A topological semigroup S has the congruence extension property if closed congruences on closed subsemigroups extend to closed congruences on S. A characterization of gatic (gamma-compact archimedean) semigroups with the congruence extension property(CEP) will be discussed as well as a result concerning the homomorphic image of compact semigroups with CEP.

McNeese State University, Lake Charles, LA 70509
aucoin@mcneese.edu

In her 1971 paper M.E. Rudin asked whether there are realcompact Dowker spaces (in ZFC). Two constructions will be given to show that the answer is yes.

Miami University, Oxford, Ohio
ztbalogh@miavx1.acs.muohio.edu

Interaction between individuals in a social network may be modeled by matrices representing various relationships. Derived (2nd order) relations may be constructed via matrix multiplication. Correlation coefficients are typically used to identify nearly equivalent relations for the purpose of constructing a finite semigroup representation. An alternative method will be presented based on angles between the matrices when viewed as data vectors.

University of Louisville, Louisville, Kentucky 40292
grbarn01@homer.louisville.edu

We define the hyperspace of (X, T, T*) to be (2^X, L(T), U(T*)) where 2^X is the set of non-empty T-closed subsets of X, L(T) is the lower Vietoris topology for T, and U(T*) is the upper Vietoris topology for T*.

The asymmetry of this definition (including the fact that 2^X depends on T but not on T*) creates a favorable context for certain bitopological separation properties to hold in this hyperspace. Two bitopological compactness properties turn out to be important for this hyperspace: sup-compactness, and a property closely related to stability. Finally, a bitopological form of sobriety is developed, with applications for hyperspaces and compactness.

Roger Williams University
bruce@alpha.rwu.edu

It is possible to use the structure of a topological semigroup to model the activities of a small community of individuals (called a social network). In particular, it can be used to model the means by which an Academic Department conducts departmental business. The model will be demonstrated by means of an example of 18 individuals. The construction of the topological semigroup will be developed in detail.

University of Louisville, Louisville, Kentucky 40292
pbcerr01@homer.louisville.edu

For certain classes K of algebraic structures, we determine which A in K have the property that the endomorphism monoid of A satisfies a specified semigroup-theoretic condition such as regularity.

State University of New York, New Paltz, NY 12561
adamsm@matrix.newpaltz.edu

Vanderbilt University, Nashville, TN 37240
mgould@math.vanderbilt.edu

We learn in a first course in multivariable calculus that a real-valued function of two real variables x, y can be continuous along each horizontal and each vertical line in the plane without being (jointly) continuous. If X, Y are topological spaces, we consider four topologies on their Cartesian product X×Y. Three of them are the usual product topology \tau, the smallest topology \sigma making each real-valued separately continuous function continuous, and the topology \alpha such that a subset of X× Y is in \alpha whenever its trace on each horizontal fiber X×{y} and its trace on each vertical fiber {x}×Y is open in the relative topology induced by \tau for each x in X and y in Y, and the fourth one lies between \tau and \sigma. Clearly \tau \subset \sigma \subset \alpha, and it is shown that these three topologies are distinct at least when X and Y are metric spaces without isolated points. While \sigma is usually much larger than \tau, they often share the same dense subsets, and often (X×Y, \tau) connected implies (X×Y, \sigma) is connected. (The meaning of often will be made precise.) Conditions under which each \sigma-continuous f is \tau-continuous on a \tau-dense subspace or is the pointwise limit of a sequence of \tau-continuous functions are obtained and compared with classical results of Namioka and Moran. A pretty duality between the topologies \alpha and \sigma will be described. Conditions under which the Stone-Cech remainder of (X ×Y, \sigma) is connected are given. A number of results published by Knight, Pym, and Moran in 1970-71 are generalized. (This is joint work with R. G. Woods of the University of Manitoba.)

Harvey Mudd College, Claremont, CA 91711
henriksen@hmc.edu

We give a number of characterizations of the uniform weight of a completely regular space. Let \kappa be an infinite cardinal and let X be a topological space. Consider the following conditions on a collection {G_\alpha : \alpha in \kappa} of open covers of X and an open neighborhood assignment {V_\alpha(p) : \alpha in \kappa} for X.

AU(\kappa)

(1) for all p in X, { \st(p,G_\alpha) } is a local base for p;
(2) for all \alpha in \kappa, there exists \beta in \kappa such that G_\beta <_r G_\alpha.

AU^\Delta(\kappa)

(1) for all p in X, { \st(p,G_\alpha) } is a local base for p;
(2) for all \alpha in \kappa, there exists \beta in \kappa such that G_\beta <_\Delta G_\alpha.

UF(\kappa)

for all \alpha in \kappa, there exists \beta in \kappa such that for all p in X, if V_\beta(p) \cap V_\beta(x) \ne \emptyset, then V_\beta(x) \subseteq V_\alpha(p).

UN(\kappa)

for all \alpha in \kappa, there exists \beta in \kappa such that for all p in X, if V_\beta(p) \cap V_\beta(x) \ne \emptyset, then x in V_\alpha(p).

UNDEV(\kappa)

for all \alpha in \kappa, there exists \beta in \kappa such that for all p in X, if p in V_beta(x), then V_\beta(x) \subseteq V_\alpha(p).

Theorem. Let \kappa\ge\omega; the following are equivalent for any space X.

1. There exist open covers {G_\alpha : \alpha in \kappa} that satisfy AU(\kappa);
2. There exist open covers {G_\alpha : \alpha in \kappa} that satisfy AU^\Delta(\kappa);
3. There is a local base {V_\alpha(p) : \alpha in \kappa} for X that satisfies UF(\kappa);
4. There is a local base {V_\alpha(p) : \alpha in \kappa} for X that satisfies UN(\kappa);
5. There is a local base {V_\alpha(p) : \alpha in \kappa} for X that satisfies UNDEV(\kappa);

This theorem allows one to give a neighborhood approach to uniform spaces, as opposed to the traditional entourage or uniform cover approach. Proofs using the neighborhood approach more closely resemble the proofs that one gives in the metric case. In other works, one obtains a unified approach to metrization and uniformity.

Duke University
hodel@math.duke.edu

In 1949 and 1958, I. Segal and Y. Mibu independently extended Haar measure to the Baire subsets of isogeneous uniform spaces. Work by G. Itzkowitz in 1972 and by the author in 1992 extended the theory to the Borel subsets. The complete development is given in Chapter 9 of the author's book "Modern Analysis and Topology" (Springer 1995). This extension allows the construction of a Radon-Nikodym type derivative of a measure n at a point p in a locally compact, isogeneous uniform space as the limit of the ratios n(U)/m(U) where m is Haar measure and U is a uniform neighborhood of p. We denote the function obtained in this manner by dn/dm. If dn/dm is L¹ measureable w.r.t. m we say n is L¹ differentiable. If in addition the ratios n(U)/m(U) converge uniformly to some L¹ measureable function g (w.r.t. m), we say n is uniformly differentiable and that g is the uniform derivative of n w.r.t. m.

It can be shown that if n is absolutely continuous w.r.t. m on a locally compact, sigma compact, isogeneous uniform space that the uniform derivative of n w.r.t. m exists and is equivalent to the Radon-Nikodym derivative. If in addition the underlying space is metric, it is possible to relax the hypothesis to L¹ differentiable measures and the theorem still holds. Many question arise naturally concerning relaxing the hypothesis of this theorem and others in the development of Haar measure, uniform measure and uniform derivatives. Other questions arise in trying to extend more of analysis from topological groups to isogeneous uniform spaces. The central question, however, is if there exists a Fourier type transform for L² measureable functions on locally compact isogeneous uniform spaces. The existence of Haar measure provides a natural way to do this on locally compact topological groups. With the above mentioned tools it should be possible to extend more analysis on topological groups to isogeneous uniform spaces. These problems and results will be discussed in this talk.

Institute for Defense Analyses 1801 N. Beauregard Street, Alexandria, VA 22311
nhowes@ida.org

Quasi-metrizability of a T₁ space X is the availability on X of a decreasing neighbourhood base <(x)_n> at every x in X, so constituted that, for every A\subset X and n in \omega, (writing, for each B\subset X and each m in \omega, (B)_m for \bigcup{(x)_m:x in B}) we have ((A)_\nu)_\nu\subset(A)_n for some \nu in \omega (dependent on A and n). By comparison, metrizability of T₀-spaces X is that on X of a decreasing sequence <(x)_n> of neighbourhoods at every x in X, so constituted that, for every A\subset X, we have \bigcap{(A)_n:n in \omega} = \operatorname{Cl} A = \bigcap{\operatorname{Cl}\,{(A)_n} : n in \omega} (1977).

Concordia University, Montreal, Quebec, Canada H4B 1R6

The category of bicomplete quasi-uniform frames and the category of totally bounded quasi-uniform frames are coreflective subcategories of the category of quasi-uniform frames. A technique due to B. Banaschewski and A. Pultr is employed to give an alternative description of the bicomplete coreflection of a quasi-uniform frame. The Samuel compactification of a quasi-uniform frame is the bicompletion of the totally bounded coreflection of that quasi-uniform frame.

Southern Illinois University at Carbondale Carbondale, IL 62901
hunsaker@math.siu.edu

Recently, the authors considered a notion of prime initial set in preordered sets and showed that every initial set is the intersection of prime initial sets. Here, we repeat and augment this development; then, we define two different preorders for semigroups. Each of these spotlights a class of ideals that are prime initial sets; not surprisingly, these are each "prime-like" ideals: completely prime and an apparently new class of ideals - here called quasiprime ideals. Every ideal is the intersection of quasiprime ideals. We use the theory of prime initial sets to present a unified, yet very easy, development of known results in the theory of semigroup ideals and to produce results concerning the new class of ideals and its place in the scheme of things. The class of prime semigroup ideals of a semigroup is used to induce a preorder on the semigroup; this preorder has its own families of initial sets and prime initial sets, which are ideals. Very little more is known about these objects. Finally, the theory of semilattice homomorphic images of a semigroup is easily developed in this setting, and we present a characterization of those semigroups that are expressible as semilattices of archimedean subsemigroups.

Glen Rock, NJ

Given a cardinal function f and a space X we define the f-spectrum of X, in symbols Sp(f,X), as the set of f-values taken on all infinite closed subspaces of X. In the first part of our talk we present a number of results concerning the cardinality and weight spectra of compact T₂ spaces. In particular, we study which cardinals can be omitted by these spectra. One of our main results says that under GCH a cardinal k may be omitted by the weight spectrum of a compact T₂ space if and only if k is singular.

In the second part of the talk we study the image spectrum Im(f,X,C) which, by definition, is the set of f-values taken on all (infinite) continuous images of X from the class of spaces C. We show for instance that if T is the class of Tychonov spaces and X is in T then no cardinal k < w(X) can be omitted by Im(w,X,T) provided k is regular or has countable cofinality. The question whether this is true for all cardinals k remains open.

We also characterize those regular cardinals k that can be omitted by an image spectrum of the form Im(d,X,K), where d is the density function, K is the class of compact Hausdorff spaces and X is in K.

Mathematical Institute of the Hungarian Academy of Sciences, and Miami University
juhaszi@miavx1.acs.muohio.edu

The hull-kernel topology (also known as the Stone topology or the Zariski topology) on a family of prime ideals in a ring is an important tool in algebra and analysis.

After briefly reviewing one construction of the h-k topology and noting that the derivation has little to do with rings, I will use some simple notions from the theory of ordered sets to investigate h-k and other topologies on families of subsets of a given set.

New Mexico State University

Distance spaces are a generalization of metric spaces, obtained by replacing the nonnegative integers as distances by elements of a partially ordered, commutative monoid. Examples are sets as distances, partially ordered distances, Hottentot metrics, Manhattan metrics and their vector refinements, and median distance spaces. Median distance spaces are defined by a property of Manhattan metrics: any three points have a unique median. They are proven to be equivalent to median algebras of all cardinalities. More generally, to any subdirect decomposition of an algebra is associated a distance space. With this, and in a manner similar to that for median distance spaces, two other easily defined classes of distance spaces are found that characterize Boolean algebras and elementary Abelian 2-groups. An open problem is to find simple axioms on distance spaces which will capture other more common classes of algebras, such as rings and lattices. A broader question is whether such axiomatizations are possible in general.

856 Buffalo Valley Road, Cookeville, TN 38501
knoebel@math.vanderbilt.edu

We give some applications which take advantage of notational and category-theoretic differences between quasiuniform spaces and continuity spaces, such as nonexpansive maps.

A continuity space is a set X with a (not necessarily symmetric) distance function valued in an ordered abelian semigroup (\cal V,+,\leq) with least element and identity 0, greatest absorbing element \infty, and satisfying d(x,x)=0, d(x,z)\leq d(x,y)+d(y,z). This semigroup has additional structure; for this talk we assume (V, \leq) is a completely distributive lattice, \infty\not=0 and for all p in V, S\subseteq V, p + \bigwedge S = \bigwedge_{s_in_S}(p+s)

We also assume that if v,w\succ 0 then v\wedge w\succ 0, where v\succ w if whenever \bigwedge S\leq w, there is an s in S such that s\leq v. Such a semigroup is called a value quantale, and v in V is positive if v\succ 0.

City College, CUNY, New York, NY, 10031
rdkcc@cunyvm.cuny.edu

\def\wto{\overset{W}\to\rightarrow}

Let S be a locally compact topological semigroup. The set of all regular probability measures defined on S is denoted by P(S). In this paper, the necessary and sufficient conditions for strong uniform convergence of composition sequence of probability measures on topological semigroups are studied. Our main results are:

Theorem 1 Let N be a simple semigroup of measures on S and E(N)\ne\emptyset. Denote that G₁=\cup_{\mu in N} S_\mu and G₁=\overline{\cup_{\mu in N} S_\mu}. Then there exists a simple minimal ideal K of G₁ such that G₁\subset\overline{K}\subset G₂.

Corollary 1.1 Let N be a completely simple semigroup of measures on S. Then G₁=G₂ and both are completely simple semigroups.

Corollary 1.2 Let {\mu_n} be a sequence in P(S) such that for k>0, the sequence \mu_k,n = \mu_k+1*\mu_k+2*...*\mu_n \wto \lambda_k (as n-->\infty). Let N₁ be a set of all weak cluster points of the sequence {\lambda_k}. Suppose that {\mu_k,n} is tight. Then S₁=\overline{\cup_{\mu in N1} S_\mu} is a completely simple subsemigroup of S.

Theorem 2 Let S be a locally compact L-X topological semigroup (S is called a L-X semigroup if ex=xe for every e in E(S) and x in S), and \mu_n in P(S), n=1,2,3,.... If, for k>0, \mu_k,n\wto\lambda_k, then S₁ is a closed subgroup of S.

Theorem 3 Let {\mu_n} be a sequence in P(S) such that for k>0, the \mu_k,n\wto\lambda_k. Then \lambda_k\wto Haar measure \omega if and only if S₁ is a subgroup of S.

Theorem 4 Let S be a locally compact L-X topological semigroup and \mu_n in P(S), n=1,2,3,.... If, for k>0, \mu_k,n\wto\lambda_k, and {\mu_k,n} is tight, then \lambda_k\wto Haar measure \omega.

Department of Statistics, Shandong Institute of Economics, Jinan, Shandong 250014, China
Supported partly by the National Natural Science Foundation of China.

The group Inn S of all the inner automorphisms of a semigroup S of transformations of a finite n-element set X_n may be identified with a subgroup G of the symmetric group S_n on X_n. This talk addresses the problem of describing S given that Inn S = G for certain specific G.

University of Louisville Louisville, Kentucky 40292
i0levi01@ulkyvm.louisville.edu

The space Id(A) of closed ideals of a C^*-algebra A admits two natural topologies: \omega (= lower or hull-kernel topology) and \sigma (= Scott topology) so that (Id(A), \omega, \sigma) forms a well behaved joincompact bitopological space. The separate topological spaces (Id(A), \omega) and (Id(A), \sigma) are naturally occurring examples of skew compact, asymmetric (non-Hausdorff) topological spaces as studied in R. Kopperman, Asymmetry and duality in topology, Top \& Appl. 66(1995), 1-39.

Recall that when A is commutative, the maximal ideal space of A, under the hull-kernel topology, is locally compact Hausdorff. For arbitrary C^*-algebras, non-commutativity of A, generally, induces asymmetric (non-Hausdorff) topologies on all of the naturally occurring subspaces of (Id(A), \omega). Thus the non-commutativity of A and the asymmetry of topologies on the ideal spaces of A are intimately intertwined.

The properties of (Id(A), \omega) are described in the context of an involution algebra (*-algebra) that has no pre-assigned norm.

University of Kentucky, Lexington, KY 40506-0027
mack@ms.uky.edu

A nearring is a triple (G,+,*) where (G,+) is a group, (G,*) is a semigroup and (a+b)*c = a*c+b*c for all a,b,c in G. The nearring (G,+,*) is a topological nearring if, in addition, both + and * are continuous functions from G×G into G. We begin by presenting a general result which tells how to get all the multiplications * such that (G,+,*) is a topological nearring whenever (G,+) is a locally compact Hausdorff group. We apply this result to the n-dimensional torus group Tⁿ and, indeed, we are able to completely determine all the multiplications * such that (Tⁿ,+,*) is a topological nearring. We then turn our attention to the n-dimensional Euclidean group Rⁿ and we apply the general result to this case as well. However, a difficulty in applying the result arises when one attempts to verify that the multiplication * is associative. It turns out that this problem is equivalent to the problem of solving a system of n² functional equations involving n² continuous selfmaps of Rⁿ. We do not determine all the multiplications * such that (Rⁿ,+,*) is a topological nearring (except when n = 1, we get them all in this case), but we do produce a large class of such multiplications. Each of these is induced by a continuous function \lambda from Rⁿ to R which satisfies a certain condition and the resulting nearring is denoted by N_\lambda(Rⁿ). We show that each of these nearrings is either simple or contains a unique maximal proper ideal. All this comprises the first portion of the paper. In the second portion, we investigate the nearring, N_\lambda(X, Rⁿ), of all continuous functions from a compact Hausdorff space into the topological nearring N_\lambda(Rⁿ) under the pointwise operations. We characterize the maximal ideals of N_\lambda(X,Rⁿ) whenever Z(\lambda) = {0} where Z(\lambda) = \lambda^-1(0). In the case where Z(\lambda)\ne{0}, we characterize a subset of maximal ideals and this characterization is sufficient to derive the isomorphism theorem. Specifically, if X and Y are two compact Hausdorff spaces and \lambda satisfies a rather mild condition, then the nearrings N_\lambda(X, Rⁿ) and N_\lambda(Y, Rⁿ) are isomorphic if and only if X and Y are homeomorphic. We then investigate, in detail, the homomorphisms from one nearring of continuous realvalued functions to another. Some of the results concerning the nearrings N_\lambda(X, Rⁿ) are reminiscent of results in the classical rings of continuous functions, but even when the results look the same, the proofs are often quite different, due to the vast difference between the structure of the topological nearring N_\lambda(Rⁿ) and the field of real numbers.

SUNY at Buffalo, Buffalo, NY
mthmagil@ubvms.cc.buffalo.edu

A D-class of a semigroup S is usually visualized as an "eggbox" in which the rows correspond to R-classes and the columns to G-classes. In this talk we describe the shape of the D-classes of certain regular semigroups of transformations on a finite set.

University of Louisville, Louisville, Kentucky 40292
rbmcfa01@homer.louisville.edu

By a semigroup of a topological space X we mean S(X), the semigroup of all continuous selfmaps on X, under the operation composition. In this talk we will describe all those Peano Continua whose semigroups have the property that their regular \Cal J-classes form a finite lattice. In an earlier paper (Semigroup Forum, Vol. 50, 1995, 265-285) we have characterized the dendrites whose semigroups possess the same property.

SUNY at Buffalo, Buffalo, NY
mthmagil@ubvms.cc.buffalo.edu

The College of Staten Island, CUNY, Staten Island, NY
misra@postbox.csi.cuny.edu

A partially ordered space is a triple (X,\tau,\le) consisting of a set, a topology on the set, and a partial order which is compatible with the topology in the sense that the topology has a base of convex sets. A theory of ordered quotient maps and their applications to ordered compactifications will be presented. Of particular interest will be conditions under which the identification of the components of remainders results in totally disconnected, order totally disconnected, zero-dimensional, or order zero-dimensional remainders.

Western Kentucky University, Bowling Green, KY 42101
mooney@pulsar.cs.wku.edu

Semigroups (S,\cdot) and (S,*) are interassociative if (x \cdot y) * z = x \cdot (y * z) and (x * y) \cdot z = x * (y \cdot z) for all x, y, z in S. When this occurs, we say that (S,\cdot) interassociates with (S,*), or that the semigroups are interassociates of each other. For certain classes K of semigroups, we characterize all interassociates of members of K.

David Lipscomb University, Nashville, TN 37204-3951
nelsonal@dlu.edu

A number of topologies have been defined on trees in general. Best known to general topologists is the interval topology, often called simply "the tree topology." Its base is the set of all intervals (a,b] = {t: a < t \leq b} where a is allowed to be -\infty, a point below all points of T. A focal point of the talk will be the classification of Aronszajn trees with the interval topology under various set-theoretic axioms. The coarse wedge topology, which gives a compact zero-dimensional monotonically normal topology to complete trees, is the other main topology discussed. Its base is the Boolean algebra generated by all wedges V_t = {s: s \geq t } such that t is on a successor level. Omitting this last requirement gives the fine wedge topology, which will be briefly discussed as will the logician's wedge topology.

University of South Carolina, Columbia, SC
nyikos@math.sc.edu

1. We generalize Katetov's Lemma so that we don't have to 'apply the argument of Lemma 1 and 2 for each case ...' in order to apply the lemma.
2. It was claimed that Katetov's Lemma 'does not appear to be readily adaptable to a proof of' the Blair's Insertion Theorem which yields many known results. Using the generalized result, we show that there is actually a very simple Katetov-type proof to the theorem.
3. We use the generalized result to obtain a simple Katetov-type proof of Kubiak's result on the insertion property of monotonically normal spaces.

University of South Carolina, Columbia, SC 29208
pan@milo.math.sc.edu

The work of Souslin and Kurepa made it clear how important it is to know whether a given linearly ordered or generalized ordered space has a dense subspace with special metric related propertied. We study the relationship among the following four properties on generalized ordered spaces:

1. X has a dense subspace that is the union of countably many closed, discrete subspaces of X;
2. X has a dense metrizable subspace;
3. X has Property III, i.e., there are open sets U(n) and relatively closed discrete subsets D(n) \subset U(n) such that if G is open and p in G, then for some n \geq 1, we have p in U(n) and G \cap D(n) \neq \emptyset;
4. X has a dense subset that is the union of countably many discrete (but not necessarily closed) subspaces.

Barry University
purisch@dominic.barry.edu

A Banach space has the Dunford-Pettis property if and only if the composition mapping m:l₁(X)× K(X) --> l₁(X) is jointly weakly sequentially continuous. Here, l₁(X) is the l₁-sequence of elements in X and K(X) is the space of Compact operators on X.

Miami University Middletown, 4200 East University Bvd., Middletown, Ohio

Ordered compactifications which have zero-dimensional remainders are discussed, with particular attention to the case of compactifications of totally ordered spaces. Every ordered compactification of a totally ordered space X has zero-dimensional remainder, and thus the topological compactification underlying the Stone-Cech ordered compactification of X is \leq the Freudenthal compactification of the underlying topological space. Conditions under which these two compactifications are equal will be examined.

Western Kentucky University, Bowling Green, KY 42101
Richmond@pulsar.cs.wku.edu

Three abstract versions of the Banach Category Theorem are compared when an arbitrary ideal replaces the \sigma-ideal of meager sets. Every ideal is contained in a unique smallest ideal for which a given Banach Category Theorem holds. The behavior of these ideal extensions is also investigated.

Southeastern College of the Assemblies of God, Lakeland, FL 33801-6099
dar0se@aol.com

Let K be a nonempty compact convex subset of a Hausdorff locally convex topological vector space. Let f:K--> K be any function (not necessarily continuous or measurable). Then there exists a point \xi in K which is an "approximate fixed point" of f in the following sense:

Vanderbilt University, Nashville TN, 37240
schectex@math.vanderbilt.edu

The concept of strong normal semicontinuity arises naturally in the attempt to generalize the properties of normality and weak cb. A function f is said to be strongly normal upper-semicontinuous (snusc) if cl_X{x : f(x)>k} is a support for every k. The characteristic function of a support, for example, is snusc. We define a space X to be almost weak cb if every locally bounded, normal lower semicontinuous function on X is bounded above by a continuous function. We say a space X is meekly normal if disjoint supports are completely separated.

With some trivial changes to proofs of well-known theorems of Mack and Lane, the concept of strong normal semicontinuity leads to many equivalences of meek normality and almost weak cb. At least one expected "theorem" however, if it is true at all, may very well be non-trivial:

University of Tennessee at Martin, Martin, TN 38238
jschomme@utm.edu

Holub in his interesting paper on Shift Operators (Canad.Math.Bull.31(1988), 85-94) raised the problem of characterizing compact Hausdorff Spaces X such that the Banach Space C(X) of complex valued continuous functions (or real valued continuous functions) admits a backward shift. My joint efforts in collaboration with Professor M. Rajagopalan resulted in the complete resolution of this problem.

Theorem. If X is an infinite compact Hausdorff space, then the Banach Space C(X) in either case does not admit a back ward shift.

Related results are also discussed.

Cleveland State University, Cleveland, OH
kondagunta@lserver.math.csuohio.edu

If G is a nilpotent Lie group and \Gamma is a uniform (discrete cocompact) subgroup of G, \Gamma\backslash G is called a nilmanifold. In this case \Gamma turns out to be the fundamental group of the compact nilmanifold \Gamma\backslash G. The quasi regular representation U_\Gamma of G on L²(\Gamma\backslash G) acts on G by right translations. For each f in L²(\Gamma\backslash G), x in G, y in \Gamma\backslash G, (U_\Gamma(x)f)(y)=f(yx).

The unitary spectrum of a nilmanifold \Gamma\backslash G is determined by the corresponding quasi regular representation U_\Gamma. Two nilmanifolds are said to have the same unitary spectrum if the corresponding quasi regular representations are unitarily equivalent. This leads to the following question: If two nilmanifolds have the same unitary spectrum, do they have to be homeomorphic? The answer to this question is no but this phenomenon has been considered to be rare and the first example in the whole class of solvable lie groups appeared in the literature around 1994.

In this talk the speaker will show that this phenomenon occurs surprisingly often, by establishing a theorem (a new result obtained in 1995) that would contain this example as a special case. If time permits, the speaker would propose some open questions that arise from this result.

State Technical Institute at Memphis

The notion of a P-space, that is, a space in which every prime ideal is maximal, is discussed in the setting of frames. We show that a natural definition for a "P-frame" is a frame whose cozero part is complemented. Some rather elementary properties of these frames will be discussed as will the interaction with measurable frames. This is based on work done, in the spatial setting, by L. Gillman and M. Henricksen on P-spaces, and A. Hager on measurable spaces. The speaker would like to emphasize that the work on P-frames has just begun, and is very much "work in progress".

University of Denver
sjwayland@aol.com

This is joint work with Prof. Robert Raphael and Ms. Ruth McCoosh of Concordia University, Montreal.

The epimorphic hull EH(A) of a commutative semiprime ring A with one can be characterized as the unique smallest regular ring of quotients of A. This object always exists.

Let X be a realcompact (Tychonoff) space and let C(X) denote its ring of continuous real-valued functions. In this talk we investigate conditions under which there exists a realcompact space Y for which EH(C(X))=C(Y). We characterize those realcompact spaces X for which C(X) will have a regular ring of quotients of the form C(Y), and we determine what Y must be if EH(C(X))=C(Y) . Although we do not characterize those X for which this happens, we provide necessary conditions and some illuminating examples.

University of Manitoba
rgwoods@cc.umanitoba.ca