Tensor Products of Convex Cones, Part I: Mapping Properties, Faces, and Semisimplicity
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Projective Tensor Products, Injective Tensor Products, and Dual Relations on Operator Spaces
University of Windsor Scholarship at UWindsor Electronic Theses and Dissertations Theses, Dissertations, and Major Papers 1-1-2006 Projective tensor products, injective tensor products, and dual relations on operator spaces. Fuhua Chen University of Windsor Follow this and additional works at: https://scholar.uwindsor.ca/etd Recommended Citation Chen, Fuhua, "Projective tensor products, injective tensor products, and dual relations on operator spaces." (2006). Electronic Theses and Dissertations. 7096. https://scholar.uwindsor.ca/etd/7096 This online database contains the full-text of PhD dissertations and Masters’ theses of University of Windsor students from 1954 forward. These documents are made available for personal study and research purposes only, in accordance with the Canadian Copyright Act and the Creative Commons license—CC BY-NC-ND (Attribution, Non-Commercial, No Derivative Works). Under this license, works must always be attributed to the copyright holder (original author), cannot be used for any commercial purposes, and may not be altered. Any other use would require the permission of the copyright holder. Students may inquire about withdrawing their dissertation and/or thesis from this database. For additional inquiries, please contact the repository administrator via email ([email protected]) or by telephone at 519-253-3000ext. 3208. P r o je c t iv e T e n so r P r o d u c t s , In je c t iv e T e n so r P r o d u c t s , a n d D ual R elatio ns on O p e r a t o r S paces by Fuhua Chen A Thesis Submitted to the Faculty of Graduate Studies and Research through Mathematics and Statistics in Partial Fulfillment of the Requirements of the Degree of Master of Science at the University of Windsor Windsor, Ontario, Canada 2006 © 2006 Fuhua Chen Reproduced with permission of the copyright owner. -
Grothendieck's Tensor Norms
Grothendieck's tensor norms Vandana Shivaji College, University of Delhi, India April, 2015 Vandana Grothendieck's tensor norms Overview Grothendieck's tensor norms Operator spaces Tensor products of Operator spaces Schur tensor product Vandana Grothendieck's tensor norms Grothendieck's tensor norms A Banach space is a complete normed space. For Banach spaces X and Y , X ⊗ Y = spanfx ⊗ y : x 2 X; y 2 Y g, where x ⊗ y is the functional on B(X∗ × Y ∗; C) given by x ⊗ y(f; g) = f(x)g(y) for f 2 X∗ and g 2 Y ∗. For a pair of arbitrary Banach spaces X and Y , the norm on X ⊗ Y induced by the embedding X ⊗ Y ! B(X∗ × Y ∗; C) is known as Banach space injective tensor norm. That is , for u 2 X ⊗ Y , the Banach space injective tensor norm is defined to be n n o X ∗ ∗ kukλ = sup f(xi)g(yi) : f 2 X1 ; g 2 Y1 : i=1 Vandana Grothendieck's tensor norms Grothendieck's tensor norms Question is How can we norm on X ⊗ Y ? n X kx ⊗ ykα ≤ kxkkyk, then, for u = xi ⊗ yi, by triangle's i=1 n X inequality it follows that kukα ≤ kxikkyik. Since this holds for i=1 n X every representation of u, so we have kukα ≤ inff kxikkyikg. i=1 For a pair of arbitrary Banach spaces X and Y and u an element in the algebraic tensor product X ⊗ Y , the Banach space projective tensor norm is defined to be n n X X kukγ = inff kxikkyik : u = xi ⊗ yi; n 2 Ng: i=1 i=1 X ⊗γ Y will denote the completion of X ⊗ Y with respect to this norm. -
Distinguished Property in Tensor Products and Weak* Dual Spaces
axioms Article Distinguished Property in Tensor Products and Weak* Dual Spaces Salvador López-Alfonso 1 , Manuel López-Pellicer 2,* and Santiago Moll-López 3 1 Department of Architectural Constructions, Universitat Politècnica de València, 46022 Valencia, Spain; [email protected] 2 Emeritus and IUMPA, Universitat Politècnica de València, 46022 Valencia, Spain 3 Department of Applied Mathematics, Universitat Politècnica de València, 46022 Valencia, Spain; [email protected] * Correspondence: [email protected] 0 Abstract: A local convex space E is said to be distinguished if its strong dual Eb has the topology 0 0 0 0 b(E , (Eb) ), i.e., if Eb is barrelled. The distinguished property of the local convex space Cp(X) of real- valued functions on a Tychonoff space X, equipped with the pointwise topology on X, has recently aroused great interest among analysts and Cp-theorists, obtaining very interesting properties and nice characterizations. For instance, it has recently been obtained that a space Cp(X) is distinguished if and only if any function f 2 RX belongs to the pointwise closure of a pointwise bounded set in C(X). The extensively studied distinguished properties in the injective tensor products Cp(X) ⊗# E and in Cp(X, E) contrasts with the few distinguished properties of injective tensor products related to the dual space Lp(X) of Cp(X) endowed with the weak* topology, as well as to the weak* dual of Cp(X, E). To partially fill this gap, some distinguished properties in the injective tensor product space Lp(X) ⊗# E are presented and a characterization of the distinguished property of the weak* dual of Cp(X, E) for wide classes of spaces X and E is provided. -
Frames in 2-Inner Product Spaces
Iranian Journal of Mathematical Sciences and Informatics Vol. 8, No. 2 (2013), pp 123-130 Frames in 2-inner Product Spaces Ali Akbar Arefijamaal∗ and Ghadir Sadeghi Department of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran E-mail: [email protected] E-mail: [email protected] Abstract. In this paper, we introduce the notion of a frame in a 2- inner product space and give some characterizations. These frames can be considered as a usual frame in a Hilbert space, so they share many useful properties with frames. Keywords: 2-inner product space, 2-norm space, Frame, Frame operator. 2010 Mathematics subject classification: Primary 46C50; Secondary 42C15. 1. Introduction and preliminaries The concept of frames in Hilbert spaces has been introduced by Duffin and Schaeffer [12] in 1952 to study some deep problems in nonharmonic Fourier se- ries. Various generalizations of frames have been proposed; frame of subspaces [2, 6], pseudo-frames [18], oblique frames [10], continuous frames [1, 4, 14] and so on. The concept of frames in Banach spaces have been introduced by Grochenig [16], Casazza, Han and Larson [5] and Christensen and Stoeva [11]. The concept of linear 2-normed spaces has been investigated by S. Gahler in 1965 [15] and has been developed extensively in different subjects by many authors [3, 7, 8, 13, 14, 17]. A concept which is related to a 2-normed space is 2-inner product space which have been intensively studied by many math- ematicians in the last three decades. A systematic presentation of the recent results related to the theory of 2-inner product spaces as well as an extensive ∗ Corresponding author Received 16 November 2011; Accepted 29 April 2012 c 2013 Academic Center for Education, Culture and Research TMU 123 124 Arefijamaal, Sadeghi list of the related references can be found in the book [7]. -
On the Geometry of Projective Tensor Products
ON THE GEOMETRY OF PROJECTIVE TENSOR PRODUCTS OHAD GILADI, JOSCHA PROCHNO, CARSTEN SCHUTT,¨ NICOLE TOMCZAK-JAEGERMANN, AND ELISABETH WERNER n Abstract. In this work, we study the volume ratio of the projective tensor products `p ⊗π n n `q ⊗π `r with 1 ≤ p ≤ q ≤ r ≤ 1. We obtain asymptotic formulas that are sharp in almost all cases. As a consequence of our estimates, these spaces allow for a nearly Euclidean decomposition of Kaˇsintype whenever 1 ≤ p ≤ q ≤ r ≤ 2 or 1 ≤ p ≤ 2 ≤ r ≤ 1 and q = 2. Also, from the Bourgain-Milman bound on the volume ratio of Banach spaces in terms of their cotype 2 constant, we obtain information on the cotype of these 3-fold projective tensor products. Our results naturally generalize to k-fold products `n ⊗ · · · ⊗ `n with k 2 p1 π π pk N and 1 ≤ p1 ≤ · · · ≤ pk ≤ 1. 1. Introduction In the geometry of Banach spaces the volume ratio vr(X) of an n-dimensional normed space X is defined as the n-th root of the volume of the unit ball in X divided by the volume of its John ellipsoid. This notion plays an important role in the local theory of Banach spaces and has significant applications in approximation theory. It formally originates in the works [Sza78] and [STJ80], which were influenced by the famous paper of B. Kaˇsin[Kaˇs77] on nearly Euclidean orthogonal decompositions. Kaˇsindiscovered that for arbitrary n 2 N, 2n the space `1 contains two orthogonal subspaces which are nearly Euclidean, meaning that n their Banach-Mazur distance to `2 is bounded by an absolute constant. -
A Completely Positive Maps
A Completely Positive Maps A.1. A linear map θ: A → B between C∗-algebrasiscalledcompletely positive if θ ⊗ id: A ⊗ Matn(C) → B ⊗ Matn(C) is positive for any n ∈ N. Such a map is automatically bounded since any ∗ positive linear functional on a C -algebra is bounded. ⊗ ∈ ⊗ C An operator T = i,j bij eij B Matn( ), where the eij’s are the usual C ∗ ≥ matrix units in Matn( ), is positive if and only if i,j bi bijbj 0 for any ∈ ⊗ C b1,...,bn B. Since every positive element in A Matn( )isthesumofn ∗ ⊗ → elements of the form i,j ai aj eij,weconcludethatθ: A B is completely positive if and only if n ∗ ∗ ≥ bi θ(ai aj)bj 0 i,j=1 for any n ∈ N, a1,...,an ∈ A and b1,...,bn ∈ B. A.2. If θ: A → B is completely positive and B ⊂ B(H), then there exist a Hilbert space K, a representation π: A → B(K) and a bounded operator V : H → K such that θ(a)=V ∗π(a)V and π(A)VH = K. The construction is as follows. Consider the algebraic tensor product AH, and define a sesquilinear form on it by (a ξ,c ζ)=(θ(c∗a)ξ,ζ). Complete positivity of θ means exactly that this form is positive definite. Thus taking the quotient by the space N = {x ∈ A H | (x, x)=0} we get a pre-Hilbert space. Denote its completion by K. Then define a representation π: A → B(K)by π(a)[c ζ]=[ac ζ], where [x] denotes the image of x in (A H)/N . -
Absorbing Representations with Respect to Closed Operator Convex
ABSORBING REPRESENTATIONS WITH RESPECT TO CLOSED OPERATOR CONVEX CONES JAMES GABE AND EFREN RUIZ Abstract. We initiate the study of absorbing representations of C∗-algebras with re- spect to closed operator convex cones. We completely determine when such absorbing representations exist, which leads to the question of characterising when a representation is absorbing, as in the classical Weyl–von Neumann type theorem of Voiculescu. In the classical case, this was solved by Elliott and Kucerovsky who proved that a representation is nuclearly absorbing if and only if it induces a purely large extension. By considering a related problem for extensions of C∗-algebras, which we call the purely large problem, we ask when a purely largeness condition similar to the one defined by Elliott and Kucerovsky, implies absorption with respect to some given closed operator convex cone. We solve this question for a special type of closed operator convex cone induced by actions of finite topological spaces on C∗-algebras. As an application of this result, we give K-theoretic classification for certain C∗-algebras containing a purely infinite, two- sided, closed ideal for which the quotient is an AF algebra. This generalises a similar result by the second author, S. Eilers and G. Restorff in which all extensions had to be full. 1. Introduction The study of absorbing representations dates back, in retrospect, more than a century to Weyl [Wey09] who proved that any bounded self-adjoint operator on a separable Hilbert space is a compact perturbation of a diagonal operator. This implies that the only points in the spectrum of a self-adjoint operator which are affected by compact perturbation, are the isolated points of finite multiplicity. -
Chapter 2 C -Algebras
Chapter 2 C∗-algebras This chapter is mainly based on the first chapters of the book [Mur90]. Material bor- rowed from other references will be specified. 2.1 Banach algebras Definition 2.1.1. A Banach algebra C is a complex vector space endowed with an associative multiplication and with a norm k · k which satisfy for any A; B; C 2 C and α 2 C (i) (αA)B = α(AB) = A(αB), (ii) A(B + C) = AB + AC and (A + B)C = AC + BC, (iii) kABk ≤ kAkkBk (submultiplicativity) (iv) C is complete with the norm k · k. One says that C is abelian or commutative if AB = BA for all A; B 2 C . One also says that C is unital if 1 2 C , i.e. if there exists an element 1 2 C with k1k = 1 such that 1B = B = B1 for all B 2 C . A subalgebra J of C is a vector subspace which is stable for the multiplication. If J is norm closed, it is a Banach algebra in itself. Examples 2.1.2. (i) C, Mn(C), B(H), K (H) are Banach algebras, where Mn(C) denotes the set of n × n-matrices over C. All except K (H) are unital, and K (H) is unital if H is finite dimensional. (ii) If Ω is a locally compact topological space, C0(Ω) and Cb(Ω) are abelian Banach algebras, where Cb(Ω) denotes the set of all bounded and continuous complex func- tions from Ω to C, and C0(Ω) denotes the subset of Cb(Ω) of functions f which vanish at infinity, i.e. -
Stinespring's Theorem for Maps on Hilbert C
STINESPRING’S THEOREM FOR MAPS ON HILBERT C∗−MODULES Shefali Gupta Supervisor - Dr. G. Ramesh A Thesis Submitted to Indian Institute of Technology Hyderabad In Partial Fulfillment of the Requirements for The Degree of Master of Science in Mathematics Department of Mathematics April 2016 2 3 4 Acknowledgement The success of this work is accredited to many. Firstly, thanking the god and my par- ents for everything, my next acknowledgement goes to my supervisor-cum-guide, Dr. G. Ramesh, because of whom, I got to learn, understand thoroughly and moreover appreci- ate, a whole new branch of mathematics, Operator Algebras. Not just this but under his expert guidance and motivation, we could successfully come up with results we aimed to acquire since the beginning of this project. I am highly grateful to him for his strong support and understanding. I am also thankful to my friends in IIT Hyderabad, who stood by me through all the thicks and thins throughout the course of my M.Sc. and for being there whenever I needed them. Last but not the least, I would like to thank my classmates who have always supported me in every matter. 5 6 Abstract Stinespring’s representation theorem is a fundamental theorem in the theory of com- pletely positive maps. It is a structure theorem for completely positive maps from a C∗−algebra into the C∗−algebra of bounded operators on a Hilbert space. This theo- rem provides a representation for completely positive maps, showing that they are simple modifications of ∗−homomorphisms. One may consider it as a natural generalization of the well-known Gelfand-Naimark-Segal thoerem for states on C∗−algebras. -
Geometry in Tensor Products
Universita` degli Studi di Napoli \Federico II" Facolt`adi Scienze Matematiche Fisiche e Naturali Dipartimento di Matematica e Applicazioni \R.Caccioppoli" Dottorato di Ricerca in Scienze Matematiche Daniele Puglisi Geometry in tensor products TESI DI DOTTORATO XIX CICLO 2003-2007 Tutors: Prof. Paolo De Lucia Coordinatore: Prof. Salvatore Rionero Prof. Joseph Diestel ii Acknowledgements First of all I would like to thank Prof. Joseph Diestel, from Kent State Uni- versity, for his precious suggestions and many many helps. To sit near Joe when he is talking about math in his office, in some bar or also inside some airport, is an experience every young mathematicians should have. I really want to thank him to invited me and for giving me a comfortable permanence in Kent (I have no words to thank you!). Then I thank my advisor Prof. Paolo De Lucia for accepting me as his stu- dent and for giving me the opportunity to visit the Kent State University. Thanks to Prof. Richard Aron for his excellent mathematical support, for helping me in many situation, and for any time that he took me to hear the Cleveland Orchestra. I also want to dedicate part of my dissertation to Roberto Lo Re, Giovanni Cutolo, and Chansung Choi, they influenced strongly my life. Thanks to Prof. Artem Zvavitch for his support, and thanks to Giuseppe Saluzzo, Assunta Tataranni for the many helps. I want to give thanks the secretary support to the Department of Mathemat- ical Sciences at at Kent State University and at Napoli. Specially, I wish to thank Virginia Wright, Misty Tackett, Luciana Colmayer and Luisa Falanga. -
Algebras of Operators on Banach Spaces, and Homomorphisms Thereof
Algebras of operators on Banach spaces, and homomorphisms thereof Bence Horváth Department of Mathematics and Statistics Lancaster University This dissertation is submitted for the degree of Doctor of Philosophy May 2019 „S en-sorsom legsötétebb, mert színe s alja poltron, Élet s mű között bolyong, mint aggály-aszú lélek: Életben mű riongat, a műben lét a sorsom — S a sír-írásom ez: se égimű, se földi élet.” — Szentkuthy Miklós Declaration The research presented in this thesis has not been submitted for a higher degree elsewhere and is, to the best of my knowledge and belief, original and my own work, except as acknowledged herein. • Chapter 2 contains joint work with Y. Choi and N. J. Laustsen [12]. • Chapter 3 contains work submitted for publication [34]. • Chapter 4 contains work accepted for publication [33]. Bence Horváth May 2019 Acknowledgements First and foremost I would like to thank my brilliant supervisors, Dr Yemon Choi and Dr Niels Jakob Laustsen for all their support throughout these three and a half years. Not only did they share their incredible knowledge and mathematical insight with me, but their continued encouragement, patience, and generosity with their time helped me to get much further in my mathematical career than I ever expected. Niels and Yemon; thank you for making my PhD studies such an amazing part of my life! I would like to thank Professor Gábor Elek for the many fun conversations (math- ematical or otherwise) and for his (futile) efforts to talk some sense into me about picking up an interest in more fashionable areas of mathematics. -
Valued Functions Via Schauder Decompositions
Received: 14 April 2019 Revised: 24 October 2019 Accepted: 13 December 2019 DOI: 10.1002/mana.201900172 ORIGINAL PAPER Series representations in spaces of vector-valued functions via Schauder decompositions Karsten Kruse Technische Universität Hamburg, Institut für Mathematik, Am Schwarzenberg- Abstract Campus 3, 21073 Hamburg, Germany It is a classical result that every ℂ-valued holomorphic function has a local power Correspondence series representation. This even remains true for holomorphic functions with val- Karsten Kruse, Technische Universität ues in a locally complete locally convex Hausdorff space over ℂ.Motivatedby Hamburg, Institut für Mathematik, Am this example we try to answer the following question. Let be a locally con- Schwarzenberg-Campus 3, 21073 Ham- (Ω) burg, Germany. vex Hausdorff space over a field ,let be a locally convex Hausdorff space Email: [email protected] of -valued functions on a set Ω and let (Ω, ) be an -valued counterpart of (Ω) (where the term -valued counterpart needs clarification itself). For which spaces is it possible to lift series representations of elements of (Ω) to elements of (Ω, )? We derive sufficient conditions for the answer to be affirmative using Schauder decompositions which are applicable for many classical spaces of func- tions (Ω) having an equicontinuous Schauder basis. KEYWORDS injective tensor product, Schauder basis, Schauder decomposition, series representation, vector-valued function MSC (2020) Primary: 46E40; Secondary: 46A32; 46E10 1 INTRODUCTION The purpose of this paper is to lift series representations known from scalar-valued functions to vector-valued functions and its underlying idea was derived from the classical example of the (local) power( ) series representation of a holomorphic function.