Existence and uniqueness of an invariant probability for a class of Feller Markov chains

We consider the class of Feller Markov chains on a phase spaceX whose kernels mapC ₀ (X), the space of bounded continuous functions that vanish at infinity, into itself. We provide a necessaryand sufficient condition for the existence of an invariant probability measure using a generalized Farkas Lemma. This condition is a Lyapunov type criterion that can be checked in practice. We also provide a necessaryand sufficient condition for existence of aunique invariant probability measure. When the spaceX is compact, the conditions simplify.     

The Tracial Topological Rank of C*-Algebras

We introduce the notion of tracial topological rank for C*-algebras.In the commutative case, this notion coincides with the coveringdimension. Inductive limits of C*-algebrasof the form PM_n(C(X))P,where X is a compact metric space with dim X k, and P is aprojection in M_n(C(X)), have tracial topological rank no morethan k. Non-nuclear C*-algebras can have small tracial topologicalrank. It is shown that if A is a simple unital C*-algebra withtracial topological rank k (< ), then

(i) A is quasidiagonal,

(ii) A has stable rank 1,

(iii) A has weakly unperforatedK₀(A),

(iv) A has the following Fundamental Comparabilityof Blackadar:if p, q A are two projections with (p) < (q)for all tracialstates on A, then p q

. 2000 MathematicsSubject Classification: 46L05, 46L35.     

An explicit solution for an instantaneous two-phase Stefan problem with nonlinear thermal coefficients

We consider a nonlinear heat conduction problem for a semi-infinitematerial x > 0, with phase-change temperature T₁, an initialtemperature T₂ (> T₁) and a heat flux of the type q (t) =q₀/t imposed on the fixed face x = 0. We assume that the volumetricheat capacity and the thermal conductivity are particular nonlinearfunctions of the temperature in both solid and liquid phases. We determine necessary and/or sufficient conditions on the parametersof the problem in order to obtain the existence of an explicitsolution for an instantaneous nonlinear twophase Stefan problem(solidification process).     

Density averaging for Markov processes and the invariant measures problem

A necessary and sufficient condition is given for the existence of a finite invariant measure equivalent to a given reference measure for a discrete time, general state Markov process. The condition is an extension of one given by D. Maharam in the deterministic case and involves an averaging method (called by Maraham ‘density averaging’) applied to the Radon-Nikodym derivatives with respect to the reference measure of the usual sequence of measures induced by the Markov process acting on the fixed reference     

Finiteness Properties of Certain Metabelian Arithmetic Groups in the Function Field Case

Consider the group scheme where R is an arbitrary commutative ring with 1 0 and a unitx R* acts on R by multiplication. We will study the finiteness properties of subgroups of G(O_S)where O_S is an S-arithmetic subring of a global function field.The subgroups we are interested in are of the form where Q is a subgroup of O_S*. The finiteness propertiesof these metabelian groups can be expressed in terms of the-invariant due to R. Bieri and R. Strebel. Theorem A. Let S be a finite set of places of a global functionfield (regarded as normalized discrete valuations) and O_S thecorresponding S-arithmetic ring. Let Q be a subgroup of O_S*.Then Q is finitely generated and for all integers n 1 the followingare equivalent:

(1) O_S Q is of type FP_n;

(2) O_S is n-tameas a ZQ-module;

(3) each p S restricts to a non-trivial homomorphism and the set is n-tame.

If these conditions hold for at least one n 1 then the identity holds.} Theorem B. Let r denote the rank of Q. Then the followinghold:

(1) the group O_S Q is not of type FP_r+1};

(2) if Qhas maximum rank r = |S| –1 then the group O_S Q is oftype FP_r.

In particular, is of type FP_{|S| –1} but not of type FP_|S|. 1991 Mathematics SubjectClassification: 20E08, 20F16, 20G30, 52A20.     

Finite Time Equipartition for Second-order Hyperbolic Systems

A sufficient condition for equipartition of energy for secondorder hyperbolic systems in three space variables is given.The condition states that the system should evolve in such away that the time derivative of a solution of the form (u₁,0)^T is connected with the space derivatives of a solution ofthe form (0, u₂)^T and the time derivative of (0, u₂)^T is connectedwith the space derivatives of (u₁, 0)^T.     

RANDOMLY CONNECTED DYNAMICAL SYSTEMS ON BANACH SPACES

We give sufficient conditions for asymptotic stability of Markov operators governing the evolution of measures due to the action of randomly chosen dynamical systems on Banach spaces. We show that the existence of an invariant measure for the transition operator implies the existence of an invariant measure for a semigroup generated by the considered systems.     

Quantum Markov Processes with a Christensen-Evans Generator in a Von Neumann Algebra

Let A be a unital von Neumann algebra of operators on a complexseparable Hilbert space H₀, and let {T_t, t 0} be a uniformlycontinuous quantum dynamical semigroup of completely positiveunital maps on A. The infinitesimal generator L of {T_t} is abounded linear operator on the Banach space A. For any Hilbertspace K, denote by B(K) the von Neumann algebra of all boundedoperators on K. Christensen and Evans [3] have shown that Lhas the form [formula] where is a representation of A in B(K) for some Hilbert spaceK, R: H₀ K is a bounded operator satisfying the ‘minimality’condition that the set {(RX–(X)R)u, uH₀, XA} is totalin K, and K₀ is a fixed element of A. The unitality of {T_t}implies that L(1) = 0, and consequently K₀=iH–R^*R, whereH is a hermitian element of A. Thus (1.1) can be expressed as [formula] We say that the quadruple (K, , R, H) constitutes the set ofChristensen–Evans (CE) parameters which determine theCE generator L of the semigroup {T_t}. It is quite possible thatanother set (K', ', R', H') of CE parameters may determine thesame generator L. In such a case, we say that these two setsof CE parameters are equivalent. In Section 2 we study thisequivalence relation in some detail. 1991 Mathematics SubjectClassification 81S25, 60J25.     

Some Open Problems and Conjectures Associated with the Invariant Subspace Problem

There is a subtle difference as far as the invariant subspace problem is concerned for operators acting on real Banach spaces and operators acting on complex Banach spaces. For instance, the classical hyperinvariant subspace theorem of Lomonosov [Funktsional. Anal. nal. i Prilozhen 7(3)(1973), 55–56. (Russian)], while true for complex Banach spaces is false for real Banach spaces. When one starts with a bounded operator on a real Banach space and then considers some “complexification technique” to extend the operator to a complex Banach space, there seems to be no pattern that indicates any connection between the invariant subspaces of the “real” operator and those of its “complexifications.” The purpose of this note is to examine two complexification methods of an operator T acting on a real Banach space and present some questions regarding the invariant subspaces of T and those of its complexifications Mathematics Subject Classification 1991: 47A15, 47C05, 47L20, 46B99 Y.A. Abramovich: 1945–2003 The research of Aliprantis is supported by the NSF Grants EIA-0075506, SES-0128039 and DMI-0122214 and the DOD Grant ACI-0325846     

Topological conditions for the existence of absorbing Cantor sets

This paper deals with strange attractors of S-unimodal maps . It generalizes earlier results in the sense that very general topological conditions are given that either

i)

guarantee the existence of an absorbing Cantor set provided the critical point of is sufficiently degenerate, or

ii)

prohibit the existence of an absorbing Cantor set altogether.

As a by-product we obtain very weak topological conditions that imply the existence of an absolutely continuous invariant probability measure for .

     

On the existence of an invariant measure for Markov-Feller operators

Let X be a Polish space and P a Markov operator acting on the space of Borel measures on X. We will prove the existence of an invariant measure with respect to P, provided that P satisfies some condition of a Prokhorov type and that the family of functions is equi-continuous with respect to the Prokhorov distance at some point of the space X. Moreover, we will construct a counterexample which show that the above equi-continuity condition cannot be dropped.     

A trichotomy theorem for o-minimal structures

Let M = M, <, ... be alinearly ordered structure. We defineM to be o-minimal if every definable subset of M is a finiteunion of intervals. Classical examples are ordered divisibleabelian groups and real closed fields. We prove a trichotomytheorem for the structure that an arbitraryo-minimal M can induceon a neighbourhood of any a in M. Roughly said, one of the followingholds:

(i) a is trivial (technical term), or

(ii) a has aconvex neighbourhood on which M induces the structureof anordered vector space, or

(iii) a is contained in an open intervalon which M inducesthe structure of an expansion of a real closedfield.

The proof uses ‘geometric calculus’ whichallows one to recover a differentiable structure by purely geometricmethods. 1991 Mathematics Subject Classification: primary 03C45;secondary 03C52, 12J15, 14P10.     

Random invariant measures for Markov chains,and independent particle systems

Summary Let P be the transition operator for a discrete time Markov chain on a space S. The object of the paper is to study the class of random measures on S which have the property that MP=M in distribution. These will be called random invariant measures for P. In particular, it is shown that MP=M in distribution implies MP=M a.s. for various classes of chains, including aperiodic Harris recurrent chains and aperiodic irreducible random walks. Some of this is done by exploiting the relationship between random invariant measures and entrance laws. These results are then applied to study the invariant probability measures for particle systems in which particles move independently in discrete time according to P. Finally, it is conjectured that every Markov chain which has a random invariant measure also has a deterministic invariant measure.Research supported in part by N.S.F. Grant No. MCS 77-02121     

Numerical Radius Norms on Operator Spaces

We introduce a numerical radius operator space (X, W_n). Theconditions to be a numerical radius operator space are weakerthan Ruan's axiom for an operator space (X, O_n). Let w(·)be the numerical radius on B(H). It is shown that, if X admitsa norm W_n(·) on the matrix space M_n(X) which satisfiesthe conditions, then there is a complete isometry, in the senseof the norms W_n(·) and w_n(·), from (X, W_n) into(B(H), w_n). We study the relationship between the operator space(X, O_n) and the numerical radius operator space (X, W_n). Thecategory of operator spaces can be regarded as a subcategoryof numerical radius operator spaces.     

An Exceptional Set in the Ergodic Theory of Markov Maps of the Interval

It is known that a Markov map T of the unit interval preservesa measure µ, say, equivalent to Lebesgue measure, andthat almost every point of the interval has a forward orbitunder T that is uniformly distributed with respect to µ.In the opposite direction the main result of this paper statesthat there is a set of points having Hausdorff dimension 1 whoseforward orbits are in a certain sense very far from being sodistributed. 1991 Mathematics Subject Classification: 58F08,28A80.     

Intersections of the Left and Right Essential Spectra of 2 x 2 Upper Triangular Operator Matrices

In this paper, perturbations of the left and right essentialspectra of 2 x 2 upper triangular operator matrix M_C are studied,where is an operator acting on the Hilbert space H K. For given operators A and B, thesets and are determined, where _le(T) and _re(T) denote, respectively,the left essential spectrum and the right essential spectrumof an operator T. 2000 Mathematics Subject Classification 47A10,47A55.     

On the West Decomposition of Riesz Operators

Let T be a Riesz operator on a Banach space and let {_n} be anenumeration of the nonzero eigenvalues of T, each repeated accordingto algebraic multiplicity. Then a sufficient condition thatT admit a West decomposition T = C + Q, where C is compact andQ is quasinilpotent, is that n[_n] < . Furthermore, C andQ can be chosen to have a quasinilpotent commutator.     

The Generic Local Spectrum of Any Operator is The Full Spectrum

Let T=(T₁, ..., T_n) be a commuting n-tuple of bounded linearoperators acting on some complex Banach space X. We show thatif T has the single-valued extension property, then the localspectrum _T(x) coincides with the spectrum (T), for all vectorsx X, except on a set of the first Baire category. 1991 MathematicsSubject Classification 47A11, 47A13.     

Lumpability and Commutativity of Markov Processes

Abstract

We introduce the concepts of lumpability and commutativity of a continuous time discrete state space Markov process, and provide a necessary and sufficient condition for a lumpable Markov process to be commutative. Under suitable conditions we recover some of the basic quantities of the original Markov process from the jump chain of the lumped Markov process.