An algebroid function K(z) is the set-valued function obtained by taking the zeroes of a polynomial whose coefficients are holomorphic functions of z. We present a sharpened version of the Schwarz lemma for algebroid functions, and discuss it in the context of the spectral Nevanlinna-Pick problem.

We say a multifunction T : X ® 2^X* is monotone provided that for any x,y Î X, and x^* Î T(x), y^* Î T(y),

References

[1]

J. M. Borwein, Maximal Monotonicity via Convex Analysis. J. Convex Analysis (Special issue in memory of Simon Fitzpatrick) 13 (June 2006). [D-drive Preprint 281].

We will show that the space Oq of multipliers of temperate distributions can be expressed as the inductive limit of certain Hilbert spaces.

Joint work with Jan Kucera.

Let H and K be Banach spaces, let B(H,K) denote the set of bounded linear operators from H to K, and abbreviate B(H,H) to B(H). For the operators A Î B(H), B Î B(K) and C Î B(K,H), let M_C denote the operator matrices in B(HÅK) defined with

In this talk we will describe spectrum, Weyl's and Browder's spectrum of operator matrices M_C using spectral property of operators A and B.

We prove an atomic decomposition for all the Borel measures that arise as boundary limits of Banach-valued harmonic functions on a Lipschitz domain D, whose non-tangential maximal function is integrable with respect to harmonic measure of the boundary of D. As in the case of the disk, the existence of non-tangential boundary limits of all these harmonic functions characterizes the Radon-Nikodym property of the Banach space.

Given a complex sequence s = {a_n}, the discrete Cesaro operator T assigns to it the sequence Ts = {b_n}, where b_n = [(a₀+ ¼+ a_n)/(n+1)], n = 0,1,... . If s is a convergent sequence, we prove that {Tⁿ s} converges if, and only if, a₁ = lim_{n® ¥} a_n. We also establish a corresponding result for the continuous Cesaro operator defined on C[0,1].

We prove that the condition of the empty slice property (EIS), which is a generalization of uniform smoothness, implies the fixed point property. That is, in a Banach space with EIS, every nonexpansive map from a weakly compact convex set into itself has a fixed point. Furthermore, the EIS property is stable under finite l^p sums of Banach spaces. We also give some examples.

(Joint work with Helga Fetter)

We prove an extension of Ekeland's variational principle to locally complete spaces which uses subbaditive, strictly increasing continuous functions as perturbations.

We present some recent results on multipliers of a Banach algebra and their applications to topological centre problems.

The talk is based on joint work with Matthias Neufang and Zhong-Jin Ruan.

Geophysical applications demand a mathematical modeling of physical processes that respect minimum phase conditions. Essentially, this states that energy in a signal is concentrated near the beginning of the onset of a signal. We present a mathematical definition of minimum phase, develop robust calculation of equivalent minimum phase signals, and examine the class of linear operators on Hilbert space that preserve minimum phase. Properties are closely connected to factorization problems in Hardy space.

An entropy lemma states that if we control the diameter of a body on a subspace then we control the covering of the body. More precisely, given two centrally-symmetric bodies K and L, satisfying K Ì AL and KÇE Ì a L for a k-codimensional subspace E, one has N (K, 2r L) £ ( 4A/(r-a) )^k for every r > a. That means that, surprisingly, the covering numbers of K behave in the same way as the covering numbers of a cylinder with the base aLÇE. We prove this lemma and discuss its applications to the Gelfand numbers and to the Sudakov inequality.

This talk is based on joint works with A. Pajor and N. Tomczak-Jaegermann and with V. Milman, A. Pajor, and N. Tomczak-Jaegermann.

For many C^*-algebras A, techniques have been developed to show that all elements which have trace zero with respect to all tracial states can be written as a sum of finitely many commutators, and that the number of commutators required depends only upon the algebra, and not upon the individual elements. In this paper, we show that if the same holds for q A q whenever q is a "sufficiently small" projection in A, then every element that is a sum of finitely many commutators in A is in fact a sum of two. We use these results to show that many C^*-algebras are linearly spanned by their projections.

Using eigenmatrices, we characterize when a bounded operator in Hilbert space commutes with a finite-rank operator. We use this characterization to prove that if an operator commutes with a finite-rank operator, then it must commute with an operator of rank one. As a corollary of this, we show that (classical) Toeplitz operators do not commute with operators of finite rank.

The notion of character amenability of Banach algebras will be discussed. It will be shown that for a locally compact group G, the amenability of either of the group algebra L¹(G) or the Fourier algebra A(G) is equivalent to the amenability of the underlying group G.

We also discuss some cohomological implications of character amenability. In particular we show that if A is a commutative character amenable Banach algebra, then Hⁿ (A,E) = {0} for all finite-dimensional Banach A-bimodules E, and all n Î N. This in particular implies that all finite-dimensional extensions of such Banach algebras split strongly. This extends earlier results of H. Steiniger and myself on Fourier and generalized Fourier algebras to the larger class of commutative character amenable Banach algebras.

Let G = (M, G, j, y) be a co-amenable locally compact quantum group. In recent work with M. Junge and Z.-J. Ruan, we have constructed and studied a completely isometric representation of the algebra of completely bounded (right) multipliers of L₁(G) on B (L₂(G) ). This extends and unifies earlier work by F. Ghahramani, E. Størmer and myself in the case M = L_¥(G), and by Z.-J. Ruan, N. Spronk and myself for M = VN(G), where G is a locally compact group. We have shown that the multiplier algebra can in fact be identified with the algebra of completely bounded, normal [^(M)]-bimodule maps on B ( L₂(G) ) leaving M invariant. The part of the latter algebra consisting of completely positive maps provides a natural class of quantum channels which, from the viewpoint of quantum computing, are of particular interest in the case of finite-dimensional quantum groups.

In this talk, we shall discuss various applications of the above representation to quantum information theory. Indeed, several properties of our channels are highly desirable with regard to quantum error correction: the bimodule property means precisely that the channels are noiseless for [^(M)]; moreover, every such channel has a symbol which is easy to retrieve, and the completely bounded minimal entropy (cb-entropy) can be calculated explicitly. Note that the cb-entropy has recently be shown to be additive (I. Devetak, M. Junge, C. King, and M. B. Ruskai); proving additivity of the bounded minimal entropy is a major open problem in quantum information theory.

This is joint work with Marius Junge, David Kribs and Zhong-Jin Ruan.

Let G be a locally compact group. The Fourier-Stieltjes algebra B(G) is the dual space of the group C^*-algebra C^*(G), and it can be naturally be made into Banach algebra which can be identified with a subalgebra of the bounded continuous functions on G. If G is abelian, then B(G) is exactly the algebra of Fourier-Stieltjes transforms of measures on the dual group. As such, B(G) is a large commutative Banach algebra and, frequently, has an intractable spectrum and is not regular.

We consider the closed span of the idempotents in B(G), B_I(G). Even for totally disconnected groups, B_I(G) is a regular Banach algebra. M. Ilie and I have computed the spectrum of B_I(G), and characterised, for another locally compact group H, when B_I(G) is isometrically algebraically isomorphic to B_I(H). We have also computed some examples. This represents an application of the "spine" of B(G), which we defined previously, and has a nice application in amenability theory.

The method of minimal vectors was developed to find invariant subspaces of certain classes of operators on Hilbert spaces. We describe applications of this method to Banach spaces, Banach lattices, and algebras of operators.