Yu. I. Zhuraev, F. Sharipov
In the present paper the notion of a Hilbert module over a locally C*-algebra is discussed and some results are obtained on this matter. In particular, we give a detailed proof of the known result that the set of adjointable endomorphisms of such modules is itself a locally C*-algebra.
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We study induced representations of Hilbert modules over locally C*-algebras and their non-degeneracy. We show that if $V$ and $W$ are Morita equivalent Hilbert modules over locally C*-algebras $A$ and $B$, respectively, then there exists a bijective correspondence between equivalence classes of non-degenerate representations of $V$ and $W$.
Operator Models for Hilbert Locally $C^*$-Modules
本文通过局部有界算子刻画了局部 $C^*$-代数上的具体 Hilbert 模概念，为所有 Hilbert 局部 $C^*$-模构造了算子模型，并基于一般膨胀定理导出了外张量积的直接构造及两个 Stinespring 型定理。
In this article, we study tensor product of Hilbert $C^*$-modules and Hilbert spaces. We show that if $E$ is a Hilbert $A$-module and $F$ is a Hilbert $B$-module, then tensor product of frames (orthonormal bases) for $E$ and $F$ produce frames (orthonormal bases) for Hilbert $A\otimes B$-module $E\otimes F$, and we get more results. For Hilbert spaces $H$ and $K$, we study tensor product of frames of subspaces for $H$ and $K$, tensor product of resolutions of the identities of $H$ and $K$, and tensor product of frame representations for $H$ and $K$.
In this paper, we define the notion of induced representations of a Hilbert $C^{*}$-module and we show that Morita equivalence of two Hilbert modules (in the sense of Moslehian and Joita), implies the equivalence of categories of non-degenerate representations of two Hilbert modules.
Analogues for Hilbert C*-modules of classical results of Fourier series theory in Hilbert spaces are considered. Relations between different properties of orthogonal and orthonormal systems for Hilbert C*-modules are studied with special attention paid on the differences with the well-known Hilbert space situation.
A Hilbert $C^*$-quad module of finite type has a multi structure of Hilbert $C^*$-bimodules with two finite bases. We will construct a $C^*$-algebra from a Hilbert $C^*$-quad module of finite type and prove its universality subject to certain relations among generators. Some examples of the $C^*$-algebras from Hilbert $C^*$-quad modules of finite type will be presented.
We provided an analogue Banach-Alaoglu theorem for Hilbert $H^*$-module. We construct a $Λ$-weak$^*$ topology on a Hilbert $H^*$-module over a proper $H^*$-algebra $Λ$, such that the unit ball is compact with respect to $Λ$-weak$^*$ topology.
In this paper we view some fundamentals of the theory of Hilbert C*-modules and examine some ways in which Hilbert C*-modules differ from Hilbert spaces.
It is shown that the metric on the union of the sets $X$ and $Y$ defines a Hilbert $C^*$-module over the uniform Roe algebra of the space $X$ with a fixed metric $d_X$. A number of examples of such Hilbert $C^*$-modules are described.
We generalize some results on compact operators on Hilbert spaces to "compact" operators on some Hilbert right W*-modules. We present in this frame the Schatten decomposition of the compact operators, the trace, the Banach Lp-spaces and their duals, the Hilbert-Schmitt operators, and the integral operators as an example of Hilbert-Schmitt operators.
In this paper we introduce the concepts of atomic systems for operators and K-frames in Hilbert C*-modules and we establish some results.
In this paper, we characterize a class of additive maps on Hilbert $C^\ast$-modules which maps a "rank one" adjointable operators to another rank one operators.
Theory of extensions of Hilbert C*-modules was developed by D. Bakic and B. Guljas. An easy observation shows that in the case, when the underlying C*-algebra extension is commutative and the Hilbert C*-modules are projective of finite type, the algebraic properties of the corresponding Busby invariant allow to identify extensions with isometric maps of the corresponding vector bundles. When the Hilbert C*-modules are free of rank one, we evaluate the set of extensions in topological terms.
In this paper we study the unitary equivalence between Hilbert modules over a locally C*-algebra. Also, we prove a stabilization theorem for countably generated modules over an arbitrary locally C*-algebra and show that a Hilbert module over a Frechet locally C*-algebra is countably generated if and only if the locally C*-algebra of all ''compact'' operators has an approximate unit.
Rigged modules over an operator algebra are a generalization of Hilbert modules over a $C^{\star}$-algebra. We characterize the rigged modules over an operator algebra $\mathcal A$ which are orthogonally complemented in $C_\infty(\mathcal A),$ the space of infinite columns with entries in $\mathcal A.$ We show that every such rigged module `restricts' to a bimodule of Morita equivalence between appropriate stably isomorphic operator algebras.
The authors have recently shown how direct limits of Hilbert spaces can be used to construct multi-resolution analyses and wavelets in $L^2(\R)$. Here they investigate similar constructions in the context of Hilbert modules over $C^*$-algebras. For modules over $C(\T^n)$, the results shed light on work of Packer and Rieffel on projective multi-resolution analyses for specific Hilbert $C(\T^n)$-modules of functions on $\R^n$. There are also new applications to modules over $C(C)$ when $C$ is the infinite path space of a directed graph.
Direct integral of locally Hilbert spaces
本文提出局部希尔伯特空间的直积分概念，定义两类局部有界算子，证明可分解算子构成局部冯·诺依曼代数，可对角化算子构成阿贝尔冯·诺依曼代数，且互为换位子。
We provide a characterization for operator valued completely bounded linear maps on Hilbert $C^*$-modules in terms of $φ$-maps. Also, we show that for every operator valued completely positive map $φ$ on a $C^*$-algebra $\mathcal{A}$, there is a unique (up to multiplication by a unitary operator) non-degenerate $φ$-map on each Hilbert $\mathcal{A}$-module.
The aim of this article is to extend the results of Asadi M.B, B.V.R. Bhat, G. Ramesh, K. Sumesh about completely positive maps on Hilbert C*-modules. We prove a Stinespring type theorem for a finite family of completely positive maps on Hilbert C*-modules. We also show that any two minimal Stinespring representations are unitarily equivalent.
In this paper, we show that every completely semi-$ϕ$-map on a submodule of a Hilbert $C^*$-module has a completely semi-$ϕ$-map extension on the whole of module. We also investigate the extendability of $ϕ$-maps and provide examples of $ϕ$-maps which has no $ϕ$-map extension. Finally, we introduce a category of Hilbert $C^*$-module and determine injective objects in this category.
It's well known that the functional Hilbert space over the unit ball in $B_{d} \in C^d$, with kernel function $K(z,w)=\frac{1}{1-z_{1}w_{1}-... -z_{d}w_{d}}$, admits a natural $A(B_{d})$-module structure. We show the rank of a nonzero submodule is infinity if and only if the submodule is of infinite codimension. Together with Arveson's dilation theory, our result shows that Hilbert modules stand in stark contrast with Hilbert basis theorem for algebraic modules. This result answers a question of Arveson.
We characterize the generators of dynamical systems on Hilbert modules as those generators of one-parameter groups of Banach space isometries which are ternary derivations. We investigate in how far a similar condition can be expressed in terms of generalized derivations.
Direct Integral and Decompoisitions of Locally Hilbert spaces
本文引入局部Hilbert空间的直积分概念，定义可分解与可对角化的局部有界算子，证明可对角化算子类构成交换的局部von Neumann代数，并对一类交换的局部von Neumann代数肯定地回答了相应的分解逆问题。
We prove that an arbitrary (not necessarily countably generated) Hilbert $G$-$\cla$ module on a G-C^* algebra $\cla$ admits an equivariant embedding into a trivial $G-\cla$ module, provided G is a compact Lie group and its action on $\cla$ is ergodic.
Let F be a right Hilbert C*-module over a C*-algebra B, and suppose that F is equipped with a left action, by compact operators, of a second C*-algebra A. Tensor product with F gives a functor from Hilbert C*-modules over A to Hilbert C*-modules over B. We prove that under certain conditions (which are always satisfied if, for instance, A is nuclear), the image of this functor can be described in terms of coactions of a certain coalgebra canonically associated to F. We then discuss several examples that fit into this framework: parabolic induction of tempered group representations; Hermitian c…
We reflect on the notions of positivity and square roots. We review many examples which underline our thesis that square roots of positive maps related to *-algebras are Hilbert modules. As a result of our considerations we discuss requirements a notion of positivity on a *-algebra should fulfill and derive some basic consequences.
On Hilbert C*-modules with Hilbert dual and C*-Fredholm operators
本文研究具有Hilbert对偶结构的Hilbert C*-模，证明了单调完备C*-代数上自对偶模的同构条件，并据此得出A-Fredholm算子指标可按核与余核之差计算。
Direct integral of locally Hilbert spaces over a locally measure space
本文引入局部测度空间上局部希尔伯特空间的直接积分，证明其构成局部希尔伯特空间，并研究可分解与可对角化局部有界算子，得出可对角化算子代数等于可分解算子代数的交换子。
We show that, when $A$ is a separable C*-algebra, every countably generated Hilbert $A$-module is projective (with bounded module maps as morphisms). We also study the approximate extensions of bounded module maps. In the case that $A$ is a $σ$-unital simple C*-algebra with strict comparison and every strictly positive lower semicontinuous affine function on quasitraces can be realized as the rank of an element in Cuntz semigroup, we show that the Cuntz semigroup is the same as unitarily equivalent class of countably generated Hilbert $A$-modules if and only if $A$ has stable rank one.
We derive Paschke's GNS construction for completely positive maps on unital pro-C*-algebras from the KSGNS construction, presented by M. Joita [J. London Math. Soc. {\bf 66} (2002), 421--432], and then we deduce an analogue of Stinespring theorem for Hilbert modules over pro-C*-algebras. Also, we obtain a Radon-Nikodym type theorem for operator valued completely positive maps on Hilbert modules over pro-C*-algebras.
In this note, a general version of Bessel multipliers in Hilbert $C^*$-modules is presented and then, many results obtained for multipliers are extended. Also the conditions for invertibility of generalized multipliers are investigated in details. The invertibility of multipliers is very important because it helps us to obtain more reconstruction formula.
We strengthen Mohammad B. Asadi's analogue of Stinespring's theorem for certain maps on Hilbert C*-modules. We also show that any two minimal Stinespring representations are unitarily equivalent. We illustrate the main theorem with an example.
Apart from presenting some new insights and results, one of our main purposes is to put some records in the development of von Neumann modules straight. The von Neumann or $W^*$-objects among the Hilbert ($C^*$-)modules are around since the first papers by Paschke (1973) and Rieffel (1974) that lift Kaplansky's setting (1953) to modules over noncommutative $C^*$-algebras. While the formal definition of $W^*$-modules} is due to Baillet, Denizeau, and Havet (1988), the one of von Neumann modules as strongly closed operator spaces started with Skeide (2000). It has been paired with the definition…
In this paper, we investigate the structure of the multiplier module of a Hilbert module over a locally C*-algebra and the relationship between the set of all adjointable operators from a Hilbert A-module E to a Hilbert A-module F and the set of all adjointable operators from the multiplier module M(E)of E to the multiplier module M(F) of F.
The Morita equivalence of m-regular involutive quantales in the context of the theory of Hilbert $A$-modules is presented. The corresponding fundamental representation theorems are shown. We also prove that two commutative m-regular involutive quantales are Morita equivalent if and only if they are isomorphic
Kernels of Hilbert Module Maps: A Counterexample
本文构造 Hilbert 模上非零有界右线性映射的反例，证明其核的正交补可为平凡，且核不等于双重正交补，从而解答长期悬而未决的问题。
On Essential and Topologically Essential Submodules of Hilbert C*-Modules
本文研究了希尔伯特C*-模闭子模的两种大规模性概念：本质性和拓扑本质性，证明了闭右理想中这两种概念等价，并扩展至希尔伯特C*-模的闭子模。
Co-actions, Isometries and isomorphism classes of Hilbert modules
本文证明了Hilbert A-模的A-线性映射由酉算子诱导的条件，并基于乘法酉算子理论计算了C*-代数量子群上Hilbert模的等价类，发展了可用于证明co-actions不存在性的理论。
A Radon-Nikodym theorem for CP-maps on Hilbert pro-C*-modules
该研究针对pro-C*-代数上Hilbert模间的完全正映射，引入等价关系并分析其Stinespring构造，定义预序关系，最终得到一类Radon-Nikodym型定理，拓展了相关算子代数理论的研究范畴。
Trace-class operators on Hilbert modules and Haagerup tensor products
本文证明交换C*-代数上Hilbert模的迹类算子空间与其算子论伴随的Haagerup张量积完全等距同构，推广了Hilbert空间的经典性质，并给出迹类算子两种定义等价性的新证明。
We show that the gluing construction for Hilbert modules introduced by Raeburn in his computation of the Picard group of a continuous-trace C*-algebra (Trans. Amer. Math. Soc., 1981) can be applied to arbitrary C*-algebras, via an algebraic argument with the Haagerup tensor product. We put this result into the context of descent theory by identifying categories of gluing data for Hilbert modules over C*-algebras with categories of comodules over C*-coalgebras, giving a Hilbert-module version of a standard construction from algebraic geometry. As a consequence we show that if two C*-algebras ha…
It is shown that if A is a stably finite C*-algebra and E is a countably generated Hilbert A-module, then E gives rise to a compact element of the Cuntz semigroup if and only if E is algebraically finitely generated and projective. It follows that if E and F are equivalent in the sense of Coward, Elliott and Ivanescu (CEI) and E is algebraically finitely generated and projective, then E and F are isomorphic. In contrast to this, we exhibit two CEI-equivalent Hilbert modules over a stably finite C*-algebra that are not isomorphic.
We study the stability of Fredholm property for regular operators on Hilbert $C^*$-modules under some certain perturbations. We treat this problem when perturbing operators are (relatively) bounded or relatively compact. We also consider the perturbations of regular Fredholm operators in terms of the gap metric. In particular, we prove that the space of all regular Fredholm operators on a Hilbert $C^*$-module $E$ is open in the space of all regular operators on $E$ with respect to the gap metric. As an application, we construct some continuous paths of selfadjoint regular Fredholm operators wi…
A notion of curvature is introduced in multivariable operator theory and an analogue of the Gauss-Bonnet-Chern theorem is established for graded (contractive) Hilbert modules over the complex polynomial algebra in d variables, d=1,2,3,.... The curvature invariant, Euler characteristic, and degree are computed for some explicit examples based on varieties in (multidimensional) complex projective space, and applications are given to the structure of graded ideals in C[z_1,...,z_d] and to the existence of "inner sequences" for closed submodules of the free Hilbert module H^2(C^d).
In this paper, firstly we investigate conditions under which the action of an operator on a $K$-frame, remain again a $K$-frame for Hilbert module E. We also give a generalization of Douglas Theorem and we shall use it to prove the sum of two $K$-frame under certain condition is again a $K$-frame. Finally, we characterize the $K$-frame generators for a unitary system in terms of operators.
Functional Models of Abelian Locally von Neumann Algebras and Direct Integrals of Locally Hilbert Spaces
本文研究了可积局部 von Neumann 代数的功能模型，通过局部本质有界函数构建，探讨了局部 Hilbert 空间直接积分及局部可分解与可对角化的算子，证明其构成互为对易的局部 von Neumann 代数。
The two reference lists contain 54/22 references of papers and preprints concerned with the theory and/or various applications of Hilbert modules over Hilbert $*$-algebras and over (non-self-adjoint) operator algebras. They are far from being complete, but they give additional information about two research fields which are closely related to the theory of Hilbert C*-modules, i.e. they are complements to the reference guide about this circle of problems. Any additions, corrections and forthcoming information are welcome.
Continuing the research on the Banach-Saks and Schur properties started in (cf. M. Frank, A. A. Pavlov, Banach-Saks properties of C*-algebras and Hilbert C*-modules (submitted)) we investigate analogous properties in the module context. As an environment serves the class of Hilbert C*-modules. Some properties of weak module topologies on Hilbert C*-modules are described. Natural module analogues of the classical weak Banach-Saks and the classical Schur properties are defined and studied. A number of useful characterizations of properties of Hilbert C*-modules is obtained. In particular, some i…
We introduce the concept of frame of multipliers in Hilbert modules over pro-C*-algebras and show that many properties of frames in Hilbert C*-modules are valid for frames of multipliers in Hilbert modules over pro-C*-algebras.
In this paper, we introduce controlled frames in Hilbert $C^*$-modules and we show that they share many useful properties with their corresponding notions in Hilbert space. Next, we give a characterization of controlled frames in Hilbert $C^*$-module. Also multiplier operators for controlled frames in Hilbert $C^*$-modules will be defined and some of its properties will be shown. Finally, we investigate weighted frames in Hilbert $C^*$-modules and verify their relations to controlled frames and multiplier operators.
Woronowicz introduced the functional calculus for normal operators in Hilbert C*-modules. The aim of this paper is to translate, if possible, some basic properties of the functional calculus in Hilbert spaces to the Hilbert C*-module framework.
Frame theory has a great revolution in recent years. This new Theory have been extended from Hilbert spaces to Hilbert C*-modules. In this paper, we introduce the notion of dual *-K-g-frames in Hilbert A-modules. Lastly we study *-K-g-frames in tensor product of Hilbert C*-Modules and we establish some new results.
We show that every infinite-dimensional commutative unital C*-algebra has a Hilbert C*-module admitting no frames. In particular, this shows that Kasparov's stabilization theorem for countably generated Hilbert C*-modules can not be extended to arbitrary Hilbert C*-modules.
We construct an example of a Hilbert C*-module which shows that Troitsky's theorem on the geometrical essence of A-compact operators between Hilbert C*-modules is not extendable to a not countably generated module case (even in the case of a stronger uniform structure, which is also introduced). In addition, the constructed module admits no frames.
Frame Theory has a great revolution in recent years, this Theory have been extended from Hilbert spaces to Hilbert $C^{\ast}$-modules. The purpose of this paper is the introduction and the study of the concept of Controlled Continuous g-Frames in Hilbert $C^{\ast}$-Modules. Also we give some properties.
Motivated from the theory of Hilbert-Schmidt morphisms between Hilbert C*-modules over commutative C*-algebras by Stern and van Suijlekom \textit{[J. Funct. Anal., 2021]}, we introduce the notion of p-absolutely summing morphisms between Hilbert C*-modules over commutative C*-algebras. We show that an adjointable morphism between Hilbert C*-modules over monotone closed commutative C*-algebra is 2-absolutely summing if and only if it is Hilbert-Schmidt. We formulate version of Pietsch factorization problem for p-absolutely summing morphisms and solve partially
The aim of this paper is to present a unified framework in the setting of Hilbert $C^*$-modules for the scalar- and vector-valued reproducing kernel Hilbert spaces and $C^*$-valued reproducing kernel spaces. We investigate conditionally negative definite kernels with values in the $C^*$-algebra of adjointable operators acting on a Hilbert $C^*$-module. In addition, we show that there exists a two-sided connection between positive definite kernels and reproducing kernel Hilbert $C^*$-modules. Furthermore, we explore some conditions under which a function is in the reproducing kernel module and …
Locally adjointable operators on Hilbert $C^*$-modules
本文研究 Hilbert $C^*$-模间的局部可伴随算子，并在重要情形下给出其显式描述。
In this paper we introduce a notion of Morita equivalence for Hilbert C*-modules in terms of the Morita equivalence of the algebras of compact operators on Hilbert C*-modules. We investigate some properties of the new version of Morita equivalence and obtain some results. We then applied our results to study the continuous actions of locally compact groups on full Hilbert C*-modules. We present an extension of Green's theorem in the context of Hilbert C*-modules as well.
We prove a covariant version of the Stinespring theorem for Hilbert C*-modules.
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