Quasi-Modular Preserving Rank One Maps on Hilbert C*-Modules

In this paper,we characterize a class of quasi-modular maps on Hilbert C~*-modules which map a "rank one" adjointable operator to another rank one operator.     

Unitary Operators on Hilbert C*-Modules

For module mappings between Hilbert C^*-modules, the unitarymaps are those that are isometric and surjective.     

Hilbert C*-Modules over Monotone Complete C*-Algebras

The aim of the present paper is to describe self-duality and C*-reflexivity of Hilbert A-modules ?? over monotone complete C*-algebras A by the completeness of the unit ball of ?? with respect to two types of convergence being defined, and by a structural criterion. The derived results generalize earlier results ofH. Widom [Duke Math. J. 23, 309-324, MR 17 # 1228] and W. L. Paschke [Trans. Amer. Mat. Soc. 182 , 443-468, MR 50 # 8087, Canadian J. Math. 26, 1272-1280, MR 57 # 10433]. For Hilbert C*-modules over commutative AW*-algebras the equivalence of the self-duality property and of the Kaplansky-Hilbert property is reproved, (cf. M. Ozawa [J. Math. Soc. Japan 36, 589-609, MR 85 # 46068]). Especially, one derives that for a C*-algebra A the A-valued inner product of every Hilbert A-module ?? can be continued to an A-valued inner product on it's A-dual Banach A-module ??' turning ??' to a self-dual Hilbert A-module if and only if A is monotone complete (or, equivalently, additively complete) generalizing a result of M. Hamana [Internat. J. Math. 3 (1992), 185 - 204]. A classification of countably generated self-dual Hilbert A-modules over monotone complete C*-algebras A is established. The set of all bounded module operators End ′(??) on self-dual Hilbert A-modules ?? over monotone complete C*-algebras A is proved again to be a monotone complete C*-algebra. Applying these results a Weyl-Berg type theorem is proved.     

Dilation of Dual Frame Pairs in Hilbert C*-Modules

We investigate the geometric properties for Hilbert C*-modular frames. We show that any dual frame pair in a Hilbert C*-module is an orthogonal compression of a Riesz basis and its canonical dual for some larger Hilbert C*-module. This generalizes the Hilbert space dual frame pair dilation theory due to Casazza, Han and Larson to dual Hilbert C*-modular frame pairs. Additionally, for any Hilbert C*-modular dual frame pair induced by a group of unitary operators, we show that there is a dilated dual pair which inherits the same group structure.     

Solutions to Operator Equations on Hilbert C*-Modules II

In this paper, we study the solvability of the operator equations A*X + X*A = C and A*XB + B*X*A = C for general adjointable operators on Hilbert C*-modules whose ranges may not be closed. Based on these results we discuss the solution to the operator equation AXB = C, and obtain some necessary and sufficient conditions for the existence of a real positive solution, of a solution X with B*(X* + X)B ≥ 0, and of a solution X with B*XB ≥ 0. Furthermore in the special case that R(B) í [`(R(A*))]{R(B)\subseteq\overline{R(A*)}} we obtain a necessary and sufficient condition for the existence of a positive solution to the equation AXB = C. The above results generalize some recent results concerning the equations for operators with closed ranges.     

Ordered C*-Modules

In this first part of a study of ordered operator spaces, wedevelop the basic theory of ‘ordered C^*-bimodules’.A crucial role is played by ‘open tripotents’, aJB^*-triple variant of Akemann's notion of open projection. 2000Mathematics Subject Classification 46L08, 47L07 (primary), 46L07,47B60, 47L05 (secondary).     

Representations of Certain Banach C*-Modules

The possibility of extending the well known Gelfand–Naimark–Segal representation of *-algebras to certain Banach C*-modules is studied. For this aim the notion of modular biweight on a Banach C*-module is introduced. For the particular class of strict pre CQ*-algebras, two different types of representations are investigated.     

Compact Operators and Uniform Structures in Hilbert $$C^*$$ -Modules

Functional Analysis and Its Applications - Quite recently a criterion for the $$\mathcal{A}$$ -compactness of an ajointable operator $$F\colon {\mathcal M} \to\mathcal{N}$$ between Hilbert $$C^*$$...     

Hilbert C*-模上框架(强)可补的充要条件

Hilbert C*-模上框架的框架变换的实质是将该模进行膨胀,使得该框架变换的值域存在标准正交基,以便于Hilbert C*-模上不同框架之间关系的研究.受此启发,本文引入了Hilbert C*-模上框架(强)可补的概念,给出并证明了Hilbert C*-模上有限个框架(强)可补的充要条件.     

Derivations,Local and 2-Local Derivations on Some Algebras of Operators on Hilbert C*-Modules

For a commutative C*-algebra \({\mathcal {A}}\) with unit e and a Hilbert \({\mathcal {A}}\)-module \({\mathcal {M}}\), denote by End\(_{{\mathcal {A}}}({\mathcal {M}})\) the algebra of all bounded \({\mathcal {A}}\)-linear mappings on \({\mathcal {M}}\), and by End\(^*_{{\mathcal {A}}}({\mathcal {M}})\) the algebra of all adjointable mappings on \({\mathcal {M}}\). We prove that if \({\mathcal {M}}\) is full, then each derivation on End\(_{{\mathcal {A}}}({\mathcal {M}})\) is \({\mathcal {A}}\)-linear, continuous, and inner, and each 2-local derivation on End\(_{{\mathcal {A}}}({\mathcal {M}})\) or End\(^{*}_{{\mathcal {A}}}({\mathcal {M}})\) is a derivation. If there exist \(x_0\) in \({\mathcal {M}}\) and \(f_0\) in \({\mathcal {M}}^{'}\), such that \(f_0(x_0)=e\), where \({\mathcal {M}}^{'}\) denotes the set of all bounded \({\mathcal {A}}\)-linear mappings from \({\mathcal {M}}\) to \({\mathcal {A}}\), then each \({\mathcal {A}}\)-linear local derivation on End\(_{{\mathcal {A}}}({\mathcal {M}})\) is a derivation.     

C*-模上的准模保秩一映射

In this paper,we characterize a class of quasi-modular maps on Hilbert C~*-modules which map a "rank one" adjointable operator to another rank one operator.     

A Hilbert bundle characterization of Hilbert C*-modules

The category of Hilbert C*-modules over a given C*-algebra is shown to be equivalent to a certain simply described category of Hilbert bundles (i.e., continuous fields of Hilbert spaces) over the space of pure states of the C*-algebra with the zero functional adjoined.

     

Hilbertian Versus Hilbert W*-Modules and Applications to L2- and other invariants

Hilbert(ian) A-modules over finite von Neumann algebras with a faithful normal trace state (from global analysis) and Hilbert W*-modules over A (from operator algebra theory) are compared and a categorical equivalence is established. The correspondence between these two structures sheds new light on basic results in L ²-invariant theory providing alternative proofs. We indicate new invariants for finitely generated projective B-modules, where B is a unital C*-algebra (usually the full group C*-algebra C*() of the fundamental group =₁(M) of a manifold M).     

Extension Operators Preserving Biholomorphic Mappings on Hartogs Domains*

In this paper, the authors extend the Roper-Suffridge operator on the generalized Hartogs domains. They mainly research the properties of the extended operator.By the characteristics of Hartogs domains and the geometric properties of subclasses of spirallike mappings, they obtain the extended Roper-Suffridge operator preserving almost starlikeness of complex order λ, almost spirallikeness of type β and order α, parabolic spirallikeness of type β and order ρ on the Hartogs domains in different co...     

Geometrical Aspects of Hilbert C*-modules

The aim of the present paper is to solve some major open problems of Hilbert C^*-module theory by applying various aspects of multiplier C^*-theory. The key result is the equivalence established between positive invertible quasi-multipliers of the C^*-algebra of compact operators on a Hilbert C^*-module {, ., } and A-valued inner products on , inducing an equivalent norm to the given one. The problem of unitary isomorphism of C^*-valued inner products on a Hilbert C^*-module is considered and new criteria are formulated. Countably generated Hilbert C^*-modules turn out to be unitarily isomorphic if they are isomorphic as Banach C^*-modules. The property of bounded module operators on Hilbert C^*-modules of being compact and/or adjointable is unambiguously connected to operators with respect to any choice of the C^*-valued inner product on a fixed Hilbert C^*-module if every bounded module operator possesses an adjoint operator on the module. Every bounded module operator on a given full Hilbert C^*-module turns out to be adjointable if the Hilbert C^*-module is orthogonally complementary. Moreover, if the unit ball of the Hilbert C^*-module is complete with respect to a certain locally convex topology, then these two properties are shown to be equivalent to self-duality.     

保零积可加映射

给出了Ｂ（Ｘ）上保零积可加映射的刻画．     

Bessel-type inequalities in Hilbert C *-modules

Regarding the generalizations of the Bessel inequality in Hilbert spaces which are due to Bombieri and Boas–Bellman, we obtain a version of the Bessel inequality and some generalizations of this inequality in the framework of Hilbert C *-modules.     

Hilben C*-模之间的同构定理

获得了Hilbert C^*-模之间的两个同构定理．作为推论，证明了C^*-代数上每个可数生成的Hilbert C^*-模均稳定酉等价于A。