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Author: Gilles Pisier Publisher: Springer ISBN: 3662215373 Category : Mathematics Languages : en Pages : 170
Book Description
These notes revolve around three similarity problems, appearing in three dif ferent contexts, but all dealing with the space B(H) of all bounded operators on a complex Hilbert space H. The first one deals with group representations, the second one with C* -algebras and the third one with the disc algebra. We describe them in detail in the introduction which follows. This volume is devoted to the background necessary to understand these three open problems, to the solutions that are known in some special cases and to numerous related concepts, results, counterexamples or extensions which their investigation has generated. For instance, we are naturally lead to study various Banach spaces formed by the matrix coefficients of group representations. Furthermore, we discuss the closely connected Schur multipliers and Grothendieck's striking characterization of those which act boundedly on B(H). While the three problems seem different, it is possible to place them in a common framework using the key concept of "complete boundedness", which we present in detail. In some sense, completely bounded maps can also be viewed as spaces of "coefficients" of C*-algebraic representations, if we allow "B(H) valued coefficients", this is the content of the fundamental factorization property of these maps, which plays a central role in this volume. Using this notion, the three problems can all be formulated as asking whether "boundedness" implies "complete boundedness" for linear maps satisfying cer tain additional algebraic identities.
Author: Gilles Pisier Publisher: Springer ISBN: 3662215373 Category : Mathematics Languages : en Pages : 170
Book Description
These notes revolve around three similarity problems, appearing in three dif ferent contexts, but all dealing with the space B(H) of all bounded operators on a complex Hilbert space H. The first one deals with group representations, the second one with C* -algebras and the third one with the disc algebra. We describe them in detail in the introduction which follows. This volume is devoted to the background necessary to understand these three open problems, to the solutions that are known in some special cases and to numerous related concepts, results, counterexamples or extensions which their investigation has generated. For instance, we are naturally lead to study various Banach spaces formed by the matrix coefficients of group representations. Furthermore, we discuss the closely connected Schur multipliers and Grothendieck's striking characterization of those which act boundedly on B(H). While the three problems seem different, it is possible to place them in a common framework using the key concept of "complete boundedness", which we present in detail. In some sense, completely bounded maps can also be viewed as spaces of "coefficients" of C*-algebraic representations, if we allow "B(H) valued coefficients", this is the content of the fundamental factorization property of these maps, which plays a central role in this volume. Using this notion, the three problems can all be formulated as asking whether "boundedness" implies "complete boundedness" for linear maps satisfying cer tain additional algebraic identities.
Author: Donald G. Babbitt Publisher: American Mathematical Soc. ISBN: 9780821896709 Category : Mathematics Languages : en Pages : 762
Book Description
This second volume of Featured Reviews makes available special detailed reviews of some of the most important mathematical articles and books published from 1997 through 1999. Also included are excellent reviews of several classic books and articles published prior to 1970. Among those reviews, for example, are the following: Homological Algebra by Henri Cartan and Samuel Eilenberg, reviewed by G. Hochschild; Faisceaux algebriques coherents by Jean-Pierre Serre, reviewed by C. Chevalley; and On the Theory of General Partial Differential Operators by Lars Hormander, reviewed by J. L. Lions. In particular, those seeking information on current developments outside their own area of expertise will find the volume very useful. By identifying some of the best publications, papers, and books that have had or are expected to have a significant impact in applied and pure mathematics, this volume will serve as a comprehensive guide to important new research across all fields covered by MR.
Author: Laurent Bartholdi Publisher: Springer Science & Business Media ISBN: 3764374470 Category : Mathematics Languages : en Pages : 419
Book Description
This book offers a panorama of recent advances in the theory of infinite groups. It contains survey papers contributed by leading specialists in group theory and other areas of mathematics. Topics include amenable groups, Kaehler groups, automorphism groups of rooted trees, rigidity, C*-algebras, random walks on groups, pro-p groups, Burnside groups, parafree groups, and Fuchsian groups. The accent is put on strong connections between group theory and other areas of mathematics.
Author: David P. Blecher Publisher: Oxford University Press ISBN: 0191523569 Category : Mathematics Languages : en Pages :
Book Description
This invaluable reference is the first to present the general theory of algebras of operators on a Hilbert space, and the modules over such algebras. The new theory of operator spaces is presented early on and the text assembles the basic concepts, theory and methodologies needed to equip a beginning researcher in this area. A major trend in modern mathematics, inspired largely by physics, is toward `noncommutative' or `quantized' phenomena. In functional analysis, this has appeared notably under the name of `operator spaces', which is a variant of Banach spaces which is particularly appropriate for solving problems concerning spaces or algebras of operators on Hilbert space arising in 'noncommutative mathematics'. The category of operator spaces includes operator algebras, selfadjoint (that is, C*-algebras) or otherwise. Also, most of the important modules over operator algebras are operator spaces. A common treatment of the subjects of C*-algebras, nonselfadjoint operator algebras, and modules over such algebras (such as Hilbert C*-modules), together under the umbrella of operator space theory, is the main topic of the book. A general theory of operator algebras, and their modules, naturally develops out of the operator space methodology. Indeed, operator space theory is a sensitive enough medium to reflect accurately many important noncommutative phenomena. Using recent advances in the field, the book shows how the underlying operator space structure captures, very precisely, the profound relations between the algebraic and the functional analytic structures involved. The rich interplay between spectral theory, operator theory, C*-algebra and von Neumann algebra techniques, and the influx of important ideas from related disciplines, such as pure algebra, Banach space theory, Banach algebras, and abstract function theory is highlighted. Each chapter ends with a lengthy section of notes containing a wealth of additional information.
Author: Gelu Popescu Publisher: American Mathematical Soc. ISBN: 0821843966 Category : Mathematics Languages : en Pages : 105
Book Description
This paper concerns unitary invariants for $n$-tuples $T:=(T_1,\ldots, T_n)$ of (not necessarily commuting) bounded linear operators on Hilbert spaces. The author introduces a notion of joint numerical radius and works out its basic properties. Multivariable versions of Berger's dilation theorem, Berger-Kato-Stampfli mapping theorem, and Schwarz's lemma from complex analysis are obtained. The author studies the joint (spatial) numerical range of $T$ in connection with several unitary invariants for $n$-tuples of operators such as: right joint spectrum, joint numerical radius, euclidean operator radius, and joint spectral radius. He also proves an analogue of Toeplitz-Hausdorff theorem on the convexity of the spatial numerical range of an operator on a Hilbert space, for the joint numerical range of operators in the noncommutative analytic Toeplitz algebra $F_n^\infty$.
Author: Aleksandr I︠A︡kovlevich Khelemskiĭ Publisher: American Mathematical Soc. ISBN: 082185254X Category : Mathematics Languages : en Pages : 264
Book Description
Interpreting ""quantized coefficients"" as finite rank operators in a fixed Hilbert space allows the author to replace matrix computations with algebraic techniques of module theory and tensor products, thus achieving a more invariant approach to the subject.
Author: Keri A. Kornelson Publisher: American Mathematical Soc. ISBN: 1470410400 Category : Mathematics Languages : en Pages : 192
Book Description
This volume contains the proceedings of the AMS Special Session on Harmonic Analysis of Frames, Wavelets, and Tilings, held April 13-14, 2013, in Boulder, Colorado. Frames were first introduced by Duffin and Schaeffer in 1952 in the context of nonharmonic Fourier series but have enjoyed widespread interest in recent years, particularly as a unifying concept. Indeed, mathematicians with backgrounds as diverse as classical and modern harmonic analysis, Banach space theory, operator algebras, and complex analysis have recently worked in frame theory. Frame theory appears in the context of wavelets, spectra and tilings, sampling theory, and more. The papers in this volume touch on a wide variety of topics, including: convex geometry, direct integral decompositions, Beurling density, operator-valued measures, and splines. These varied topics arise naturally in the study of frames in finite and infinite dimensions. In nearly all of the papers, techniques from operator theory serve as crucial tools to solving problems in frame theory. This volume will be of interest not only to researchers in frame theory but also to those in approximation theory, representation theory, functional analysis, and harmonic analysis.
Author: Gilles Pisier
Author: Gilles Pisier
Author: Vern Paulsen
Author: Donald G. Babbitt
Author: Laurent Bartholdi
Author: David P. Blecher
Author: 
Author: Gilles Pisier
Author: Gelu Popescu
Author: Aleksandr I︠A︡kovlevich Khelemskiĭ
Author: Keri A. Kornelson