Department of Mathematical and Statistical Sciences

**Tuesday, January 22 (at 3 pm in CAB 369)**: Vladimir G. Troitsky (University of Texas at Austin; USA).*Title*: The invariant subspace problem in Banach spaces: some recent advances.*Abstract*: We discuss various extensions and limitations of Lomonosov's theorem. We also present some results on invariant subspaces of adjoint operators, operators with an invariant cone, and transitive algebras.- January 24
**(at 1 pm in CAB 377)**: Laurent W. Marcoux.*Title*: On the linear spans of projections in certain*C*-algebras.^{*}*Abstract*: We show that if a*C*-algebra^{*}*A*admits a certain 3 by 3 matrix decomposition, then every commutator in*A*can be written as a linear combination of at most 84 projections in*A*. In certain*C*-algebras, this is sufficient to allow us to show that every element of^{*}*A*is a linear combination of a fixed finite number of projections which lie in the algebra. - January 24
**(at 2 pm in CAB 377)**: William G. Bade (University of California at Berkeley; USA).*Title*: Splittings of extensions of commutative Banach algebras.*Abstract*: Let*A*and*B*be commutative Banach algebras with*A*semisimple. We say that*B*is an extension of*A*if*A*is iomorphic to the quotient*B*/*R*, where*R*is the radical of*B*. The extension is said to*split strongly*if there is a closed subalgebra*C*of*B*isomorphic to*B*/*R*such that*B*is the direct sum of*C*and*R*. Nice radicals are nilpotent (*R*for some^{n}= {0}*n*) or uniformly topologically nilpotent (UTN) if ||*r*||^{1/n}tends to 0 uniformly on the unit ball of*R*. Solving a long standing problem, we have: If*K*is a compact Hausdorff space, then any extension*B*of*C(K)*splits strongly provided that*R*is UTN. The key is the fact that if*R*is UTN, then any bounded group of invertibles in*B*/*R*can be lifted to a unique bounded group of invertibles of*B*. We do not know if a similar theorem holds with*C(K)*replaced by*L*, where^{1}(G)*G*is a locally compact abelian group. Partial results and open problems will be discussed. - January 31: Roman Vershynin.
*Title*: Combinatorial aspects of dimension.*Abstract*: For a convex body*K*in*R*, there is an alternative notion of dimension (called Vapnik-Chernovenkis dimension) that carries more information about the geometry of^{n}*K*than the usual, algebraic, dimension*n*. The VC dimension is well studied in combinatorics. One of the central problems in empirical processes ("the Glivenko-Cantelli problem") is reduced to the question whether the VC dimension controls the entropy of*K*(which is trivially true for the algebraic dimension). We will complete the solution of this problem. This also completes the work on the isomorphic Elton's theorem. - February 7: Mahmoud Filali (University of Oulu; Finnland).
*Title*: On the dimension of right ideals in*L*.^{1}(G)^{**}*Abstract*: For a locally compact group*G*, we see that usually the dimension of any nonzero right ideal in*L*is infinite. In particular, without assuming that^{1}(G)^{**}*G*is amenable, the radical is of infinite dimension when it is nonzero. - February 28
**(at 2:30 pm)**: Günter Schlichting (Technische Universität München; Germany).*Title*: Some remarks on neutral subgroups.*Abstract*: The canonical model of a transitive, topological transformation group*(G,X,)*is multiplication on a coset space*G/H*. We will discuss conditions for*(G,X,)*to be isomorphic to a canonical model. - Not a seminar, but an interesting colloquium:
**Tuesday, March 5, 2002 (at 3:30 pm in CAB 377)**: Ernst Albrecht (Universität des Saarlandes; Germany).*Title*: Local spectral properties for systems of linear differential operators on*L*-spaces.^{p}*Abstract*: On*L*, every constant coefficient linear partial differential operator^{2}(R^{N})*P(D)*is normal and hence has a spectral measure and a rich functional calculus. For*p 2*, there exists a spectral measure for*P(D)*on*L*if and only if^{p}(R^{N})*P(D)*is of order zero. In many cases (e.g. for elliptic operators), the operator*P(D)*is still decomposable and has some sufficiently rich functional calculus. In this lecture, we consider perturbations of systems of ordinary linear differential operators by lower order differential operators with variable coefficients. Assuming a certain decay of these coefficients, we still obtain at least locally - with more restrictive conditions even globally - spectral decomposition properties for the perturbed operator and locally a functional calculus. The results have been obtained in joint work with Werner Ricker (Katholische Universität Eichstätt).**Refreshments will be served at 3:00 pm in CAB 649.** - March 7: Ernst Albrecht (Universität des Saarlandes; Germany).
*Title*: Local spectral theory for operators with thin spectra.*Abstract*: We consider bounded linear operators*T*on a Banach space such that the spectrum*(T)*of*T*has Lebesgue measure zero (or locally: such that for some open set the intesection of that set with*(T)*has measure zero). Using Domar' criteria for the existence of subharmonic majorants, we obtain decomposability criteria for operators satisfying certain growth conditions near the spectrum (or near the thin part of the spectrum). In particular, this gives criteria for the existence of non-trivial hyperinvariant subspaces. It is shown how these methods allow to prove the non-quasi analyticity for certain algebras of functions on some compact fractal sets of measure zero. **Tuesday, March 12, 2002 (at 1 pm in CAB 229)**: Carsten Schütt (Christian Albrechts Universität zu Kiel; Germany).*Title*: Best and random approximation of convex dodies by polytopes.*Abstract*: Let*K*be a convex body in*R*with boundary^{n}*S*, and let*f: SR*be a continuous, positive function such that the integral of_{+}*f*over*S*with respect to the surface measure*µ*on_{S}*S*equals one. Let**P**be the probability measure on_{f}*S*whose density with respect to*µ*is_{S}*f*. Let be the (generalized) Gauß-Kronecker curvature, and let**E***(f,N)*be the expected volume of the convex hull of*N*randomly chosen points on*S*with respect to**P**. Then, under some regularity conditions on_{f}*S*, we have

lim vol_{n}*(K)*-**E***(f,N) (1/N)*^{2/(n-1)}= c_{n}_{S}(x)^{1/(n-1)}/f(x)^{2/(n-1)}dµ_{S}(x)

as*N*, where*c*is a constant depending on the dimension_{n}*n*only.- March 21: Martin Mathieu (Queen's University Belfast; Northern Ireland).
*Title*: Central bimodule homomorphisms of*C*-algebras.^{*}*Abstract*: An effective method to study operators on*C*-algebras consists in studying them on special classes of^{*}*C*-algebras such as simple, primitive, or prime ones first and then using some sort of reduction or representation theory to put the information obtained in homomorphic images together to a global picture. This technique works well, for example, for elementary operators, (generalised) derivations, hermitian operators, and other classes of bounded linear operators compatible with representation theory. In this talk we introduce and discuss a class of operators, the central bimodule homomorphisms, that are amenable to such an approach by definition. This allows us to unify a number of existing results in the literature and to extend them to wider classes. The results are part of the speaker's forthcoming book^{*}*Local Multipliers of*(Springer Verlag, 2002) with Pere Ara (Barcelona).*C*-Algebras^{*} - March 28: Shahar Mendelson (Australian National University).
*Title*: Geometric parameters in learning theory.*Abstract*: We will present several geometric problems which appear naturally when one attempts to estimate the "complexity" of a learning problem. We focus on questions involving the combinatorial parameters and the Rademacher averages of the given class. The aim of this talk is to show that such problems are not only interesting from the machine learning point of view but are also intriguing as purely theoretical questions. - April 4: Stefano Ferri.
*Title*: Covering numbers and universal semigroup compactifications.*Abstract*: Given a topological group*G*, its uniform compactification*uG*is a semigroup compactification characterised by the universal property that the complex-valued, bounded uniformly continuous functions defined on*G*(uniformly continuous with respect to the right uniformity of*G*) are precisely the functions which can be continuously extended to*uG*. Similarly, the weakly almost periodic compactification*wG*of*G*is characterised by the universal property that the complex-valued weakly almost periodic functions defined on*G*are precisely the functions which can be continuously extended to*wG*. In this talk results concerning the relations between certain covering numbers defined for*G*and the number of left ideals, idempotents and right-cancellable elements of*uG*will be given for a large class of topological groups which includes all the locally compact as well as many non-locally compact groups. It will also be shown how techniques similar to those used to prove these results can be used to investigate algebraic properties of*wG*in the special case in which*G*is a locally compact SIN-group. These results are part of my PhD thesis under the supervision of Dr. Dona Strauss of the University of Hull (England). - April 11: Vincente Montesinos (Polytechnical University of Valencia; Spain).
*Title*: A quantitative version of Krein's theorem.*Abstract*: Krein's heorem says that, in a Banach space, the closed convex hull of a weakly compact subset is again weakly compact. Motivated by a recent characterization of subspaces of weakly compactly generated Banach spaces (characterization in terms of the existence of a countable family of subsets of the unit ball which are "nearly" weakly compact, in the sense that their weak^{*}-closure in the bidual is not far from the space), we asked the following simple and natural question: If*A*is a bounded subset of a Banach space*X*such that its weak^{*}-closure in the bidual is "close" to*X*, is it so (with the same constant) for its convex hull? We give a precise answer to this question in separable spaces, then more generally in weakly compactly generated spaces and (less precisely) in full generality. - April 25
**(at 2 pm in CAB 657)**: Roger Smith (Texas A & M University; USA).*Title*: Approximation properties for*C*-algebras.^{*}*Abstract*: The completely bounded approximation property, introduced by Haagerup, requires the existence of a uniformly bounded net of completely bounded finite rank maps on a*C*-algebra^{*}*A*converging point norm to the identity. We investigate this property in the context of crossed products by actions of locally compact groups. To accomplish this, we introduce the completely contractive factorization property for a pair*(A,B)*of*C*-algebras. For such a pair, many properties, such as nuclearity or exactness, transfer from^{*}*B*to*A*. This is joint work with May Nilsen. - An interesting colloquium on the same day by the same speaker:
**Thursday, April 25, 2002 (at 3:30 pm in CAB 657)**: Roger Smith (Texas A & M University; USA).*Title*: Maximal abelian subalgebras of von Neumann algebras.*Abstract*: Maximal abelian subalgebras in von Neumann algebras are the infinite dimensional version of the diagonal matrices in finite matrix algebras, and play a similarly important role. In a matrix algebra, the diagonal is unique up to unitary conjugation, but the situation is much richer in infinite dimensions. We will discuss the various types of masas which arise, and present some recent joint work with Allan Sinclair on their structure.**Refreshments will be served at 3:00 pm in CAB 649.**

Last update: 4/24/02.