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2004, Journal of Functional Analysis
We use the injective envelope to study quasimultipliers of operator spaces. We prove that all representable operator algebra products that an operator space can be endowed with are induced by quasimultipliers. We obtain generalizations of the Banach-Stone theorem.
Journal of Functional Analysis, 2007
Let X be an operator space, let ϕ be a product on X, and let (X, ϕ) denote the algebra that one obtains. We give necessary and sufficient conditions on the bilinear mapping ϕ for the algebra (X, ϕ) to have a completely isometric representation as an algebra of operators on some Hilbert space. In particular, we give an elegant geometrical characterization of such products by using the Haagerup tensor product. Our result makes no assumptions about identities or approximate identities. Our proof is independent of the earlier result of Blecher-Ruan-Sinclair ([6]) that solved the case when the algebra has an identity of norm one, and our result is used to give a simple direct proof of this earlier result. We also develop further the connections between quasi-multipliers of operator spaces, and shows that the quasi-multipliers of operator spaces defined in [12] coincide with their C *-algebraic counterparts.
Journal of Operator Theory, 2016
We study extreme points of the unit ball of an operator space by introducing the new notion “(approximate) quasi-identities”. More specifically, we characterize an operator algebra having a contractive (approximate) quasi– (respectively, left, right, two-sided) identity in terms of quasi-multipliers and extreme points of the unit ball (of the weak*-closure) of the underlying operator space. Furthermore, we give a necessary and sufficient condition for a given operator space to be qualified to become a C*-algebra or a one-sided ideal in a C*-algebra in terms of quasi-multipliers.
International journal of pure and applied mathematics, 2019
New notions of (φ,ψ)-multipliers and (φ,ψ)-quasi multipliers on Banach algebras are introduced, where φ and ψ are linear mappings. Examples are given to show that for most of the new notions, the corresponding class of these operators is larger than that for the old versions. General theory is developed for these notions. AMS Subject Classification: 47B48
Pacific Journal of Mathematics, 1976
Fix a sequence ^ = {P n }"=i of finite dimensional projections increasing to the identity on a separable Hubert space W and let j?(2ίf) denote the algebra of all bounded operators on 5K. The quasitriangular algebra associated with $P and denoted as Ά3~(&) is defined to be the set of those operators T in i?($?) for which || Pt TP n ||-»0. In this paper we will examine the structure of the Ά2Γ(SP) algebras. Specifically, if &t = {jR n }^= 1 is another sequence of finite dimensional projections increasing to the identity on the same Hubert space, when is Άθ'iβ) equal to °U(<P)Ί By an algebraic isomorphism between two algebras we shall mean a bijection which preserves algebraic structure: that is to say-addition, scalar multiplication, multiplication, but we do not impose any topological condition. When are two quasitriangular algebras isomorphic?
Abstract and Applied Analysis, 2011
We investigate the extent to which the study of quasimultipliers can be made beyond Banach algebras. We will focus mainly on the class of -algebras, in particular on complete -normed algebras, , not necessarily locally convex. We include a few counterexamples to demonstrate that some of our results do not carry over to general -algebras. The bilinearity and joint continuity of quasimultipliers on an -algebra are obtained under the assumption of strong factorability. Further, we establish several properties of the strict and quasistrict topologies on the algebra of quasimultipliers of a complete -normed algebra having a minimal ultra-approximate identity.
In this thesis, we investigate some results on multipliers and bounded approximate identities in Banach algebras. In doing this, we first review the general theory of Banach algebras. We define some basic and important concepts relating to Banach algebras, we give some vital examples, mention and prove some important results, and form new Banach algebras from known ones. We then define the notion of bounded approximate identities in Banach algebras, study and establish some basic results, and then consider bounded approximate identities in group algebras. Finally, we define and give examples of multipliers on Banach algebras without order (faithful) and then investigate some interesting results.
Proceedings of the American Mathematical Society, 2008
We construct a simple reproducing kernel space whose multiplier algebra does not satisfy a "corona theorem".
Mathematica Scandinavica
We use the notion of Π 2 -hyperreflexivity to construct, for a wide variety of Banach algebras A, an operator space X and a representation π : A → CB(X), such that CB(X) consists of 2-summing perturbations of π(A). This gives rise to some examples of operator spaces with interesting properties.
Journal of Mathematical Analysis and Applications, 2020
Quasi *-algebras form an essential class of partial *algebras, which are algebras of unbounded operators. In this work, we aim to construct tensor products of normed, respectively Banach quasi *-algebras, and study their capacity to preserve some important properties of their tensor factors, like for instance, *semisimplicity and full representability.
2004
This paper is a revision and an enlargement of the previous version titled "Extreme points of the unit ball of a quasi-multiplier space" which had been circulated since 2004. We study extreme points of the unit ball of an operator space by introducing the new notion (approximate) "quasi-identities". Then we characterize an operator algebra with a contractive approximate quasi- (respectively, left, right, two-sided) identity in terms of quasi-multipliers and extreme points. Furthermore, we give a very neat necessary and sufficient condition for a given operator space to become a $C^*$-algebra or a one-sided ideal in a $C^*$-algebra in terms of quasi-multipliers. An extreme point is also used to show that any TRO with predual can be decomposed to the direct sum of a two-sided ideal, a left ideal, and a right ideal in some von Neumann algebra.
In this paper, we give definition of quasiring as a new concept. Also, we introduce the notions of quasimodule and normed quasimodule defined on a quasiring. We should immediately note that a quasimodule is a generalization of quasilinear spaces. Similarly, normed quasimodule is a generalization of normed quasilinear spaces defined by Aseev, . Moreover, we obtain some results about the relationships between these concepts. We think that investigations on quasimodules may provide some important contributions to improvement of some branches of quasilinear functional analysis such as the duality theory of quasilinear spaces. Also we recognize that the notion of quasimodule is more suitable backdrop as regards theory of quasilinear spaces in examination of quasilinear functional.
2010
Quasi-multipliers for a Hilbert C * -bimodule V were introduced by L. G. Brown, J. A. Mingo, and N.-T. Shen (Canad. J. Math., 46(1994), 1150-1174) as a certain subset of the Banach bidual module V * * . We give another (equivalent) definition of quasi-multipliers for Hilbert C * -bimodules using the centralizer approach and then show that quasi-multipliers are, in fact, universal (maximal) objects of a certain category. We also introduce quasi-multipliers for bimodules in Kasparov's sense and even for Banach bimodules over C * -algebras, provided these C * -algebras act non-degenerately. A topological picture of quasi-multipliers via the quasi-strict topology is given. Finally, we describe quasi-multipliers in two main situations: for the standard Hilbert bimodule l 2 (A) and for bimodules of sections of Hilbert C * -bimodule bundles over locally compact spaces.
We study extreme points of the unit ball of the set of quasi-multipliers of an operator space by introducing the new notion: (approximate) quasi-identities. We give a necessary and sufficient condition for an operator space to become an operator algebra with a contractive approximate quasi-(respectively, left, right, two-sided) identity in terms of extreme points of contractive quasi-multipliers. We also give a necessary sufficient condition for an operator space to become a C * -algebra. Furthermore, we answer the open question about Properties (L) and (R) raised by D. P. Blecher.
Rendiconti del Circolo Matematico di Palermo, 2003
Journal of Mathematical Analysis and Applications, 2004
Let A 1 , A 2 , . . . , A r be C * -algebras with second duals A 1 , A 2 , . . . , A r , and let X be an arbitrary Banach space. Let Γ : A 1 × A 2 × · · · × A r → X be a bounded r-linear map, and denote by Γ : A 1 × A 2 × · · ·× A r → X the Johnson-Kadison-Ringrose extension (i.e., the separately weak * to weak * continuous r-linear extension) of Γ . The problem of characterising those Γ for which Γ takes its values in X was solved by Villanueva when the algebras are all commutative. Because the Dunford-Pettis property fails for noncommutative C * -algebras, the 'obvious' extension of Villanueva's characterisation does not give the correct condition. In this paper we solve this problem for general C * -algebras. This result is then applied to obtaining a multilinear generalisation of the normal-singular decomposition of a bounded linear operator on a von Neumann algebra.
Al-Qadisiyah Journal Of Pure Science, 2021
This paper studies concept of a quasi-inner product space and its completeness to get and prove some properties of quasi-Hilbert spaces. The best examples of this notion are spaces where 0
Journal of Mathematical Analysis and Applications, 1991
Journal of the Australian Mathematical Society, 2004
We show that the operator-valued Marcinkiewicz and Mikhlin Fourier multiplier theorem are valid if and only if the underlying Banach space is isomorphic to a Hilbert space.
2009
We provide an operator space version of Maurey's factorization theorem. The main tool is an embedding result of independent interest. Applications for operator spaces and noncommutative Lp spaces are included.
Journal of Functional Analysis, 2008
The little Grothendieck theorem for Banach spaces says that every bounded linear operator between C(K) and ℓ 2 is 2-summing. However, it is shown in [7] that the operator space analogue fails. Not every cb-map v : K → OH is completely 2-summing. In this paper, we show an operator space analogue of Maurey's theorem : Every cb-map v : K → OH is (q, cb)-summing for any q > 2 and hence admits a factorization v(x) ≤ c(q) v cb axb q with a, b in the unit ball of the Schatten class S 2q .
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