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Journal articles on the topic 'Hilbert C*-module'

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Published: 4 June 2021

Last updated: 1 February 2022

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1

Frank, Michael. "Characterizing C*-algebras of compact operators by generic categorical properties of Hilbert C*-modules." Journal of K-theory 2, no. 3 (March 4, 2008): 453–62. http://dx.doi.org/10.1017/is008001031jkt035.

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AbstractC*-algebras A of compact operators are characterized as those C*-algebras of coefficients of Hilbert C*-modules for which (i) every bounded A-linear operator between two Hilbert A-modules possesses an adjoint operator, (ii) the kernels of all bounded A-linear operators between Hilbert A-modules are orthogonal summands, (iii) the images of all bounded A-linear operators with closed range between Hilbert A-modules are orthogonal summands, and (iv) for every Hilbert A-module every Hilbert A-submodule is a topological summand. Thus, the theory of Hilbert C*-modules over C*-algebras of compact operators has similarities with the theory of Hilbert spaces. In passing, we obtain a general closed graph theorem for bounded module operators on arbitrary Hilbert C*-modules.

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2

Murphy, Gerard J. "Positive definite kernels and Hilbert C*-modules." Proceedings of the Edinburgh Mathematical Society 40, no. 2 (June 1997): 367–74. http://dx.doi.org/10.1017/s0013091500023804.

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A theory of positive definite kernels in the context of Hilbert C*-modules is presented. Applications are given, including a representation of a Hilbert C*-module as a concrete space of operators and a construction of the exterior tensor product of two Hilbert C*-modules.

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3

Rashidi-Kouchi, M., and A. Rahimi. "On controlled frames in Hilbert C∗-modules." International Journal of Wavelets, Multiresolution and Information Processing 15, no. 04 (May 7, 2017): 1750038. http://dx.doi.org/10.1142/s0219691317500382.

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In this paper, we introduce controlled frames in Hilbert [Formula: see text]-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 [Formula: see text]-module. Also multiplier operators for controlled frames in Hilbert [Formula: see text]-modules will be defined and some of its properties will be shown. Finally, we investigate weighted frames in Hilbert [Formula: see text]-modules and verify their relations to controlled frames and multiplier operators.

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4

Frank, Michael, and Alexander A. Pavlov. "Module weak Banach-Saks and module Schur properties of Hilbert C*-modules." Journal of Operator Theory 70, no. 1 (July 24, 2103): 53–73. http://dx.doi.org/10.7900/jot.2011apr21.1933.

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5

HAMANA, MASAMICHI. "MODULES OVER MONOTONE COMPLETE C*-ALGEBRAS." International Journal of Mathematics 03, no. 02 (April 1992): 185–204. http://dx.doi.org/10.1142/s0129167x92000059.

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The main result asserts that given two monotone complete C*-algebras A and B, B is faithfully represented as a monotone closed C*-subalgebra of the monotone complete C*-algebra End A(X) consisting of all bounded module endomorphisms of some self-dual Hilbert A-module X if and only if there are sufficiently many normal completely positive maps of B into A. The key to the proof is the fact that each pre-Hilbert A-module can be completed uniquely to a self-dual Hilbert A-module.

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6

ASADI, M. B., M. FRANK, and Z. HASSANPOUR-YAKHDANI. "FRAME-LESS HILBERT C*-MODULES." Glasgow Mathematical Journal 61, no. 1 (February 7, 2018): 25–31. http://dx.doi.org/10.1017/s0017089518000010.

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AbstractWe show that if A is a compact C*-algebra without identity that has a faithful *-representation in the C*-algebra of all compact operators on a separable Hilbert space and its multiplier algebra admits a minimal central projection p such that pA is infinite-dimensional, then there exists a Hilbert A1-module admitting no frames, where A1 is the unitization of A. In particular, there exists a frame-less Hilbert C*-module over the C*-algebra $K(\ell^2) \dotplus \mathbb{C}I_{\ell^2}$.

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7

Jiang, Runliang. "The irreducibility of C*-algebras acting on Hilbert C*-modules." Filomat 30, no. 9 (2016): 2425–33. http://dx.doi.org/10.2298/fil1609425j.

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Let B be a C*-algebra, E be a Hilbert B module and L(E) be the set of adjointable operators on E. Let A be a non-zero C*-subalgebra of L(E). In this paper, some new kinds of irreducibilities of A acting on E are introduced, which are all the generalizations of those associated to Hilbert spaces. The difference between these irreducibilities are illustrated by a number of counterexamples. It is concluded that for a full Hilbert B-module, these irreducibilities are all equivalent if and only if the underlying C*-algebra B is isomorphic to the C*-algebra of all compact operators on a Hilbert space.

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8

Frank, Michael, Alexander S. Mishchenko, and Alexander A. Pavlov. "Orthogonality-preserving, C⁎-conformal and conformal module mappings on Hilbert C⁎-modules." Journal of Functional Analysis 260, no. 2 (January 2011): 327–39. http://dx.doi.org/10.1016/j.jfa.2010.10.009.

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9

Rashidi-Kouchi, Mehdi, Akbar Nazari, and Massoud Amini. "On stability of g-frames and g-Riesz bases in Hilbert C*-modules." International Journal of Wavelets, Multiresolution and Information Processing 12, no. 06 (November 2014): 1450036. http://dx.doi.org/10.1142/s0219691314500362.

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In this paper, we develop the basic stability theory for g-frames and g-Riesz bases in Hilbert C*-module. Also, we study stability of the dual g-frames in Hilbert C*-module. We extend the Casazza–Christensen perturbation theorem to g-frames in Hilbert C*-module but this is not valid for g-Riesz bases. We prove some characterizations of g-frames and g-Riesz bases in Hilbert C*-module.

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10

Li, Hanfeng. "A Hilbert C*-module admitting no frames." Bulletin of the London Mathematical Society 42, no. 3 (February 3, 2010): 388–94. http://dx.doi.org/10.1112/blms/bdp109.

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11

Irmatov, Anwar A., and Alexandr S. Mishchenko. "On Compact and Fredholm Operators over C*-algebras and a New Topology in the Space of Compact Operators." Journal of K-Theory 2, no. 2 (April 17, 2008): 329–51. http://dx.doi.org/10.1017/is008004001jkt034.

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AbstractIt is well-known that bounded operators in Hilbert C*-modules over C*-algebras may not be adjointable and the same is true for compact operators. So, there are two analogs for classical compact operators in Hilbert C*-modules: adjointable compact operators and all compact operators, i.e. those not necessarily having an adjoint.Classical Fredholm operators are those that are invertible modulo compact operators. When the notion of a Fredholm operator in a Hilbert C*-module was developed in [6], the first analog was used: Fredholm operators were defined as operators that are invertible modulo adjointable compact operators.In this paper we use the second analog and develop a more general version of Fredholm operators over C*-algebras. Such operators are defined as bounded operators that are invertible modulo the ideal of all compact operators. The main property of this new class is that a Fredholm operator still has a decomposition into a direct sum of an isomorphism and a finitely generated operator.The special case of Fredholm operators (in the sense of [6]) over the commutative C*-algebra C(K) of continuous functions on a compact topological space K was also considered in [2]. In order to describe general Fredholm operators (invertible modulo all compact operators over C(K)) we construct a new IM-topology on the space of compact operators on a Hilbert space such that continuous families of compact operators generate the ideal of all compact operators over C(K).

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12

KHOSRAVI, AMIR, and BEHROOZ KHOSRAVI. "g-FRAMES AND MODULAR RIESZ BASES IN HILBERT C*-MODULES." International Journal of Wavelets, Multiresolution and Information Processing 10, no. 02 (March 2012): 1250013. http://dx.doi.org/10.1142/s0219691312500130.

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In this paper we introduce modular Riesz basis, modular g-Riesz basis in Hilbert C*-modules in a very natural way and we show that they share many properties with Riesz basis and g-Riesz basis in Hilbert spaces. We also found that by using the fact that every finitely or countably generated Hilbert C*-module over a unital C*-algebra has a standard Parseval frame, we characterize g-frames, modular Riesz bases and modular g-Riesz bases. Finally we obtain a perturbation result for modular g-Riesz bases.

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13

LEUNG, CHI-WAI, CHI-KEUNG NG, and NGAI-CHING WONG. "LINEAR ORTHOGONALITY PRESERVERS OF HILBERT BUNDLES." Journal of the Australian Mathematical Society 89, no. 2 (October 2010): 245–54. http://dx.doi.org/10.1017/s1446788710001515.

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AbstractA ℂ-linear map θ (not necessarily bounded) between two Hilbert C*-modules is said to be ‘orthogonality preserving’ if 〈θ(x),θ(y)〉=0 whenever 〈x,y〉=0. We prove that if θ is an orthogonality preserving map from a full Hilbert C0(Ω)-module E into another Hilbert C0(Ω) -module F that satisfies a weaker notion of C0 (Ω) -linearity (called ‘localness’), then θ is bounded and there exists ϕ∈Cb (Ω)+ such that 〈θ(x),θ(y)〉=ϕ⋅〈x,y〉 for all x,y∈E.

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14

Bui, Huu Hung. "Full coactions on Hilbert C*-modules." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 59, no. 3 (December 1995): 409–20. http://dx.doi.org/10.1017/s1446788700037307.

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AbstractWe introduce a natural notion of full coactions of a locally compact group on a Hilbert C*-module, and associate each full coaction in a natural way to an ordinary coaction. We also introduce a natural notion of strong Morita equivalence of full coactions which is sufficient to ensure strong Morita equivalence of the corresponding crossed product C*-algebras.

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15

Alijani, A. "Generalized frames with C*-valued bounds and their operator duals." Filomat 29, no. 7 (2015): 1469–79. http://dx.doi.org/10.2298/fil1507469a.

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Certain facts about frames and generalized frames are extended for the new g-frames, referred as *-g-frames, in a Hilbert C*-modules. As a matter of fact, some relations are establish between *-frames and *-g-frames in a Hilbert C*-module. Furthermore, the paper studies the operators associated to a given *-g-frame, the construction of new *-g-frames. Moreover, the operator duals for a *-g-frame are introduced and their properties are investigated. Finally, operator duals of a *-g-frame are characterized.

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16

Clare, Pierre, Tyrone Crisp, and Nigel Higson. "ADJOINT FUNCTORS BETWEEN CATEGORIES OF HILBERT -MODULES." Journal of the Institute of Mathematics of Jussieu 17, no. 2 (June 30, 2016): 453–88. http://dx.doi.org/10.1017/s1474748016000074.

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Let$E$be a (right) Hilbert module over a$C^{\ast }$-algebra$A$. If$E$is equipped with a left action of a second$C^{\ast }$-algebra$B$, then tensor product with$E$gives rise to a functor from the category of Hilbert$B$-modules to the category of Hilbert$A$-modules. The purpose of this paper is to study adjunctions between functors of this sort. We shall introduce a new kind of adjunction relation, called a local adjunction, that is weaker than the standard concept from category theory. We shall give several examples, the most important of which is the functor of parabolic induction in the tempered representation theory of real reductive groups. Each local adjunction gives rise to an ordinary adjunction of functors between categories of Hilbert space representations. In this way we shall show that the parabolic induction functor has a simultaneous left and right adjoint, namely the parabolic restriction functor constructed in Clareet al.[Parabolic induction and restriction via$C^{\ast }$-algebras and Hilbert$C^{\ast }$-modules,Compos. Math.FirstView(2016), 1–33, 2].

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17

Hashemi Sababe, Saeed, Ali Ebadian, and Shahram Najafzadeh. "On reproducing property and 2-cocycles." Tamkang Journal of Mathematics 49, no. 2 (June 30, 2018): 143–53. http://dx.doi.org/10.5556/j.tkjm.49.2018.2553.

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In this paper, we study reproducing kernels whose ranges are subsets of a $C^*$-algebra or a Hilbert $C^*$-module. In particular, we show how such a reproducing kernel can naturally be expressed in terms of operators on a Hilbert $C^*$-module. We focus on relative reproducing kernels and extend this concept to such spaces associated with cocycles.

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18

Labrigui, Hatim, Abdeslam Touri, Mohamed Rossafi, and Samir Kabbaj. "Controlled ∗ -Operator Frames on Hilbert C ∗ -Modules." Journal of Mathematics 2021 (April 8, 2021): 1–8. http://dx.doi.org/10.1155/2021/5530498.

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In this paper, we study the concept of controlled ∗ -operator frames for the space of all adjointable operators on a Hilbert C ∗ -module H. Also, we discuss characterizations of controlled ∗ -operator frames and we give some properties. Some illustrative examples are provided to advocate the usability of our results.

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19

Kabbaj, Samir, Abdellatif Chahbi, Ahmed Charifi, and Nourdine Bounader. "The generalized of Selberg’s inequalities in C*-module." Filomat 28, no. 8 (2014): 1585–92. http://dx.doi.org/10.2298/fil1408585k.

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We obtain a Generalized of Selberg?s type inequalities in Hilbert spaces and their extensions in operators algebras, in C*-modules and in algebras of adjointable A-linear maps. Some applications for improving the Bessel type inequality result are given.

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20

Kolarec, Biserka. "Inequalities for the C^✻-valued norm on a Hilbert C^✻-module." Mathematical Inequalities & Applications, no. 4 (2009): 745–51. http://dx.doi.org/10.7153/mia-12-57.

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21

Keckic, Dragoljub, and Zlatko Lazovic. "Measures of noncompactness on the standard hilbert C*-module." Filomat 33, no. 12 (2019): 3683–95. http://dx.doi.org/10.2298/fil1912683k.

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We define a measure of noncompactness ? on the standard Hilbert C*-module l2(A) over a unital C*-algebra, such that ?(E) = 0 if and only if E is A-precompact (i.e. it is ?-close to a finitely generated projective submodule for any ? > 0) and derive its properties. Further, we consider the known, Kuratowski, Hausdorff and Istr??escu measure of noncompactnes on l2(A) regarded as a locally convex space with respect to a suitable topology, and obtain their properties as well as some relationship between them and introduced measure of noncompactness ?.

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22

Asadi, Mohammad B., and A. Khosravi. "A Hilbert $C^*$-module not anti-isomorphic to itself." Proceedings of the American Mathematical Society 135, no. 01 (August 2, 2006): 263–67. http://dx.doi.org/10.1090/s0002-9939-06-08474-7.

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23

Wei, Changguo, Chenchen Liu, and Shudong Liu. "Note on compact operators on a Hilbert C⁎-module." Journal of Mathematical Analysis and Applications 490, no. 1 (October 2020): 124137. http://dx.doi.org/10.1016/j.jmaa.2020.124137.

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24

Moslehian, Mohammad Sal, and Ali Zamani. "Mappings preserving approximate orthogonality in Hilbert $C^*$-modules." MATHEMATICA SCANDINAVICA 122, no. 2 (April 8, 2018): 257. http://dx.doi.org/10.7146/math.scand.a-102945.

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We introduce a notion of approximate orthogonality preserving mappings between Hilbert $C^*$-modules. We define the concept of $(\delta , \varepsilon )$-orthogonality preserving mapping and give some sufficient conditions for a linear mapping to be $(\delta , \varepsilon )$-orthogonality preserving. In particular, if $\mathscr {E}$ is a full Hilbert $\mathscr {A}$-module with $\mathbb {K}(\mathscr {H})\subseteq \mathscr {A} \subseteq \mathbb {B}(\mathscr {H})$ and $T, S\colon \mathscr {E}\to \mathscr {E}$ are two linear mappings satisfying $|\langle Sx, Sy\rangle | = \|S\|^2|\langle x, y\rangle |$ for all $x, y\in \mathscr {E}$ and $\|T - S\| \leq \theta \|S\|$, then we show that $T$ is a $(\delta , \varepsilon )$-orthogonality preserving mapping. We also prove whenever $\mathbb {K}(\mathscr {H})\subseteq \mathscr {A} \subseteq \mathbb {B}(\mathscr {H})$ and $T\colon \mathscr {E} \to \mathscr {F}$ is a nonzero $\mathscr {A}$-linear $(\delta , \varepsilon )$-orthogonality preserving mapping between $\mathscr {A}$-modules, then \[ \bigl \|\langle Tx, Ty\rangle - \|T\|^2\langle x, y\rangle \bigr \|\leq \frac {4(\varepsilon - \delta )}{(1 - \delta )(1 + \varepsilon )} \|Tx\|\|Ty\|\qquad (x, y\in \mathscr {E}). \] As a result, we present some characterizations of the orthogonality preserving mappings.

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25

ANDRUCHOW, ESTEBAN, GUSTAVO CORACH, and DEMETRIO STOJANOFF. "PROJECTIVE SPACE OF A C*-MODULE." Infinite Dimensional Analysis, Quantum Probability and Related Topics 04, no. 03 (September 2001): 289–307. http://dx.doi.org/10.1142/s0219025701000516.

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Let X be a right Hilbert C*-module over A. We study the geometry and the topology of the projective space [Formula: see text] of X, consisting of the orthocomplemented submodules of X which are generated by a single element. We also study the geometry of the p-sphere Sp(X) and the natural fibration [Formula: see text], where Sp(X) = {x ∈ X: <x, x> = p}, for p ∈ A a projection. The projective space and the p-sphere are shown to be homogeneous differentiable spaces of the unitary group of the algebra ℒA(X) of adjointable operators of X. The homotopy theory of these spaces is examined.

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26

Ghahramani, Hoger, and Saman Sattari. "The first cohomology group of some operator algebras on Hilbert C*-modules." Studia Scientiarum Mathematicarum Hungarica 57, no. 1 (March 2020): 54–67. http://dx.doi.org/10.1556/012.2020.57.1.1455.

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Abstract Let X be a Hilbert C*-module over a C*-algebra B. In this paper we introduce two classes of operator algebras on the Hilbert C*-module X called operator algebras with property and operator algebras with property ℤ, and we study the first (continuous) cohomology group of them with coefficients in various Banach bimodules under several conditions on B and X. Some of our results generalize the previous results. Also we investigate some properties of these classes of operator algebras.

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27

Heo, Jaeseong. "Stochastic Processes and Spectral Analysis for Hilbert $$C^*$$ C ∗ -Module-Valued Maps." Bulletin of the Malaysian Mathematical Sciences Society 41, no. 1 (November 3, 2015): 191–206. http://dx.doi.org/10.1007/s40840-015-0270-6.

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28

Mahmoudieh, Mohammad, Gholamreza Tabadkan, and Aliakbar Arefijamaal. "Sum of K-frames in Hilbert C*-modules." Filomat 34, no. 6 (2020): 1771–80. http://dx.doi.org/10.2298/fil2006771m.

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In this paper, we investigate some 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 to prove that the sum of two K-frames under certain condition is again a K-frame. Finally, we characterize the K-frame generators in terms of operators.

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29

Arambasic, Ljiljana. "On approximately dual frames for Hilbert C*-modules." Filomat 33, no. 12 (2019): 3869–75. http://dx.doi.org/10.2298/fil1912869a.

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In this paper we discuss approximately dual frames for a frame or an outer frame of a Hilbert C*-module. We show that every frame or an outer frame, up to a scalar multiple, is approximately dual to itself. This enables us to get a canonical dual frame of a given frame (xn)n as a limit of approximately dual frames defined by (xn)n.

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30

Arveson, W. "The curvature of a Hilbert module over C[z1, ... , zd]." Proceedings of the National Academy of Sciences 96, no. 20 (September 28, 1999): 11096–99. http://dx.doi.org/10.1073/pnas.96.20.11096.

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31

Fang, Xiaochun, and Jing Yu. "Compatibility and Schur complements of operators on Hilbert C*-module." Chinese Annals of Mathematics, Series B 32, no. 1 (December 28, 2010): 69–88. http://dx.doi.org/10.1007/s11401-010-0623-2.

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32

Eshkabilov, Yu Kh, and R. Kucharov. "Partial integral operators of Fredholm type on Kaplansky-Hilbert module over $L_0$." Владикавказский математический журнал, no. 3 (September 23, 2021): 80–90. http://dx.doi.org/10.46698/w5172-0182-0041-c.

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The article studies some characteristic properties of self-adjoint partially integral operators of Fredholm type in the Kaplansky-Hilbert module $L_{0}\left[L_{2}\left(\Omega_{1}\right)\right]$ over $L_{0}\left(\Omega_{2}\right)$. Some mathematical tools from the theory of Kaplansky-Hilbert module are used. In the Kaplansky-Hilbert module $L_{0}\left[L_{2}\left(\Omega_{1}\right)\right]$ over $ L_{0} \left (\Omega _ {2} \right)$ we consider the partially integral operator of Fredholm type $T_{1}$ ($ \Omega_{1} $ and $\Omega_{2} $ are closed bounded sets in $ {\mathbb R}^{\nu_{1}}$ and $ {\mathbb R}^{\nu_{2}},$ $\nu_{1}, \nu_{2} \in {\mathbb N} $, respectively). The existence of $ L_{0} \left (\Omega _ {2} \right) $ nonzero eigenvalues for any self-adjoint partially integral operator $T_{1}$ is proved; moreover, it is shown that $T_{1}$ has finite and countable number of real $L_{0}(\Omega_{2})$-eigenvalues. In the latter case, the sequence $ L_{0}(\Omega_{2})$-eigenvalues is order convergent to the zero function. It is also established that the operator $T_{1}$ admits an expansion into a series of $\nabla_{1}$-one-dimensional operators.

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33

Pavlov, Alexander, Ulrich Pennig, and Thomas Schick. "Quasi-multipliers of Hilbert and Banach $C^*$-bimodules." MATHEMATICA SCANDINAVICA 109, no. 1 (September 1, 2011): 71. http://dx.doi.org/10.7146/math.scand.a-15178.

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Quasi-multipliers for a Hilbert $C^*$-bimodule $V$ were introduced by L. G. Brown, J. A. Mingo, and N.-T. Shen [3] 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.

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34

Vujosevic, Biljana. "The index of product systems of Hilbert modules: Two equivalent definitions." Publications de l'Institut Math?matique (Belgrade) 97, no. 111 (2015): 49–56. http://dx.doi.org/10.2298/pim141114001v.

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We prove that a conditionally completely positive definite kernel, as the generator of completely positive definite (CPD) semigroup associated with a continuous set of units for a product system over a C*-algebra B, allows a construction of a Hilbert B?B module. That construction is used to define the index of the initial product system. It is proved that such definition is equivalent to the one previously given by Keckic and Vujosevic [On the index of product systems of Hilbert modules, Filomat, to appear, ArXiv:1111.1935v1 [math.OA] 8 Nov 2011]. Also, it is pointed out that the new definition of the index corresponds to the one given earlier by Arveson (in the case B = C).

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35

Kucerovsky, D. "Functional calculus and representations of C0(C) on a Hilbert module." Quarterly Journal of Mathematics 53, no. 4 (December 1, 2002): 467–77. http://dx.doi.org/10.1093/qjmath/53.4.467.

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36

Kolarec, Biserka. "Morphisms out of a split extension of a Hilbert C*-module." Glasnik Matematicki 41, no. 2 (December 15, 2006): 309–15. http://dx.doi.org/10.3336/gm.41.2.13.

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37

Wang, Ruofei, Changguo Wei, and Shudong Liu. "On the ideal of compact operators on a Hilbert C⁎-module." Journal of Mathematical Analysis and Applications 474, no. 1 (June 2019): 441–51. http://dx.doi.org/10.1016/j.jmaa.2019.01.053.

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38

Chavan, Sameer. "Irreducible Tuples Without the Boundary Property." Canadian Mathematical Bulletin 58, no. 1 (March 1, 2015): 9–18. http://dx.doi.org/10.4153/cmb-2014-051-0.

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AbstractWe examine spectral behavior of irreducible tuples that do not admit the boundary property. In particular, we prove under some mild assumption that the spectral radius of such an m-tuple (T1,...Tm)must be the operator norm of . We use this simple observation to ensure the boundary property for an irreducible, essentially normal, joint q-isometry, provided it is not a joint isometry. We further exhibit a family of reproducing Hilbert C[z1, ...zm]-modules (of which the Drury–Arveson Hilbert module is a prototype) with the property that any two nested unitarily equivalent submodules are indeed equal.

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39

Ivkovic, Stefan. "On upper triangular operator 2 x 2 matrices over C*-algebras." Filomat 34, no. 3 (2020): 691–706. http://dx.doi.org/10.2298/fil2003691i.

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We study adjointable, bounded operators on the direct sum of two copies of the standard Hilbert C*-module over a unital C*-algebra A that are given by upper triangular 2 by 2 operator matrices. Using the definition of A-Fredholm and semi-A-Fredholm operators given in [3], [4], we obtain conditions relating semi-A-Fredholmness of these operators and that of their diagonal entries, thus generalizing the results in [1], [2]. Moreover, we generalize the notion of the spectra of operators by replacing scalars by the elements in the C*-algebra A: Considering these new spectra in A of bounded, adjointable operators on Hilbert C*-modules over A related to the classes of A-Fredholm and semi-A-Fredholm operators, we prove an analogue or a generalized version of the results in [1] concerning the relationship between the spectra of 2 by 2 upper triangular operator matrices and the spectra of their diagonal entries.

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40

Xu, Qingxiang, and Guanjie Yan. "Harmonious projections and Halmos' two projections theorem for Hilbert C⁎-module operators." Linear Algebra and its Applications 601 (September 2020): 265–84. http://dx.doi.org/10.1016/j.laa.2020.05.012.

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41

Bui, Huu Hung. "A Hilbert $C^{*}$-module method for Morita equivalence of twisted crossed products." Proceedings of the American Mathematical Society 125, no. 7 (1997): 2109–13. http://dx.doi.org/10.1090/s0002-9939-97-03792-1.

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42

Heo, Jaeseong. "Hilbert {$C\sp *$}-module representation on Haagerup tensor products and group systems." Publications of the Research Institute for Mathematical Sciences 35, no. 5 (1999): 757–68. http://dx.doi.org/10.2977/prims/1195143422.

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43

Mehrazin, Marzieh, Maryam Amyari, and Mohsen Erfanian Omidvar. "A new type of numerical radius of operators on Hilbert $$C^*$$-module." Rendiconti del Circolo Matematico di Palermo Series 2 69, no. 1 (November 10, 2018): 29–37. http://dx.doi.org/10.1007/s12215-018-0385-3.

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44

TEIMOURI-AZADBAKHT, T., and A. G. GHAZANFARI. "Some Grüss Type Inequalities for Fréchet Differentiable Mappings." Kragujevac Journal of Mathematics 44, no. 4 (December 2020): 571–79. http://dx.doi.org/10.46793/kgjmat2004.571t.

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Let X be a Hilbert C∗-module on C∗-algebra A and p ∈ A. We denote by Dp(A,X) the set of all continuous functions f : A → X, which are Fréchet diﬀerentiable on a open neighborhood U of p. Then, we introduce some generalized semi-inner products on Dp(A,X), and using them some Grüss type inequalities in semi-inner product C∗-module Dp(A,X) and Dp(A,Xn) are established.

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45

Blecher, David P. "On Morita's fundamental theorem for $C^*$-algebras." MATHEMATICA SCANDINAVICA 88, no. 1 (March 1, 2001): 137. http://dx.doi.org/10.7146/math.scand.a-14319.

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We give a solution, via operator spaces, of an old problem in the Morita equivalence of $C^*$-algebras. Namely, we show that $C^*$-algebras are strongly Morita equivalent in the sense of Rieffel if and only if their categories of left operator modules are isomorphic via completely contractive functors. Moreover, any such functor is completely isometrically isomorphic to the Haagerup tensor product (= interior tensor product) with a strong Morita equivalence bimodule. An operator module over a $C^*$-algebra $\mathcal A$ is a closed subspace of some B(H) which is left invariant under multiplication by $\pi(\mathcal\ A)$, where $\pi$ is a*-representation of $\mathcal A$ on $H$. The category $_{\mathcal{AHMOD}}$ of *-representations of $\mathcal A$ on Hilbert space is a full subcategory of the category $_{\mathcal{AOMOD}}$ of operator modules. Our main result remains true with respect to subcategories of $OMOD$ which contain $HMOD$ and the $C^*$-algebra itself. It does not seem possible to remove the operator space framework; in the very simplest cases there may exist no bounded equivalence functors on categories with bounded module maps as morphisms (as opposed to completely bounded ones). Our proof involves operator space techniques, together with a $C^*$-algebra argument using compactness of the quasistate space of a $C^*$-algebra, and lowersemicontinuity in the enveloping von Neumann algebra.

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46

BHAT, B. V. RAJARAMA, and MICHAEL SKEIDE. "TENSOR PRODUCT SYSTEMS OF HILBERT MODULES AND DILATIONS OF COMPLETELY POSITIVE SEMIGROUPS." Infinite Dimensional Analysis, Quantum Probability and Related Topics 03, no. 04 (December 2000): 519–75. http://dx.doi.org/10.1142/s0219025700000261.

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In this paper we study the problem of dilating unital completely positive (CP) semigroups (quantum dynamical semigroups) to weak Markov flows and then to semigroups of endomorphisms (E0-semigroups) using the language of Hilbert modules. This is a very effective, representation free approach to dilation. In this way we are able to identify the right algebra (maximal in some sense) for endomorphisms to act. We are led inevitably to the notion of tensor product systems of Hilbert modules and units for them, generalizing Arveson's notions for Hilbert spaces. In the course of our investigations we are not only able to give new natural and transparent proofs of well-known facts for semigroups on [Formula: see text], but also extend the results immediately to much more general setups. For instance, Arveson classifies E0-semigroups on [Formula: see text] up to cocycle conjugacy by product systems of Hilbert spaces.5 We find that conservative CP-semigroups on arbitrary unital C*-algebras are classified up to cocycle conjugacy by product systems of Hilbert modules. Looking at other generalizations, it turns out that the role played by E0-semigroups on [Formula: see text] in dilation theory for CP-semigroups on [Formula: see text] is now played by E0-semigroups on [Formula: see text], the full algebra of adjointable operators on a Hilbert module E. We have CP-semigroup versions of many results proved by Paschke27 for CP maps.

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Zamani, Ali, and Mohammad Sal Moslehian. "Exact and Approximate Operator Parallelism." Canadian Mathematical Bulletin 58, no. 1 (March 1, 2015): 207–24. http://dx.doi.org/10.4153/cmb-2014-029-4.

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AbstractExtending the notion of parallelism we introduce the concept of approximate parallelism in normed spaces and then substantially restrict ourselves to the setting of Hilbert space operators endowed with the operator norm. We present several characterizations of the exact and approximate operator parallelism in the algebra B(ℋ) of bounded linear operators acting on a Hilbert space H . Among other things, we investigate the relationship between the approximate parallelism and norm of inner derivations on B(ℋ). We also characterize the parallel elements of a C*-algebra by using states. Finally we utilize the linking algebra to give some equivalent assertions regarding parallel elements in a Hilbert C*-module.

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ZHANG, SHUANG. "K-THEORY AND HOMOTOPY OF CERTAIN GROUPS AND INFINITE GRASSMANN SPACES ASSOCIATED WITH C*-ALGEBRAS." International Journal of Mathematics 05, no. 03 (June 1994): 425–45. http://dx.doi.org/10.1142/s0129167x94000243.

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We determine, in terms of [Formula: see text] and [Formula: see text], the homotopy groups of certain groups of invertibles and of certain equivalence classes in the infinite Grassmann space on a Hilbert C*-[Formula: see text]-module. These results provide various interpretations of [Formula: see text].

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MIGNACO, J. A., C. SIGAUD, F. J. VANHECKE, and A. R. DA SILVA. "CONNES–LOTT MODEL BUILDING ON THE TWO-SPHERE." Reviews in Mathematical Physics 13, no. 01 (January 2001): 1–28. http://dx.doi.org/10.1142/s0129055x01000582.

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In this work we examine generalized Connes–Lott models, with C⊕C as finite algebra, over the two-sphere. The Hilbert space of the continuum spectral triple is taken as the space of sections of a twisted spinor bundle, allowing for nontrivial topological structure (magnetic monopoles). The finitely generated projective module over the full algebra is also taken as topologically non-trivial, which is possible over S2. We also construct a real spectral triple enlarging this Hilbert space to include "particle" and "anti-particle" fields.

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Lambert, Alan. "A HILBERT C*-MODULE VIEW OF SOME SPACES RELATED TO PROBABILISTIC CONDITIONAL EXPECTATION." Quaestiones Mathematicae 22, no. 2 (June 1999): 165–70. http://dx.doi.org/10.1080/16073606.1999.9632070.

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