Functional Analysis Seminar (2002-2004)

Oct. 3, 2002.
Jan Rychtar, University of Alberta
Title: "On dual W^*LUR and URED norms"
Abstract: It will be shown that C(K)^* admits an equivalent dual URED norm whenever K is descriptive compact, that is whenever C(K)^* admits an equivalent dual W^*LUR norm. On the other hand there will be given an example of a compact space K such that C(K)^* admits an equivalent dual URED norm but no dual W^*LUR norm.

Oct. 10, 2002.
Peter A. Loeb, University of Illinois, Champaign-Urbana
Title: "Limit Theorems via Local Maximal Functions"
Abstract: A general approach to limit theorems is first illustrated with an elementary proof of the Lebesgue Differentiation Theorem using a local version of the classical maximal function. We then sketch the general result for limit theorems in analysis and probability theory showing that one need only consider sets where the input vanishes.

Oct. 17, 2002.
Fereidoun Ghahramani, University of Manitoba, Winnipeg
Title: "Derivations from Segal algebrs"
Abstract: In this talk I will show that continuous derivations from certain Segal algebras into their dual modules or into the algebrs are approximately inner. Thus evrery Segal algebra on an abelian group is weakly amenable. I will also discuss multipliers and derivations of Segal algebras on compact groups. This is joint work with Tony Lau.

Oct. 24, 2002.
Volker Runde, University of Alberta
Title: "The Fourier algebra is always operator biflat"
Abstract: Let $G$ be a locally compact group. Then its Fourier algebra $A(G)$ is biflat in the sense of quantized Banach homology. This is joint work with Nicolaas Spronk and Oleg Yu. Aristov.

Oct. 31, 2002.
Roman Vershynin, University of Alberta
Title: "Randomized Principle of Restricted Invertibility"
Abstract: I will discuss old and new results on the principle of restricted invertibility of operators on Hilbert space H, which is due to J.Bourgain and L.Tzafriri. The Principle says that there is a large subset of the orthogonal basis of H where a given operator T : H -> H is invertible. How large is the family of such subsets? (does a random set work?) Along with sharp positive results in this direction, we will also review some applications to Harmonic Analysis and Convex Geometry.

Nov. 7, 2002.
Garth Dales, University of Leeds
Title: "Modules over group algebras"
Abstract: Let A be a Banach algebra, and let E be a Banach left A-module. Then there are standard definitions (to be recalled) that tell us when E is projective or injective in the appropriate category. Let us fix A to be the group algebra L^1(G) of a locally compact group G, and think of various favourite examples of Banach left L^1(G)-modules: for example E = L^1(G), E = M(G), E = C_0(G), E = L^{\infty}(G), or E = L^p(G). We try to determine the conditions on G for the module to be projective or injective. We succeed rather often, but not always. This is work with Maksim Polyakov

Nov. 21, 2002.
Peter Hajek, Czech Academy of Sciences, Prague
Title: "Gateaux differentiable functions"
Abstract: We will show a simple example of a Gateaux differentiable Lipschitz mapping $\varphi$ from $\ell_1$ into $I\!\!R^2$ such that $\|\varphi'(x)-\varphi'(y)\| \ge 1$ for all $x,y\in\ell_1$ such that $x\not=y$. This is impossible for mappings from any Banach space into $I\!\!R^1$. A joint work with Robert Deville.

Nov. 28, 2002.
Alain Pajor, Universite de Marne-la-Vallee
Title: "Regularization of star bodies by random hyperplane cut off"
Abstract: We present a general result on regularization of arbitrary convex body (and more generally on star-body), that gives and extends a global form of well known local facts as the low $M^*$-estimate, large Euclidean section of finite volume-ratio spaces and others.

Dec. 5, 2002.
Volker Runde, University of Alberta
Title: "(Non-)amenability of Fourier and Fourier--Stieltjes algebras"
Abstract: Let $G$ be a locally compact group. We show that its Fourier algebra $A(G)$ is amenable if and only if $G$ has an abelian subgroup of finite index, and that its Fourier--Stieltjes algebra $B(G)$ is amenable if and only if $G$ has a compact, abelian subgroup of finite index.

Jan. 21, 2003.
Jan Rychtar, University of Alberta
Title: "Pointwise uniformly rotund norms"
Abstract: We will show a characterization of compact spaces carrying strictly positive measure in terms of pointwise uniformly rotund renorming of C(K). Other results connecting this renorming with topological structure of Banach spaces will be presented.

Feb. 25, 2003.
Rick Loy, , ANU
Title: "Closed ideals in the algebra of operators on a Banach space"
Abstract: The lattice of closed ideals in the algebra of operators on a Banach space is in general very complicated, but can also be very simple. After reviewing known results, we will discuss recent work on spaces which are suitable (infinite) direct sums of finite dimensional spaces. This is a class of spaces which have been studied extensively for many years with respect to other properties. Here many of the standard closed ideals can be shown to coincide, the problem is what can be said about others, in particular which are maximal. This is ongoing joint work with Niels Laustsen of Copenhagen.

Mar. 4, 2003.
Yehoram Gordon, The Technion, Israel
Title: "Local theory of convex bodies in Euclidean space"
Abstract: We shall present some results on computations of distances, volumes, and various parameters which studied in Local Theory. The tools used in the local theory are in many cases probabilistic. Some of the results presented are very recent, and some are classical.

Mar. 11, 2003.
Piotr Mankiewicz, Polish Academy of Sciences
Title: "Search for Euclidean subspaces"
Abstract: We shall discuss different methods of finding well Euclidean subspaces or quotients of finite dimensional Banach spaces. This will include the classical F. John Theorem, Dvoretzky Theorem, volume ratio argument, shrinking effect and some recent results as well.

Mar. 18, 2003.
Haryono Tandra, University of Alberta
Title: "On defining an induced trace of a discrete nilpotent group"
Abstract: Given a normal subgroup $N$ of a nilpotent group $H$ of class 2, an induced trace of $H$ by a trace of $N$ will be defined by exploiting the weakly-almost-periodicity of bounded continous functions on $H$. In this talk we shall recall the definition of characters of discrete groups, and explain the motivation why the concept of induced trace is needed.

Mar. 25, 2003.
Jon Vanderwerff, La Sierra University, California
Title: Convex sets that can be expressed as countable intersections of half-spaces"
Abstract: Recently, Azagra & Ferrera showed that if a closed convex set can be written as a countable intersection of half-spaces, then it is the set of minimizers of some $C^\infty$-smooth convex function. This talk will examine when closed convex sets can be written as countable intersections of half-spaces and relationship between the existence of closed convex sets that cannot be represented as countable intersections of half-spaces and the existence of certain uncountable "almost" biorthogonal systems as well as some other types of "Kunen-Shelah properties" as studied recently by Granero et. al.

May 1, 2003.
Alan Paterson, Univ. of Mississippi
Title: "Harmonic analysis and a Kasparov module for index theory"
Abstract: We start by recalling how the analytic index of an order $0$ elliptic pseudodifferential operator (pdo) on a compact manifold is defined in the Atiyah-Singer index theorem. It is defined in terms of the Fredholm index which has its values in the integers, $K(pt)$. This will be used to motivate a much more general structure for obtaining K-classes, that of a {\em Kasparov module}. We then discuss informally the construction of such a module for an elliptic pdo which is equivariant under the proper action of a continuous family groupoid. For illustration, we will concentrate on the locally compact group case and show how a harmonic analysis construction gives the index as an element of $K(C_{r}^{*}(G))$.

Sep. 2, 2003.
Stanimir Troyanski, University of Murcia, Spain
Title: "Uniformly smooth renorming of Banach spaces with modulus of convexity of power type 2"
Abstract: Assume that for the modulus of convexity we have $\delta_X(\varepsilon)\ge c\varepsilon^2$. we find estimates for $q(c)$ such that for every $q$ smaller than $q(c)$ there exists a norm $\|\cdot\|_q$ for which we have $\rho_{X}(\tau)\leq k \tau^q$, where $\rho$ is a modulus of smoothness.

Sep. 24, 2003.
Jan Rychtar, University of Alberta
Title: "On smoothness in WCG spaces"
Abstract: The well known Mazur's theorem says that a continuous convex function f on a separable Banach space X is Gateaux differentiable on a dense $G_\delta$ set. The same is true for a WCG space. In the talk we will show that the set of points of Gateaux smoothness of a continuous convex function on WCG space X is even bigger. Namely, if K is weakly compact symmetric convex set such that closed linear hull of K is X, and x is in X, then there is a point of Gateaux smoothness in x+K.

Oct. 8, 2003.
Shahar Mendelson, ANU
Title: "Geometric structures in Learning Theory"
Abstract: We present several connections between Statistical Learning Theory and the Local Theory of normed spaces. We focus on a comparison between the empirical and normed structures endowed on a random coordinate projection of a class of functions, and an application of this comparison to Learning Theory.

Oct. 15 and 22, 2003.
Volker Runde, University of Alberta
Title: "Operator space structure and amenability for Figa-Talamanca-Herz algebras" (Joint work with A. Lambert and M. Neufang)
Abstract: Column and row operator spaces - which we denote by COL and ROW, respectively - over arbitrary Banach spaces were introduced by Lambert in his Ph.D. thesis; for Hilbert spaces, these definitions coincide with the usual ones. Given a locally compact group $G$ and $p,p' \in (1,\infty)$ with $\frac{1}{p} + \frac{1}{p'} = 1$, we use the operator space structure on $CB(COL(L^{p'}(G)))$ to equip the Figa-Talamanca-Herz algebra $A_p(G)$ with an operator space structure, turning it into a quantized Banach algebra. Moreover, we show that, for $p \leq q \leq 2$ or $2 \leq q \leq p$ and amenable $G$, the canonical inclusion $A_q(G) \subset A_p(G)$ is completely bounded (with cb-norm at most $K_G^2$, where $K_G$ is Grothendieck's constant). As an application, we show that $G$ is amenable if and only if $A_p(G)$ is operator amenable for all - and equivalently for one - $p \in (1,\infty)$; this extends a theorem by Z.-J. Ruan.

Nov. 5, 12, and 19, 2003.
Piotr Mankiewicz, Polish Academy of Sciences
Title: "On the duality of the entropy numbers"
Abstract: Very recently an essential progress has been made in the Duality of Entropy Numbers Conjecture. Namely, S Artstein, V. Milman and S.J. Szarek verified the conjecture in the case when one of the spaces involved is a Hilbert space. The aim of my talk is to present the result, its reduction to the finite-dimensional problem concerning covering numbers for finite-dimensional convex bodies, and to provide some of the crucial arguments in the proof.

Nov. 26, 2003.
Vladimir Troitsky, University of Alberta
Title: "Strictly semi-transitive algebras and semigroups of operators"
Abstract: A collection of operators on a Banach space X is said to be strictly semi-transitive if for every non-zero x, y in X there is an operator T in the family such that either Tx=y or Ty=x. In this talk we will discuss examples and properties of strictly semi-transitive algebras and semigroups of operators on finite and infinite dimensional Banach spaces.

Jan. 9, 2004.
Stanislaw Szarek, Case Western Reserve University, Cleveland
Title: "On Knaster conjecture"

Jan. 14, 2004.
Artem Zvavitch, University of Missouri
Title: "The Busemann-Petty problem for Gaussian measures"
Abstract: "In this talk we will present a formula connecting the Minkowski functional of a convex body with the Gaussian measure of its sections via Fourier transform. Using this formula we will solve the Busemann-Petty problem for Gaussian measures, asking whether symmetric convex sets with smaller Gaussian measures of hyperplane sections necessarily have smaller Gaussian measure."