DOCSLIB.ORG

Creating a New Nest Algebra Structure for Nilpotent Lie Algebras

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

INTERNATIONAL JOURNAL OF MULTIDISCIPLINARY SCIENCES AND ENGINEERING, VOL. 12, NO. 2, MARCH 2021 Creating a New Nest Algebra Structure for Nilpotent Lie Algebras Atefeh Hasan-Zadeh Fouman Faculty of Engineering, College of Engineering, University of Tehran, Iran Abstract—In this paper a functional analysis approach to solve such that the main concepts of Lie derivations and the results the rigidity problems of the Riemannian nilmanifolds using obtained from it in nilmanifolds are consistent with the information from their geodetic flows is considered. The corresponding results in nest algebras. For this reason, we proposed method combines analytical and geometric materials to could prove one proposition of Gordon, which can be used in construct a structure of nest algebra for nilpotent Lie algebras. In this way, the main concepts of Lie derivations and the results solving the rigidity problem for 2-step nilmanifolds, in a new obtained from it in nilmanifolds are consistent with the structure, Corollary 3.1. It should be noted that this corresponding results in nest algebras. Due to this adaptation of proposition was proved once again in [4] with the use of the the main concepts, one proposition of Gordon, which can be used concepts of Lie groupoids. Lastly, in expressing the in solving the rigidity problem for 2-step nilmanifolds, has been importance of the approach used, an extension of Corollary proved in the new structure. Also, the resulting nesting algebra 3.1 to quasi-nilpotent operators has been mentioned in structure leads to an extension of that proposition to quasi- nilpotent operators. Corollary 3.2. Keywords— Nilpotent Lie Algebra, Inner Derivations, Trace II. PRELIMINARIES Class Operators and Quasi-nilpotent Operators A. Ingredients of operator algebra I. INTRODUCTION Let H be a complex Hilbert space and B()H is the algebra of all linear bounded operators on H . A nest N of N recent years, considerable attention has been paid to projections on is a chain of orthogonal projections on I solving the rigidity problems for nilmanifolds via some information about their geodesic flows: which contains 0 , I and is closed in the strong operator topology. The nest algebra TN() corresponding to the nest M Problem. ([2]) Whether two compact 2-step nilmanifolds N is the set of all operators A in such that AP= PA for all PN . Clearly, is a weak* closed operator algebra. M and are isometric or not, if they have conjugated geodesic For more details refer to [7-9]. flows? Theorem 2.1. ([7]) Let be a nest of projections on a complex Hilbert space H and B be a weak* closed Among which one can be mentioned through the works of Eberlein, Gordon, Mao and Schueth in [1-3]. This problem has subalgebra of which contains . Then a linear been already considered in [4] by an algebraic-geometric map :()TNB→ is a Lie triple derivation if and only if approach, especially in the category of Lie groupoids, and in there are an element S B and a linear functional f on [5] in notions of von Neumann algebras. Also, the symplectic with fD( )= 0 for every second commutator version of it, [3], has been extended to Poisson notions, [6]. In our ongoing research on this issue, we have come up with DXYZ= [[ , ], ] such that ()[,]()A=+ S A f A I , A TN(). an analytical approach. The method presented in this paper Corollary 2.1. ([7]) Let be a nest of infinite multiplicity uses the structure of nest algebras corresponding to the algebra on a complex Hilbert space H and let B be a weak* closed of projections on complex Hilbert spaces. In this way, by subalgebra of which contains . Let be a Lie some ingredients of operator algebra and geometry, we triple derivation from into . Then there is an construct a structure of nest algebra for an 푚-dimensional nilpotent Lie algebra, Theorem 3.1. The structure obtained is element S B such that ()[,]ASA= , A TN(). In fact, Lie derivations are special cases of Lie triple derivations. A. Hasan-Zadeh is a faculty member of Fouman Faculty of Engineering, College of Engineering, University of Tehran, Iran, P.O..Box: 43581-39115, (e-mail: [email protected]). [ISSN: 2045-7057] www.ijmse.org 8 INTERNATIONAL JOURNAL OF MULTIDISCIPLINARY SCIENCES AND ENGINEERING, VOL. 12, NO. 2, MARCH 2021 B. Ingredients of geometry e e, it can be defined the layer MMB ={ :J = 0} . By As the notions of [1-3], a Riemannian nilmanifold is a construction, each layer is a M -invariant subset of M and e quotient of a simply connected nilpotent Lie group M by M is a disjoint finite union of layers MM= B . Also, e a discrete subgroup ; together with a Riemannian metric g all the orbits contained in a given layer have the same e whose lift to is left invariant. A nilmanifold has step dimension ( card e ). Let the orbits contained in M B are 2r - dimensional. Then, there exists on size k if is -step nilpotent. M Especially, a Lie group is said to be two-step nilpotent if i) mr− 2 polynomial functions zz12,..., mr− ; its Lie algebra M satisfies [[MMM , ], ]= 0 , equivalently ii) 2r rational functions p11,..., prr , q ,..., q which are regular [,] is central in . If be a left-invariant MM M e Riemannian metric on 2-step nilpotent Lie group , then it on M B such that the polynomial functions zz12,..., mr− defines an inner product .,. on the Lie algebra of . separate the orbits contained in ; Let ZMM= [,] and V denote the orthogonal complement The first layer has additional properties: it is a Zariski dense of Z in M relative to . For zZ , a skew symmetric open subset of M , it contains only orbits of maximal linear transformation Jz():VV→ can be defined by dimension 2d , and the polynomial functions zz12,..., md− J( z ) x= ( ad ( x )) z for x V , where (ad ( x )) denotes the separating the orbits of are M -invariant. adjoint of ad() x . Equivalently, Moreover, if we identify the symmetric algebra SM() of J( z ) x , y= [ x , y ], z , x , y VZ , z (2.1) with the space of polynomial functions on M and and this process is reversible. denote by SM()M the subring of of the 푀invariant An authomorphism of M is said to be -almost inner if polynomial functions, then the quotient field of () is conjugate to for all . It is said to be almost coincides exactly with the field R( z ,..., z ) of rational inner if ()x is conjugate to x for all xM . A derivation 12md− e of the Lie algebra is said to be -almost inner functions in the zk variables. The open set M B is usually (respectively, almost inner) if ()[,]XX M for all X log called the generic set associated to the basis B , the orbits e (respectively, for all xM . The spaces of -almost inner contained in M B are the generic orbits and the automorphisms is denoted by AIA() and similarly, AID() is corresponding polynomial functions zz12,..., md− are the its infinitesimal counterpart, namely -almost inner generic invariants. derivations. Let M is any open subset of M such that the M III. MAIN STRUCTURE polynomial functions separate the orbits contained in and the vectors dz( ),..., dz ( ) are linearly Theorem 3.1. For an m-dimensional nilpotent Lie algebra 12md− , it can be constructed a structure of nest algebra TM() independent for all in . Also, since the generic corresponded to it which the notions of Lie derivations in invariants associated to any basis B generate the these two apparently different structures produce the same quotient field of SM()M , then satisfies the above meanings and results (especially Theorem 2.1 and Corollary conditions for any Jordan-Holder basis. 2.1 in this structure will also be satisfied). Corollary 3.1. ([2]) Let be a 2-step nilpotent Lie algebra Proof. Suppose that is an m -dimensional nilpotent Lie with an inner product .,. and be an almost inner algebra. Denote by M the dual space of and M by the connected and simply connected Lie group with Lie algebra derivation of continuous type on say (x )= [ ( x ), x ] with . Let MMMM01... m = be a flag of continuous on M \{0} . Let z Z() M and yker( J ( z )) . ( dimMi = i ) such that []M,Mii M −1 for all im{1,..., } . Let Then (x ), z= [ ( x ), y ], z , x M , where Jz():VV→ , also BXX= ( 1,..., m ) be a Jordan-Holder basis adapted to is a skew symmetric linear transformation defined by equation (2.1). (Mi ) that Mii=RRXX1 ... , for all i . Then, M has a i In special case, if the center of properly contains the natural layering which can be summarized as follows. derived algebra, then every almost inner derivation of For in , it can be defined the set of indexes continuous type on M is inner. Proof. From Theorem 3.1 applied to Theorem 2.1 and J= j: X ji (MM−1 + ) , where Corollary 2.1, it can be concluded that: MMM =XXYY : ,[ , ] = 0, . If J = j12 ... j r A Lie derivation (especially, almost inner derivation) then = RRXX ... . Let ={;}J . For (especially, , on the nest algebra where MM jj12r M [ISSN: 2045-7057] www.ijmse.org 9 INTERNATIONAL JOURNAL OF MULTIDISCIPLINARY SCIENCES AND ENGINEERING, VOL. 12, NO. 2, MARCH 2021 S is a weak* closed subalgebra which contains REFERENCES (especially, (x ), z= [ ( x ), y ], z , x M ) and vice versa. [1] P. Eberlein, “Geometry of two-step nilpotent groups with a left As a result, it has been archived more than we expected.
Recommended publications
  • Arxiv:1603.03426V2 [Math.FA]

    Arxiv:1603.03426V2 [Math.FA]

    THE PARABOLIC ALGEBRA ON Lp SPACES E. KASTIS Abstract. The parabolic algebra was introduced by Katavolos and Power, in 1997, as the SOT - closed operator algebra acting on L2(R) that is generated by the translation and multiplication semigroups. In particular, they proved that this algebra is reflexive and is equal to the Fourier binest algebra, that is, to the algebra of operators that leave invariant the subspaces in the Volterra nest and its analytic counterpart. We prove that a similar result holds for the corresponding algebras acting on Lp(R), where 1 <p< . In the last section, it is also shown that the reflexive closures of the Fourier binests∞ on Lp(R) are all order isomorphic for 1 <p< . ∞ 1. Introduction Let Dµ,µ R and Mλ,λ R be the groups of translation and multiplication respectively{ acting∈ } on the{ Hilbert∈ space} L2(R), given by D f(x)= f(x µ), M f(x)= eiλxf(x). µ − λ It is well-known that these 1-parameter unitary groups are continuous in the strong operator topology (SOT), that they provide an irreducible representation of the Weyl- iλµ commutation relations, MλDµ = e DµMλ, and that the SOT-closed operator algebra they generate is the von Neumann algebra B(L2(R)) of all bounded operators. (See Tay- lor [23], for example.) On the other hand it was shown by Katavolos and Power in [11] that the strongly closed nonselfadjoint operator algebra generated by the semigroups for µ 0 and λ 0 is a reflexive algebra, in the sense of Halmos [18], containing no self-adjoint≥ operators,≥ other than real multiples of the identity, and containing no nonzero finite rank p p operators.
  • The Carrier Space of a Reflexive Operator Algebra

    The Carrier Space of a Reflexive Operator Algebra

    PACIFIC JOURNAL OF MATHEMATICS Vol. 81, No. 2, 1979 THE CARRIER SPACE OF A REFLEXIVE OPERATOR ALGEBRA ALAN HOPENWASSER AND DAVID LARSON Many properties of nest algebras are actually valid for reflexive operator algebras with a commutative subspace lattice. In this paper we collect a number of such results related to the carrier space of the algebra. Included among these results are a generalization of Ringrose's criterion, a description of the partial correspondence between lattice homomorphisms of the carrier space and projections in the lattice, the construction of isometric representations of certain quotient algebras, and a direct sum decomposition of the commutant of the core modulo the intersection of the spectral ideals. Let J^ = Alg ^ where if is a commutative subspace lattice and let ^ be the intersection of all the spectral ideals in Jzf. (See §1 for definitions.) In §1 we generalize Ringrose's criterion to the commutative subspace lattice case: A e ^ if, and only if, for each e > 0 there is a finite family {Et} of mutually orthogonal intervals from & such that Σ^ = l and WE.AE.W < ε, i = 1, . , n. We also prove that ^ is the closed linear span of commutators of the form AL — LA, where A e jy and L e £f. In § 2 we describe the partial correspondence between certain projections in Sf and certain lattice homomorphisms in the carrier space X. A necessary (but not sufficient) condition for an operator A to be in the radical of *$/ is given in §3. In §4 we exhibit isometric representations as algebras of operators acting on Hubert space of each quotient algebra J^fjj^φ and of the quotient *S%f\<J\ In the nest algebra case this was done by Lance in [5].
  • Subdiagonal Subalgebras and Commutative Banach Algebras by Toeplitz Operators

    Subdiagonal Subalgebras and Commutative Banach Algebras by Toeplitz Operators

    Sudan University of Science and Technology College of Graduate Studies Subdiagonal Subalgebras and Commutative Banach Algebras by Toeplitz Operators الجبريات الجزئية القطرية الجزئية و جبريات باناخ التبديلية بواسطة مؤثرات تبوليتز A thesis Submitted in Fulfillment of the requirements for the degree of Ph.D in Mathematics By Siddig Ibrahim HassabAlla Gomer Supervisor Prof. Dr. Shawgy Hussein Abdalla 12017 Dedication To my Family I Acknowledgements I wish to acknowledge valuable support of my supervisor Prof. Dr. Shawgy Hussein Abd All for his guidance and help during writing the dissertation. My thanks are also due to the staff of Mathematics Department, College of sciences, Sudan University of Sciences and Technology. I would like to extend to thank to Dean of the College of Sciences, Sudan University of Sciences and Technology for him helpful advice & encouragement throughout this research. I would like my thanks to my wife Sulafa Abdel Rahim Mohammed. My thanks are due to Mohammed Ali at Bahri University. Finally I would like to thank my Colleagues. II Abstract We determine the eigenvalues inequalities, sums of hermitian and normal matrices, Schubert calculus, Wielant’s theorem with spectral sets and Banach algebra. The principal submatrices with noncommutative function theory and unique extensions was shown. We give applications of the Fuglede-Kadison determinant, Riesz and Szegö type factorizations theorem for noncommutative Hardy spaces and for a Helson-Szegö theorem noncommutative Hardy- Lorentz spaces. We also give a Helson-Szegö subdiagonal subalgebras with applications to Toeplitz operators. The algebraic structure of non-commutative analytic with quasi-radial quasi-homogeneous symbols and commutative Banach algebra of Toeplitz algebra and operators are presented, the structure of a commutative Banach algebra on the unit ball and quasi-nilpotent group action, generated by Toeplitz operators with quasi-radial quasi- homogeneous symbols are discussed.
  • Arxiv:1902.09989V2

    Arxiv:1902.09989V2

    HEREDITARILY ANTISYMMETRIC OPERATOR ALGEBRAS NIK WEAVER Abstract. We introduce a notion of “hereditarily antisymmetric” operator algebras and prove a structure theorem for them in finite dimensions. We also characterize those operator algebras in finite dimensions which can be made upper triangular and prove matrix analogs of the theorems of Dilworth and Mirsky for finite posets. Some partial results are obtained in the infinite dimensional case. In this paper an operator algebra is a linear subspace of some B(H) which is stable under products, where H is a complex Hilbert space. It is unital if it contains the identity operator I. Self-adjointness is not assumed. Indeed, an operator algebra A is said to be antisymmetric if A∩A∗ is {0} or C · I. This means that a nonunital antisymmetric operator algebra contains no nonzero self-adjoint operators, and the only self-adjoint operators a unital one contains are the real multiples of the identity operator (Proposition 3.1). Antisymmetric operator algebras were introduced in [9] and have attracted occasional attention, e.g., [5, 6]. n Following [10], we regard linear subspaces V of Mn = Mn(C) =∼ B(C ) as matrix or “quantum” analogs of relations on finite sets. In this picture the quantum analog of a preorder relation, i.e., a relation which is reflexive and transitive, is a unital operator algebra. This intuition behind this idea, and its relation to the physics of finite state quantum systems, is discussed below in Section 1. Partially ordered sets are preordered sets which satisfy the extra condition of antisymmetry (a ≤ b and b ≤ a implies a = b).
  • Lsometries of Nest Algebras

    Lsometries of Nest Algebras

    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF FUNCTIONAL ANALYSIS 86, 180-209 (1989) lsometries of Nest Algebras R. L. MOORE* AND T. T. TRENT* Department of Mathematics, University qf Alabama, Tuscaloosa, Alabama 35487 Communicated by D. Sarason Received October 22, 1987; revised February 8, 1988 1. INTRODUCTION In [S], Richard Kadison characterized all linear isometric maps of one C*-algebra, d, onto another one, .!3. In case d and 93 are von Neumann algebras, his result assumes a particularly simple and pleasing form. Where should one look for generalizations of Kadison’s results? One possibility is to investigate isometries acting on reflexive operator algebras, which represent a natural generalization of von Neumann algebras. In this paper, we begin such a study by considering nest algebras, which in some sense lie at the “opposite pole” (among reflexive algebras) from von Neumann algebras. To fix terminology and notation, we will throughout the paper denote by S a complex, separable, infinite-dimensional Hilbert space. An operator is a continuous linear transformation on 2, and the set of all such is &9(X)). A projection on SF is a self-adjoint idempotent operator in a(X). There is an obvious correspondence between projections and their ranges, which are always norm-closed subspaces of 2”. It will from time to time be convenient to blur the distinction between a projection and its range; thus, we might speak of a projection as being invariant for an operator. A lattice 9 of projections (or subspaces) is a collection of projections closed under the operations A and v , where E A F is the projection whose range is (range E) n (rangeF) and E v F is the projection whose range is the closure of (range E) + (range F).
  • Arxiv:2006.01134V1 [Math.FA]

    Arxiv:2006.01134V1 [Math.FA]

    BIMODULES OF BANACH SPACE NEST ALGEBRAS LU´IS DUARTE AND LINA OLIVEIRA Abstract. We extend to Banach space nest algebras the theory of essential supports and support function pairs of their bimodules, thereby obtaining Banach space counterparts of long established results for Hilbert space nest algebras. Namely, given a Banach space nest algebra A, we charaterise the maximal and the minimal A-bimodules having a given essential support function or support function pair. These characterisations are complete except for the minimal A-bimodule corresponding to a support function pair, in which case we make some headway. We also show that the weakly closed bimodules of a Banach space nest algebra are exactly those that are reflexive operator spaces. To this end, we crucially prove that reflexive bimodules determine uniquely a certain class of admissible support functions. 1. Introduction Nest algebras on Hilbert space were introduced in 1963 by J. R. Ringrose as a generalisation of a class of operator algebras including, for example, the algebra of upper triangular matrices and some of the maximal triangular algebras appearing in the work of Kadison and Singer (see [15],[25]). These algebras have attracted a vast amount of research over the years, with Davidson’s book [5] being the standard reference for a comprehensive account of their basic theory. A natural question is, of course, to what extent results in the Hilbert space setting can be seen to hold for Banach spaces, a question already hinted at in [25]. Although many authors have tackled this problem, it is still the case that nest algebras on Banach spaces are far less explored in the literature.

  • Semi-Discreteness and Dilation Theory for Nest Algebras

    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF FUNCTIONAL ANALYSIS 80, 7687 (1988) Semi-discreteness and Dilation Theory for Nest Algebras V. I. PAULSEN* Department of Mathematics, University of Houston, Houston, Texas 77004 S. C. POWER? Department of Mathematics, University of Lancaster, Lancaster, England AND J. D. WARD Department of Mathematics, Texas A&M University, College Station, Texas 77843 Communicated by the Editors Received May 7, 1987; revised June 30, 1987 In this paper it is shown that a contractive u-weakly continuous Hilbert space representation of a nest algebra admits a u-weakly continuous dilation to the containing algebra of all operators. This is accomplished by first establishing the complete contractivity of contractive representations through a semi-discreteness property for nest algebras relative to finite-dimensional nest algebras (Theorem 2.1). The semi-discreteness property is obtained by an examination of the order type, spectral type, and multiplicity of the nest, and by the construction of subalgebras that are completely isometric copies of finite-dimensional nest algebras, with good approximation properties. With complete contractivity at hand, the desired dilation follows from Arveson’s dilation theorem and auxiliary arguments. 0 1988 Academic Press, Inc. In this note we show that a contractive a-weakly continuous Hilbert space representation of a nest algebra admits a a-weakly continuous dilation to the containing algebra of all operators. Our method is to * Supported in part by an NSF grant. + Supported by a D. T. Bourgin visiting scholarship at the University of Houston and by a Fulbright travel grant.
  • Non-Selfadjoint Operator Algebras Generated by Unitary Semigroups

    Non-Selfadjoint Operator Algebras Generated by Unitary Semigroups

    Non-selfadjoint operator algebras generated by unitary semigroups Eleftherios Kastis Department of Mathematics and Statistics Lancaster University This dissertation is submitted for the degree of Doctor of Philosophy September 2017 Dedicated to my loving wife, Anna. Declaration Hereby I declare that the present thesis was prepared by me and none of its contents was obtained by means that are against the law. I also declare that the present thesis is a part of a PhD Programme at Lancaster University. The thesis has never before been a subject of any procedure of obtaining an academic degree. The thesis contains research carried out jointly: Chapter 3 forms the basis of the paper [37] co-authored with S. C. Power. Hereby I declare that I made a full contribution to all aspects of this research and the writing of this paper. Eleftherios Kastis September 2017 Acknowledgements First I want to thank my supervisor Professor S. C. Power for the unconditional support of my Ph.D study and related research, for his patience and immense knowledge. It has been an honor to be his Ph.D. student. I also wish to thank my thesis committee, Professor I. Todorov and Dr. Y. Choi, for their insightful comments and suggestions. My sincere thanks also goes to Professor A. Katavolos and Dr E.T.A. Kakariadis for the support and friendship they provided over the last decade. Last but not the least, I would like to thank my family for supporting me spiritually throughout writing this thesis and my life in general. Abstract The parabolic algebra was introduced by Katavolos and Power, in 1997, as the weak∗- closed operator algebra acting on L2(R) that is generated by the translation and multiplication semigroups.
  • Interpolation in Nest Algebra Modules

    Interpolation in Nest Algebra Modules

    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 130, Number 2, Pages 427{432 S 0002-9939(01)06074-9 Article electronically published on May 23, 2001 INTERPOLATION IN NEST ALGEBRA MODULES XIAOXIA ZHANG (Communicated by David R. Larson) Abstract. Let A be a nest algebra and LatA its invariant projection (or subspace) lattice. In this paper, using order homomorphisms of LatA,we give necessary and sufficient conditions on bounded linear operators X and Y on a Hilbert space to guarantee the existence of an operator A in a certain A-module such that AX = Y . Introduction In the study of a reflexive operator algebra A,itisusefultoknowforwhich vectors x and y there exists A 2Asatisfying Ax = y. E.C. Lance initiated the study of such problems and gave necessary and sufficient conditions on vectors x and y to guarantee the existence of an operator A in a given nest algebra AlgN such that Ax = y,wherex and y are vectors in a Hilbert space H and N is a nest. In addition, Lance specified the minimum norm such an operator can have. A. Hopenwasser [4] extended Lance's result to the case where the nest N is replaced by an arbitrary commutative invariant projection lattice. Munch [8] restricted the operator A to be a Hilbert-Schmidt operator in nest algebra AlgN .In[1]the authors considered the problem of finding A so that Ax = y and A is required to be in certain ideals contained in AlgN . All the results mentioned above are in the case of \vector interpolation". There is another problem related to the one above: for a given operator algebra A on a Hilbert space H, for which bounded operators X and Y on H does there exist an A in A satisfying AX = Y ? In [5] the authors solved this problem in nest algebra and gave a sufficient and necessary condition.
  • Arxiv:1201.0293V1 [Math.OA] 31 Dec 2011 Lers Epoeta Ih Uloeao Module Operator Dual Right a That Prove We Algebras

    Arxiv:1201.0293V1 [Math.OA] 31 Dec 2011 Lers Epoeta Ih Uloeao Module Operator Dual Right a That Prove We Algebras

    APPLICATIONS OF OPERATOR SPACE THEORY TO NEST ALGEBRA BIMODULES G. K. ELEFTHERAKIS Abstract. Recently, Blecher and Kashyap have generalized the notion of W ∗-modules over von Neumann algebras to the setting where the opera- tor algebras are σ−weakly closed algebras of operators on a Hilbert space. They call these modules weak* rigged modules. We characterize the weak* rigged modules over nest algebras. We prove that Y is a right weak* rigged module over a nest algebra Alg(M) if and only if there exists a completely isometric normal representation Φ of Y and a nest algebra Alg(N ) such that Alg(N )Φ(Y )Alg(M) ⊂ Φ(Y ) while Φ(Y ) is implemented by a contin- uous nest homomorphism from M onto N . We describe some properties which are preserved by continuous CSL homomorphisms. 1. Introduction C∗-modules are well known as a generalization of the notion of Hilbert spaces introduced by Kaplansky [12]. They are very important tools for the study of C∗-algebras. The dual version is the notion of W ∗-module. A W ∗-module over a von Neumann algebra is a C∗-module which is a dual Banach space and defines a separately weak* continuous inner product [4]. W ∗-modules were originally introduced by Paschke [14]. They possess fruit- ful properties and appear in many versions such as weak* ternary rings of operators (TROs). ∗ arXiv:1201.0293v1 [math.OA] 31 Dec 2011 The development of operator space theory makes the generalization of W - modules possible even in the case of non-selfadjoint dual operator algebras. In particular, Blecher and Kashyap provided a new characterization of W ∗- modules which enabled the definition of an analogous notion for modules over non-selfadjoint dual operator algebras [1].
  • The Operator Algebra Generated by the Translation, Dilation and Multiplication Semigroups

    The Operator Algebra Generated by the Translation, Dilation and Multiplication Semigroups

    THE OPERATOR ALGEBRA GENERATED BY THE TRANSLATION, DILATION AND MULTIPLICATION SEMIGROUPS E. KASTIS AND S. C. POWER Abstract. The weak operator topology closed operator algebra on L2(R) generated by the one-parameter semigroups for translation, dilation and multiplication by eiλx; λ ≥ 0; is shown to be a reflexive operator algebra, in the sense of Halmos, with invariant subspace lattice equal to a binest. This triple semigroup algebra, Aph, is antisymmetric in the sense ∗ that Aph \ Aph = CI, it has a nonzero proper weakly closed ideal generated by the finite- rank operators, and its unitary automorphism group is R. Furthermore, the 8 choices of semigroup triples provide 2 unitary equivalence classes of operator algebras, with Aph ∗ and Aph being chiral representatives. 1. Introduction 2 Let Dµ and Mλ be the unitary operators on the Hilbert space L (R) given by iλx Dµf(x) = f(x − µ);Mλf(x) = e f(x) where µ, λ are real. As is well-known, the 1-parameter unitary groups fDµ; µ 2 Rg and fMλ; λ 2 Rg provide an irreducible representation of the Weyl-commutation relations, iλµ MλDµ = e DµMλ, and the weakly closed operator algebra they generate is the von Neumann algebra B(L2(R)) of all bounded operators. (See Taylor [19], for example.) On the other hand it was shown by Katavolos and Power in [8] that the weakly closed nonselfadjoint operator algebra generated by the semigroups for µ ≥ 0 and λ ≥ 0 is a proper subalgebra containing no self-adjoint operators, other than real multiples of the identity, and no nonzero finite rank operators.
  • Kadison -Singer Algebras with Applications to Von Neumann Algebras

    Kadison -Singer Algebras with Applications to Von Neumann Algebras

    University of New Hampshire University of New Hampshire Scholars' Repository Doctoral Dissertations Student Scholarship Fall 2009 Kadison -Singer algebras with applications to von Neumann algebras Mohan Ravichandran University of New Hampshire, Durham Follow this and additional works at: https://scholars.unh.edu/dissertation Recommended Citation Ravichandran, Mohan, "Kadison -Singer algebras with applications to von Neumann algebras" (2009). Doctoral Dissertations. 501. https://scholars.unh.edu/dissertation/501 This Dissertation is brought to you for free and open access by the Student Scholarship at University of New Hampshire Scholars' Repository. It has been accepted for inclusion in Doctoral Dissertations by an authorized administrator of University of New Hampshire Scholars' Repository. For more information, please contact [email protected]. KADISON-SINGER ALGEBRAS WITH APPLICATIONS TO VON NEUMANN ALGEBRAS BY MOHAN RAVICHANDRAN B.Tech., Indian Institute of Technology, Chennai DISSERTATION Submitted to the University of New Hampshire in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Mathematics September 2009 UMI Number: 3383324 INFORMATION TO USERS The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleed-through, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. UMI® UMI Microform 3383324 Copyright 2009 by ProQuest LLC All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code.