Department of Mathematical and Statistical Sciences

- No seminar talk this week, but an interesting colloquium:
**Thursday, September 13, 2001 (at 3:30 pm in CAB 657)**: Hermann König (Christian-Albrechts-Universität zu Kiel; Germany).*Title*: Spherical design and sections of*l*-spaces._{p}*Abstract*: The unit balls of*l*-spaces admit spherical sections if and only if_{p}^{N}*p*is an even integer*p=2k*. This problem of imbedding euclidean spaces*l*into_{2}^{n}*l*is equivalent to the construction of spherical designs of degree_{2k}^{N}*2k*in*S*. Given integers^{n-1}*k*and*n*, the minimal number*N=N(k,n)*such that*l*imbeds into_{2}^{n}*l*yields tight designs which are of interest for integration over spheres. We outline the construction of sequences of nearly tight designs by using coding theory and Gaussian sums._{2k}^{N}**Refreshments will be served at 3:00 pm in CAB 649.** - September 19: Vaclav Zizler.
*Title*: Weakly compact generating.*Abstract*: A Banach space*X*is weakly compactly generated if there is a weakly compact set*K*in*X*whose closed linear span is*X*. We characterize all Banach spaces that are subspaces of weakly compactly generated spaces. Given a weakly compactly generated space*X*, we characterize all subspaces of*X*that are weakly compactly generated. As an application, we present a short proof of the known result that a continuous image of an Eberlein compact is an Eberlein compact. A compact space is an Eberlein compact if it is homeomorphic to a weakly compact set in*c*for some considered in its weak topology. This talk is based on joint work with M. Fabian and V. Montesinos._{0}() - September 26: Matthias Neufang.
*Title*: An invitation to operator spaces.*Abstract*: The closed self-adjoint subalgebras of the algebra*B(H)*consisting of all bounded operators on a Hilbert space*H*can be characterized abstractly via an intrinsic property of the algebra; these (abstract) algebras are called*C*-algebras. A natural question is whether there exists an analogous characterization for closed^{*}*subspaces*of*B(H)*. In his thesis (1988), Ruan solved this problem by formulating two simple conditions for a Banach space which guarantee the latter admits an isometric embedding in some*B(H)*. These spaces, born in 1988, are called operator spaces. The crucial idea underlying Ruan's characterization is to take into account not only the Banach space*X*itself, but all the matrix levels*M*, where_{n}X*M*denotes the_{n}*n*x*n*-matrices over the complex numbers. Consequently, in the category of operator spaces, the morphisms are operators which respect this matricial structure in a suitable fashion, called completely bounded operators. The objective of the talk is to present a - highly non-comprehensive - panorama view of the theory of operator spaces and completely bounded mappings, as developed by Effros-Ruan, Blecher-Paulsen, Pisier, Wittstock, Haagerup, et al. We particularly focus on the impact of the theory on classical questions in functional analysis, such as the solution of the famous Halmos problem, and discuss intriguing applications to abstract harmonic analysis. This aims at revealing the very nature of operator spaces - heading for the*C*-world while standing on the ground of Banach spaces.^{*} - October 3: Emmanuel Ngembo (Faculté Saint Jean).
*Title*: Some function vector sublattices of*C(K,E)''*.*Abstract*: Let*K*be a compact Hausdorff space. It is well known there is no canonical embedding of the real Banach lattice of all bounded, real-valued functions on*K*into*C(K)''*, the topological bidual of*C(K)*, where*C(K)*denotes the usual real Banach lattice of continuous functions on*K*. A natural and interesting question is whether there exist (non-trivial) function spaces contained in*C(K)''*. H. H. Schaefer established that the spaces of real Radon, Baire, and Borel functions can be embedded into*C(K)''*. These results are based on two fundamental properties of the real line*R*: (a)*R*contains an order unit; (b) topological convergence and order convergence for sequences in*R*coincide. I will give an extension of Schaefer's results to the normed case. More precisely, I consider a compact Hausdorff space*K*and a real Banach lattice*E*with the Lebesgue property, i.e. order continuity of the norm. (This means that every monotone net in*E*which increases to*x*converges to*x*in the norm topology.) Then, by using the lower semi-continuous functions, I construct some*E*-valued, bounded function spaces which can be considered vector sublattices of*C(K,E)''*, where*C(K,E)''*denotes the topological bidual of the Banach lattice of all*E*-valued, continuous functions on*K*. **Friday, October 12, 2001 (at 2:00 pm in CAB 365)**: Juliana Erlijman (University of Regina).*Title*: Braid type subfactors.*Abstract*: There is a natural way of constructing subfactors of hyperfinite II_{1}-factors from unitary braid group representations that satisfy certain properties (which arise in connection to quantum groups). We will describe some of these constructions and compute invariants such as their index and relative commutants.There is also an interesting colloquium on the same day (and by the same speaker):

**Friday, October 12, 2001 (at 4:00 pm in CAB 657)**: Juliana Erlijman (University of Regina).*Title*: Von Neumann factors and braid representations.*Abstract*: We give a brief introduction to Jones' theory of von Neumann subfactors and a discussion of examples of subfactors from braid representations.**Refreshments will be served at 3:30 pm in CAB 649.**- October 17: Roman Vershynin.
*Title*: The Sauer-Shelah Lemma.*Abstract*: The Sauer-Shelah lemma is a classical result about subsets*A*of a discrete cube {0,1}. If^{n}*A*has large cardinality then there exists a coordinate projection (onto many coordinates) such that the image of*A*is the full cube on these coordinates. This theorem has been used in a number of applications, including those in Banach space theory, probability and information theory. I will speak on the same problem for the*continuous*cube [0,1], where "largeness" of a subset^{n}*A*is measured by its volume. We are interested in the coordinate projections of*A*that contain a cube [0,vol*(A)*](or its multiple by an absolute constant). Not surprisingly, this result can be applied to convex geometry (take^{n}*A*to be convex). Finally, we will look for the Sauer-Shelah lemma in generic product spaces. - October 24: Alexander Litvak.
*Title*: On the asymmetry constant of a body with few vertices.*Abstract*: We show that a non-degenerated polytope in*R*with^{n}*n+k*,*1 k < n*, vertices is far from any symmetric body. We provide the asymptotically sharp estimates for the asymmetry constant of such polytopes. This is joint work with E. D. Gluskin. - October 31: Matthias Neufang.
*Title*: Amplification of completely bounded operators and Tomiyama's slice maps.*Abstract*: It is a characteristic feature of completely bounded operators on*B(H)*to admit an amplification to completely bounded operators on*B(H*, where_{2}K)*H*and*K*are Hilbert spaces. Using Wittstock's Hahn-Banach principle and Tomiyama's slice map theorem, one deduces that, more generally, any completely bounded mapping on*M*can be amplified to yield a completely bounded mapping on the von Neumann tensor product*M N*, whenever*M*and*N*are either von Neumann algebras or dual operator spaces with at least one of them sharing property*S*, as introduced by J. Kraus. Our aim is to show that there is a simple and explicit formula of an amplification of completely bounded operators for all such pairs_{}*(M,N)*- which we shall call*admissible*-, thus providing a*constructive*approach of the amplification problem. The key idea is to combine two fundamental concepts in the theory of operator algebras, one being classical, the other one pretty modern: Tomiyama's slice maps on one hand, and the description of the predual of*M N*given by Effros-Ruan in terms of the projective operator space tensor product, on the other hand. We will further discuss the question of uniqueness of such an amplification, but mainly focus on various applications of our construction, such as- a generalization of the Ge-Kadison Lemma
- the amplification of completely bounded bimodule homomorphisms
- a characterization of normality for completely bounded mappings in terms of a commutation relation for the associated amplification.

- November 7: Oleg Yu. Aristov (Obninsk Institute of Nuclear Power Engineering; Russia).
*Title*: Projective and flat quantized Banach modules.*Abstract*: We consider projectivity and flatness for quantized Banach modules (=completely contractive Banach modules after Ruan). A quantized Banach module is a Banach module with an operator space structure such that the multiplication has a completely bounded factorization through the operator space projective tensor product. Many results on Banach module homology have their parallels in the quantized case. Moreover, B. E. Johnson's and Ruan's considerations show that the operator space approach to the Fourier algebra of a locally compact group*G*is more fruitful than the classical one. There is some progress in this direction for amenable, biflat and biprojective quantized Banach algebras. We prove that the Fourier algebra*A(G)*is biprojective in the quantized sense if and only if*G*is discrete. We also discuss some non-commutative generalizations for Hopf-von Neumann algebras and explain how a theorem of Ruan and Xu on Kac algebras can be interpreted as a result on biflatness. - November 14: Piotr Mankiewicz (Polish Academy of Sciences).
*Title*: Geometry of random quotients of symmetric convex bodies.*Abstract*: We study the diameter of a family of random*n*-dimensional orthogonal projections of an arbitrary symmetric convex body in*R*, and we show that this diameter is arger than or equal to the square of Euclidean distances of random^{N}*k*-dimensional projections of the body (where*k=(1/2-)n*, for any*>0*). The drop of dimension is necessary and the formula is in a certain sense optimal. - November 21: Mark Rudelson (University of Missouri-Columbia; USA).
*Title*: Isotropic random vectors.*Abstract*: We consider a problem which originates in Computer Science. Let*K*be an*n*-dimensional convex body. It is called isotropic if its moments of inertia are the same with respect to any axis. Transformation of a body into an isotropic position is the first step in many computational algorithms related to convex bodies. However, if the shape of the body is unknown, it is impossible to transform it exactly to the isotropic position. Lovasz and Simonovits posed the following problem: How many points of a convex body do we have to know to transform it to an almost isotropic position. This problem was solved by Bourgain. Using rather delicate geometric considerations, he showed that roughly*n(*log*n)*would be enough. We present a different approach to this problem, which also allows to improve the estimate of Bourgain. We consider general random vectors which are not necessary related to a convex body. This reduces the geometric problem to the estimate of an operator-valued random process. This estimate, in turn, is obtained using a non-commutative Khinchine inequality.^{3} - November 28: Volker Runde.
*Title*: Operator amenability for Fourier-Stieltjes algebras.*Abstract*: The notion of operator amenability was introduced by Z.-J. Ruan in 1995. He showed that a locally compact group*G*is amenable if and only if its Fourier algebra*A(G)*is operator amenable. In this talk, we investigate the operator amenability of the Fourier-Stieltjes algebra*B(G)*and of the reduced Fourier-Stieltjes algebra*B*. The natural conjecture is that any of these algebras is operator amenable if and only if_{r}(G)*G*is compact. We partially prove this conjecture with mere operator amenability replaced by operator*C*-amenability for some constant*C < 5*. As a by-product, we obtain a new decomposition of*B(G)*, which can be interpreted as the non-commutative counterpart of the decomposition of*M(G)*into the discrete and the continuous measures. We further introduce a variant of operator amenability - called operator Connes-amenability - which also takes the dual space structure on*B(G)*and*B*into account. We show that_{r}(G)*B*is operator Connes-amenable if and only if_{r}(G)*G*is amenable. Surprisingly,*B(F*is operator Connes-amenable although_{2})*F*, the free group in two generators, fails to be amenable. This is joint work with N. Spronk._{2} - December 5: Matthias Neufang.
*Title*: Abstract harmonic analysis and*CB(B(H))*.*Abstract*: By means of the the GNS construction, every*C*-algebra can be represented as an algebra of operators on a suitable Hilbert space^{*}*H*. The most interesting algebras occurring in abstract harmonic analysis, however, are preduals of von Neumann algebras, such als*L*,^{1}(G)*M(G)*, and*LUC(G)*: These algebras fail to be Arens-regular and thus cannot be represented as algebras of operators on a Hilbert space. In the early eighties, E. Stoermer and F. Ghahramani constructed an isometric representation of^{*}*M(G)*on*B(L*as an algebra of completely bounded operators. This representation extends to an an isometric representation of^{2}(G))*LUC(G)*. We shall present analogues, - in the context of the two represented algebras^{*}*M(G)*and*LUC(G)*- of von Neumann's bicommutant theorem. We eventually obtain intrinsic characterizations of the image algebras, which can be viewed as quantized versions of the classical theorems of Brainerd-Edwards and Curtis-Figà-Talamanca. We conclude with the definition of a new product on the space of trace class operators on^{*}*L*, which - on the operator level - parallels the usual convolution of^{2}(G)*L*-functions.^{1} **Thursday, December 13, 2001 (at 3:00 pm in CAB 657)**: H. Garth Dales (University of Leeds; England).*Title*: Traces on Banach *-algebras.*Abstract*: Let*A*be an algebra. A*trace*on*A*is a linear functional on*A*such that*(ab-ba)=0 (a, b A)*. A unital algebra is*properly infinite*if it contains two idempotent elements which are orthogonal to each other and such that each is equivalent to the identity (so that the algebra can be split into two "equal halves"). It is well-known that the zero functional is the only trace on a unital, properly infinite*C*-algebra. We answer a question by showing that there are unital, properly infinite Banach *-algebras which do have a continuous non-zero trace. Further, we can arrange that these algebras are "close" to being a^{*}*C*-algebra. This is joint work with Niels Laustsen (Copenhagen) and Charles Read (Leeds).^{*}

Last update: 12/14/2001.