Theory Of Operators In Spaces Research Articles

C*-Algebren

The aim of the workshop C ^* -algebras was to bring together researchers from basically all areas related to operator algebra theory. This gives a unique opportunity to obtain a broader view of the subject and to create new interactions between researchers with different background. The organizers, Claire Anantharaman-Delaroche, Siegfried Echterhoff, Uffe Haagerup, and Dan Voiculescu took special care to invite a good number of young researchers, some of them already being leading experts in their fields. As a result, several contributions in this report are from researchers who, at the time of the workshop, were less then 30 years old. There were 29 lectures presented at this workshop with topics from Ergodic Theory, \mathrm{L}^2 -(co-)homology, classification of C ^* -algebras, Operator Theory, von Neumann algebras, \mathrm{KK} -theory and the Baum–Connes conjecture, quantum spaces and quantum groups, mathematical physics, non-commutative probability theory, and the theory of operator spaces. To name some special highlights we can mention the reports on recent developements in the study of “boundary actions” of quantum groups due to S. Vaes and R. Vergnioux, the new results on classification theory of amenable C ^* -algebras in terms of studying algebras which are stable under tensoring with the Jiang-Su algebra \mathcal Z (see the lectures of A. Toms and M. Rørdam), or the report by S. Popa on recent progress in the study of strong rigidity for II _1 factors associated to equivalence relations. But this is only a very small selection of the interesting lectures on new results presented at this workshop. It is a pleasure for the organizers of the conference to use this opportunity to thank all participants of the workshop for their contributions – either in lectures held at the workshop or in stimulating discussions following the lectures. We also thank the Mathematisches Forschungsinstitut Oberwolfach for providing a great environment and strong support for organizing this conference. Special thanks go to the very competent and helpful staff of the institute and to the chef de cuisine.

Duality and operator algebras: automatic weak* continuity and applications

We investigate some subtle and interesting phenomena in the duality theory of operator spaces and operator algebras, and give several applications of the surprising fact that certain maps are always weak * -continuous on dual spaces. In particular, if X is a subspace of a C * -algebra A, and if a ∈ A satisfies aX ⊂ X , then we show that the function x ↦ ax on X is automatically weak * continuous if either (a) X is a dual operator space, or (b) a * X ⊂ X and X is a dual Banach space. These results hinge on a generalization to Banach modules of Tomiyama's famous theorem on contractive projections onto a C * -subalgebra. Applications include a new characterization of the σ -weakly closed (possibly nonunital and nonselfadjoint) operator algebras, and a generalization of the theory of W * -modules to the framework of modules over such algebras. We also give a Banach module characterization of σ -weakly closed spaces of operators which are invariant under the action of a von Neumann algebra.

Applications of operator spaces to abstract harmonic analysis

We give a survey of how the relatively young theory of operator spaces has led to a deeper understanding of the Fourier algebra of a locally compact group (and of related algebras).

Contractive projections and operator spaces

Parallel to the study of finite-dimensional Banach spaces, there is a growing interest in the corresponding local theory of operator spaces. We define a family of Hilbertian operator spaces H n k H_n^k , 1 ≤ k ≤ n 1\le k\le n , generalizing the row and column Hilbert spaces R n R_n and C n C_n , and we show that an atomic subspace X ⊂ B ( H ) X\subset B(H) that is the range of a contractive projection on B ( H ) B(H) is isometrically completely contractive to an ℓ ∞ \ell ^\infty -sum of the H n k H_n^k and Cartan factors of types 1 to 4. In particular, for finite-dimensional X X , this answers a question posed by Oikhberg and Rosenthal. Explicit in the proof is a classification up to complete isometry of atomic w ∗ ^* -closed J W ∗ JW^* -triples without an infinite-dimensional rank 1 w ∗ ^* -closed ideal.

Spaces of fractional quotients, discrete operators, and their applications. II

The theory of discrete operators in spaces of fractional quotients is developed. A theorem on the stability of discrete operators under smooth perturbations is proved. On this basis, using special quadrature formulae of rectangular kind, the convergence of approximate solutions of hypersingular integral equations to their exact solutions is demonstrated and a mathematical substantiation of the method of closed discrete vortex frameworks is obtained. The same line of argument is also applied to difference equations arising in the solution of the homogeneous Dirichlet problem for a general second-order elliptic equation with variable coefficients.

Multipliers of Kac Algebras

Multipliers, in particular completely bounded multipliers, of Fourier algebras have played a very important role in the study of harmonic analysis and of group von Neumann algebras and C*-algebras. In this paper, we plan to extend this notion to a more general context, i.e. we are going to study the completely bounded multipliers of Kac algebras. We will introduce an appropriate definition for completely bounded multipliers of Kac algebras, study some of basic properties and their connection with the strong Voiculescu amenability of Kac algebras. Operator space theory will play a very important role throughout this paper.

On Indefinite Moment Problems and Resolvent Matrices of Hermitian Operators in Kreîn Spaces

AbstractLet (sj)∞j=0 be a sequence of real numbers such that the Hankel matrices (si+j)∞0, (Si+j+i)∞0 have finite numbers of negative eigenvalues. The indefinite moment problem with the moments Sj (j = 0,1,2, …) and the corresponding Stieltjes string are investigated. We use the approach via the Kreîn — Langer extension theory of symmetric operators in spaces with indefinite metric. In the framework of this approach a description of L– resolvents of a class of symmetric operators in Kreîn space and a simple formula for the calculation of the L– resolvent matrix in terms of boundary operators are given.

On M. G. Krein's papers in the theory of spaces with an indefinite metric

This is a survey of the papers by M. G. Krein (and his disciples) devoted to the theory of operators in spaces with an indefinite metric and its applications.

Analytic factorizations and completely bounded maps

We prove an analytic factorization theorem in the setting of the recently developed theory of operator spaces. We especially obtain the following result: LetA be aC*-algebra andH be a Hilbert space. Let π be an element ofH∞(CB(A, B(H))), i.e. a bounded analytic function valued in the space of completely bounded maps fromA intoB(H). Then there exist a Hilbert spaceK, a representation π:A→B(K), ϕ1 ∈1H∞(B(H,K)) and ∈2H∞(B(K,H)) such that ‖e1‖∞‖∈2‖∞ ≤ ‖∈‖∞ and:\(\forall z \in D, \forall a \in A, \varphi (z)(a) = \varphi _2 (z)\pi (a)\varphi _1 (z).\) We also prove an analogous result for completely bounded multilinear maps. The last part of the paper is devoted to a new proof of Pisier's theorem about gamma-norms.

Theory of integral operators in spaces of measurable vector functions

Theory of integral operators in spaces of measurable vector functions

A diagonalization theorem for a type of Hamiltonian using operator theory in spaces with an indefinite metric

A theorem on the diagonalization of Hermitian positive semidefinite matrices of a special type is proved much more easily by making use of para-unitary congruences. The simplification is achieved by restating the problem as a spectral problem for a special para-Hermitian matrix. Results are obtained incidentally on the expansions of any para-unitary matrix, and of a para-unitary matrix with the cross-property into the product of elementary para-unitary matrices.

Geometry of the tensor product of C*-algebras

When and ℬ are C*-algebras their algebraic tensor product ⊗ ℬ is a *-algebra in a natural way. Until recently, work on tensor products of C*-algebras has concentrated on norms α which make the completion ⊗α ℬ into a C*-algebra. The crucial role played by the Haagerup norm in the theory of operator spaces and completely bounded maps has produced some interest in more general norms (see [8; 12]). In this paper we investigate geometrical properties of algebra norms on ⊗ ℬ. By an ‘algebra norm’ we mean a norm which is sub-multiplicative: α(u.v) ≤ ≤ α(u).α(v).

Sturm-Liouville operators with an indefinite weight function

SynopsisSpectral properties of the singular Sturm-Liouville equation –(p−1y′)′ + qy = λry with an indefinite weight function r are studied in . The main tool is the theory of definitisable operators in spaces with an indefinite scalar product.

Direct and iterative methods for the solution of linear operator equations in Hilbert space

Introduction. During the last two decades the problem of justifying existing methods and finding new ones for the solution of the equation Lu = f, where f is a given vector in Hilbert space and L is a given operator, has been investigated by many authors(2). This investigation led to the development of a group of direct methods of Ritz, Galerkin, least squares, and moments; and the iterative methods of steepest descent or gradient method, the method with minimal residuals, and their variants. These methods have a long history and have been studied and extensively applied by earlier authors to integral and differential equations. Only recently, however, the Hilbert space operator theory has been used for their study. Thus, using the variational principle, Mikhlin [16] proved the convergence of Ritz method for a self-adjoint positive definite operator L and the convergence of the method of least squares for an invertible operator L. In case L is of the form L = A + B, where A is self-adjoint and positive definite and B satisfies some additional conditions, the method of Galerkin was investigated by Mikhlin [16], Polsky [19], and others [9; 13]. For similar differential operators the method of moments was studied by Kravchuk [11], Polsky [19], and Zdanov [22]. The convergence and the estimate of error of the gradient method for selfadjoint and positive definite operators L have been studied by Kantorovich [7] and Hayes [6]. The latter considered also a slightly more general problem. In case L is a finite, symmetric, and positive definite matrix the method with minimal residuals was developed by Krasnoselsky and Krein [10]. A powerful method related to direct methods was also developed by Murray [18]. The purpose of this paper is to extend the study and the applicability of these methods to a larger class of linear operator equations(3) than those considered by the above authors and to present these seemingly distinct methods in a more unified manner. In our investigation we do not use the variational principle