Interpolation between non-commutative BMO and non-commutative Lp-spaces

We prove that for 1?p<q<∞ the analogue of the classical result holds in the setting of a finite von Neumann algebra , equipped with an increasing filtration of von Neumann subalgebras. We also obtain the corresponding results for the real method of interpolation. We discuss the appropriate operator space matrix norms and show that these interpolation results hold in the category of operator spaces.     

Principle of local reflexivity respecting subspaces

We obtain a strengthening of the principle of local reflexivity in a general form. The added strength makes local reflexivity operators respect given subspaces. Applications are given to bounded approximation properties of pairs, consisting of a Banach space and its subspace.     

Operator-valued Fourier multiplier theorems on Lp(X) and geometry of Banach spaces

This paper gives the optimal order l of smoothness in the Mihlin and Hörmander conditions for operator-valued Fourier multiplier theorems. This optimal order l is determined by the geometry of the underlying Banach spaces (e.g. Fourier type). This requires a new approach to such multiplier theorems, which in turn leads to rather weak assumptions formulated in terms of Besov norms.     

Matrix-valued interpolation and hyperconvex sets

We prove that it is possible for two uniform algebras to have the same scalar interpolating sets, yet still have different matrix-valued interpolating sets.We prove a result for tensor products of uniform algebras that extends Agler's interpolation formula for the bidisk to more general product domains. This is accomplished by introducing a dual object for interpolation problems, which we call a Schur ideal, and proving that the Schur ideal for a tensor product is the intersection of the corresponding Schur ideals.Research supported in part by a grant from the NSF     

The non-commutative Gurarii space

We construct a “universal space” for 1-exact finite dimensional operator spaces (an analogue of the Gurarii space). Received: 20 June 2005     

Perturbation of orthonormal bases with an application to diagonalization

Given an orthonormal basis {e _n } _n=1 in a Hilbert spaceH, and a dense linear manifoldDH, we show that there exists a unitary operatorV onH such thatI-V is a trace-class operator with arbitrarily small trace norm, andVe _jD for allj. This result can be used to simplify certain arguments of J. Xia concerning the simultaneous diagonalization of operators on a space of square integrable functions.     

Algebraic reflexivity of some subsets of the isometry group

Let X be a compact first countable space. In this paper we show that the set of isometries of C(X) that are involutions is algebraically reflexive. As a consequence of a recent work of Botelho and Jamison this leads to the conclusion that the set of generalized bi-circular projections on C(X) is also algebraically reflexive. We also consider these questions for the space C(X,E) where E is a uniformly convex Banach space.     

On the Essential Lower Bound of Elements in von Neumann Algebras

In this paper we obtain a number of interesting results on the connection that exists between the essential lower bound of an element in a von Neumann algebra and its Fredholm properties. These connections lead to an extension of a well-known perturbation theorem on Fredholm operators of index zero. Finally we obtain for special cases necessary and sufficient conditions for elements in the closure of the group of invertible elements to be essentially topological divisor of zero     

Factorization of entire mappings of nuclear bounded type

In this paper we study the relationships between the spaces of entire mappings of bounded type, entire mappings of nuclear bounded type, entire mappings of Pietsch integral bounded type, and entire mappings of Grothendieck integral bounded type. Several results due to Alencar (Proc. Roy. Irish Acad.85A (1985) 131-138) and Cilia and Gutiérrez (J. Aust. Math. Soc.76 (2004) 269-280) for homogeneous polynomials are extended to entire mappings. In the main result we prove that an entire mapping is of nuclear bounded type if and only if it factors through an entire mapping of Pietsch integral bounded type.     

Hypercyclic subspaces of a Banach space

Recently a lot of research has been done on hypercyclicity of a bounded linear operator on a Banach space, based on the hypercyclicity criterion obtained by Kitai in 1982, and independently by Gethner and Shapiro in 1987. By combining this criterion with one extra condition, Montes-Rodríguez obtained in 1996 a sufficient condition for the operator to have a closed infinite dimensional hypercyclic subspace, with a very technical proof. Since then, this result has been used extensively to generate new results on hypercyclic subspaces. In the present paper, we give a simple proof of the result of Montes-Rodríguez, by first establishing a few elementary results about the algebra of operators on a Banach space.     

Ordered Involutive Operator Spaces

This is a companion to recent papers of the authors; here we construct the ‘noncommutative Shilov boundary’ of a (possibly nonunital) selfadjoint ordered space of Hilbert space operators. The morphisms in the universal property of the boundary preserve order. As an application, we consider ‘maximal’ and ‘minimal’ unitizations of such ordered operator spaces.     

Normality of Products of <Emphasis Type="Italic">p</Emphasis>-Hyponormal Operators and Their Aluthge Transforms

Let B(H) denote the algebra of operators on a complex separable Hilbert space H, and let A $\in$ B(H) have the polar decomposition A = U|A|. The Aluthge transform is defined to be the operator . We say that A $\in$ B(H) is p-hyponormal, . Let . Given p-hyponormal , such that AB is compact, this note considers the relationship between denotes an enumeration in decreasing order repeated according to multiplicity of the eigenvalues of the compact operator T (respectively, singular values of the compact operator T). It is proved that is bounded above by and below by for all j = 1, 2, . . . and that if also is normal, then there exists a unitary U¹ such that for all j = 1, 2, . . ..     

Embedding of C q and R q into noncommutative L p -spaces, 1≤p<q≤2

We prove that a quotient of a subspace of C_p⊕_pR_p (1≤p<2) embeds completely isomorphically into a noncommutative L_p -space, where C_p and R_p are respectively the p-column and p-row Hilbertian operator spaces. We also represent C_q and R_q (p<q≤2) as quotients of subspaces of C_p⊕_pR_p. Consequently, C_q and R_q embed completely isomorphically into a noncommutative L_p (M). We further show that the underlying von Neumann algebra M cannot be semifinite.     

Ultrapowers and semigroups of operators

We study two semigroups of operators between Banach spaces, related with the finite representability ofc ₀ and ℓ₁. We show that these semigroups are open, have nice duality properties and can be characterized in terms of compact perturbations, and in terms of the properties of their ultrapowers. We obtain analogous results for their associated dual semigroups. Supported in part by DGICYT Grant PB 94-1052 (Spain). Supported by a postdoctoral Grant of the Ministry of Spain for Education and Science     

Composition Operators on Orlicz–Lorentz Spaces

In this paper, we study the boundedness and the compactness of composition operators on Orlicz–Lorentz spaces.      

Classes of Operator-Smooth Functions - II. Operator-Differentiable Functions

This paper studies the spaces of Gateaux and Frechet Operator Differentiable functions of a real variable and their link with the space of Operator Lipschitz functions. Apart from the standard operator norm on B(H), we consider a rich variety of spaces of Operator Differentiable and Operator Lipschitz functions with respect to symmetric operator norms. Our approach is aimed at the investigation of the interrelation and hierarchy of these spaces and of the intrinsic properties of Operator Differentiable functions. We apply the obtained results to the study of the functions acting on the domains of closed *-derivations of C*-algebras and prove that Operator Differentiable functions act on all such domains.We also obtain the following modification of this result: any continuously differentiable, Operator Lipschitz function acts on the domains of all weakly closed *-derivations of C*-algebras.     

On non-surjective coarse isometries between Banach spaces

Assume that X, Y are real Banach spaces, Y has uniform convexity of type p (≥ 1), and f: X → Y is a standard coarse isometry. In this paper, we show that if

then there is a linear isometry U : X → Y so that

where is defined by

Representation properties of coarse isometries in free ultrafilter limits on are also discussed.     

Generalized Bochner theorem and Levy-Khinchine representation for completely positive definite generalized Toeplitz kernels

The so-called generalized Bochner theorem, stated by Cotlar and Sadosky, provides an integral representation of the positive definite generalized Toeplitz kernels. In this paper we derive noncommutative analogues of lifting theorems and the generalized Bochner theorem for completely positive definite generalized Toeplitz kernels, also considered by Cotlar and Sadosky, through the theory of unitary extensions of isometric operators. Moreover, in the case where the kernel is defined in×, we can associate to each unitary extension an interpolation colligation providing thus a wide class of liftings. Also a Levy-Khinchine type formula for this kind of kernels is obtained.This paper is supported by grants from the University of the Basque Country.     

Characteristic Functions for Ergodic Tuples

Motivated by a result on weak Markov dilations, we define a notion of characteristic function for ergodic and coisometric row contractions with a one-dimensional invariant subspace for the adjoints. This extends a definition given by G. Popescu. We prove that our characteristic function is a complete unitary invariant for such tuples and show how it can be computed.