then L is hyperinvariant if and only if the following two conditions are fulfilled: (1) supp _* supp is of Lebesgue measure zero; (2) for every pair A H (E E) and A _* H (E _* E _*) intertwining by , i.e., such that A =A _*, there exists a function A _F H (F F) intertwining with A by and with A _* by , i.e., such that A = A _F and A _F = A _*. Bibliography: 4 titles.     

Asymptotic Estimates for a Variational Problem Involving a Quasilinear Operator in the Semi-Classical Limit

Let be a domain of ^N . We study the infimum ₁(h) of the functional |u| ^p +h ^–p V(x)|u| ^p dx in W ^1,p () for ||u|| _LP()= 1 where h > 0 tends to zero and V is a measurable function on . When V is bounded, we describe the behaviour of ₁(h), in particular when V is radial and 'slowly' decaying to zero. We also study the limit of ₁(h) when h 0 for more general potentials and show a necessary and sufficient condition for ₁(h) to be bounded. A link with the tunelling effect is established. We end with a theorem of existence for a first eigenfunction related to ₁(h).     

Minimality of derivative chains,corresponding to a boundary value problem on a finite segment

One investigates the minimality of derivative chains, constructed from the root vectors of polynomial pencils of operators, acting in a Hilbert space. One investigates in detail the quadratic pencil of operators. In particular, for L()=L₀+L₁+²L₂ with bounded operators L₀0, L₂0 and Re L₁0, one shows the minimality in the space_173-02 of the system {x_k, _kekx_k}, where x_k are eigenvectors of L(), corresponding to the characteristic numbers _kin the deleted neighborhoods of which one has the representation L^–1()=(–_k)^–1R_K+W_K() with one-dimensional operators R_k and operator-valued functions W_K(), k=1, 2, ..., analytic for =_k.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 2, pp. 195–205, February, 1990.     

Morphisms of the Heat Equation

In this article we consider a certain class of mappings between Riemannian manifolds of the type M × R^{+ *}, which preserve two objects associated to the heat equation. First we show that maps which pull back local solutions of the heat equation to local solutions of the heat equation, called heat equation morphisms have to be the product of a homothetic submersion and an affine map of R^{+ *}. Then using the above characterisation, we study maps which pull back the heat kernel to the heat kernel. We call such maps heat kernel morphisms. We show in Theorem 3 that, in the case of compact manifolds, a map : M × M × R^{+ *} N × N × R^{+ *} of the form (x,y,t) = ( (x), (y),h(t)), with surjective, is a heat kernel morphism if and only if is a homothetic covering of N()-sheets and constant dilation such that N() = ^m (m = dim M) and h(t) = ²t. In particular, if is bijective then it must be an isometry and h(t) = t. A similar problem was considered from a different point of view in [6].     

On a fixed-point theorem on locally convex linear topological spaces

Iff is a self mapping on a closed convex subsetK of a separated quasicomplete locally convex linear topological spaceE such that (i)E is strictly convex, (ii)f (K) is contained in a compact subset ofK and (iii)f satisfies a contraction condition, then it is shown that for eachxK, the sequence of {U ⁿ (x)} _n =1 of iterates, whereU KK is defined byU (y)=f(y)+(1-) y, yK, converges to a fixed point off.     

The existence of some resolvable block designs with divisibility into groups

This paper proves the existence of resolvable block designs with divisibility into groups GD(v; k, m; ₁, ₂) without repeated blocks and with arbitrary parameters such that ₁ = k, (v–1)/(k–1) ₂ v^k–2 (and also ₁ k/2, (v–1)/(2(k–1)) ₂ v^k–2 in case k is even) k 4 andp=1 (mod k–1), k < p for each prime divisor p of number v. As a corollary, the existence of a resolvable BIB-design (v, k, ) without repeated blocks is deduced with X = k (and also with = k/2 in case of even k) k , where a is a natural number if k is a prime power and=1 if k is a composite number.Translated from Matematicheskie Zametki, Vol. 19, No. 4, pp. 623–634, April, 1976.     

On extremal perturbations of semi-Fredholm operators

Given a bounded linear operatorA in an infinite dimensional Banach space and a compact subset of a connected component of its semi-Fredholm domain, we construct a finite rank operatorF such that –A+F is bounded below (or surjective) for each ,F ²=0 and rankF=max min{dimN(–A), codimR(–A)}, if ind(–A)0 (or ind(–A)0, respectively) for each .     

Maximal Classes of Topological Spaces and Domains Determined by Function Spaces

In this paper the question of what classes A of T ₀-spaces should be paired with classes of domains in order that all function spaces [AB] for AA and B are -compact domains is considered. It is shown that core compact spaces are paired with bounded complete domains and a class of topological spaces called RW-spaces (with finitely many components) is paired with the class of -compact pointed L-domains (L-domains).