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Recently, Rashidi-Kouchi et al. devoted their efforts to giving analogs of the Casazza–Christensen general perturbation theorem of Hilbert space frames for the case of g-frames in Hilbert C*-modules. We found, however, that in the proof they did not obtain the lower bound inequality of g-frames in Hilbert C*-modules. In this paper, we establish an equivalent definition of g-frames in Hilbert C*-modules, using it we present a complete proof for the stability result given by them.

For a Hilbert C ( X ) -module V , where X is a compact metrizable space, we show that the following conditions are equivalent: (i) V is topologically finitely generated, (ii) there exists K ∈ N such that every algebraically finitely generated submodule of V can be generated with k ≤ K generators, (iii) V is canonically isomorphic to the Hilbert C ( X ) -module Γ ( E ) of all continuous sections of an (F) Hilbert bundle E = ( p , E , X ) over X , whose fibres E x have uniformly finite dimensions, and each restriction bundle of E over a set where dim E x is constant is of finite type, (iv) there exists N ∈ N such that for every Banach C ( X ) -module W , each tensor in the C ( X ) -projective tensor product V ⊗ π C ( X ) W is of (finite) rank at most N .

We characterize the Birkhoff-James orthogonality for elements of a Hilbert C*-module in terms of states of the underlying C*-algebra. We also show that the Birkhoff-James orthogonality in a Hilbert C*-module over a C*-algebra A, and orthogonality with respect to the A-valued inner product coincide if and only if A is isomorphic to C.

Hilbert C*-modules are the analogues of Hilbert spaces where a C*-algebra plays the role of the scalar field. With the advent of Kasparov's celebrated KK-theory they became a standard tool in the theory of operator algebras. While the elementary properties of Hilbert C*-modules can be derived basically in parallel to Hilbert space theory the lack of an analogue of the Projection Theorem soon leads to serious obstructions and difficulties. In particular the theory of unbounded operators is notoriously more complicated due to the additional axiom of regularity which is not easy to check. In this paper we present a new criterion for regularity in terms of the Hilbert space localizations of an unbounded operator. We discuss several examples which show that the criterion can easily be checked and that it leads to nontrivial regularity results., 31 pages, 2 figures; v3: revised version containing an erratum to v2 clarifying the contributions by Fran\c{c}ois Pierrot

Due to an example indicated to us in September 2009 we have to add one more restriction to the suppositions on the imprimitivity bimodules treated in Proposition 4.1, Theorem 5.1, Theorem 6.2 and Proposition 6.3. In the situation when the Banach-Saks property holds for the imprimitivity bimodule we can describe all possible additional examples violating the newly invented supposition. So the classification of Hilbert $C^*$-modules with the Banach-Saks property is complete. Beyond that, there is still an open problem for a certain class of imprimitivity bimodules with the weak or uniform weak Banach-Saks property which might violate the additional condition.

In this paper, we generalize the construction of a pullback diagram in the framework of Hilbert C*-modules and investigate some conditions under which a diagram of Hilbert C*-modules is pullback.

We comment on the triangle (in)equality for a C*-valued norm defined on a Hilbert C*-module V. A C*-valued norm is used to define a preorder on V. In this preorder, one can interpolate the convex combination of two elements "between" any two elements of V that satisfy certain condition.

In this paper we obtain a characterization of finite-dimensional Hilbert C*-modules. It is known that those are the modules for which both underlying C*-algebras are finite-dimensional. We show that such modules can be described by a certain property of bounded sequences of their elements. It turns out that similar property leads to another characterization of Hilbert C*-modules over C*-algebras of compact operators.

We present a new operator equality in the framework of Hilbert C*-modules. As a consequence, we get an extension of the Euler–Lagrange type identity in the setting of Hilbert bundles as well as several generalized operator Bohr's inequalities due to O. Hirzallah, W.-S. Cheung, J.E. Pecaric and F. Zhang.

In this paper, we define a Α-linear bounded operator about two Bessel sequences in Hilbert C*-Module and, by means of which we make some characterzations of the properties of modular frames in Hilbert C*-module.

Fusion frames and g-frames were recently introduced as generalizations of frames. In this note we study their properties in Hilbert spaces and Hilbert C*-modules and we generalize some of the known results in frame theory to fusion frames and g-frames. We study the behavior of these generalized frames under mall perturbations. We also show that tensor product of fusion frames (g-frames) is a fusion frame (g-frame) and tensor product of resolution of identity is a resolution of identity in Hilbert spaces and Hilbert C*-modules.

We prove two new reverse Cauchy--Schwarz inequalities of additive and multiplicative types in a space equipped with a positive sesquilinear form with values in a C*-algebra. We apply our results to get some norm and integral inequalities. As a consequence, we improve a celebrated reverse Cauchy--Schwarz inequality due to G Polya and G. Szego., Comment: 10 pages, to appear in Math. Ineq. Apl. (MIA)

The C*-valued norm is defined on a Hilbert C*-module by its standard inner product. In this paper we give generalizations of a number of classical inequalities known for either complex numbers or Hilbert space operators. In particular, we study Bohr’s inequality for the C*-valued norm on a Hilbert C*-module

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., Minor change. To appear in Bull. Lond. Math. Soc

We prove that for two elements x, y in a Hilbert C*-module V over a C*-algebra \( \mathcal{A} \) the C*-valued triangle equality |x + y| = |x| + |y| holds if and only if 〈x, y〉 = |x| |y|. In addition, if \( \mathcal{A} \) has a unit e, then for every x, y ∊ V and every ɛ > 0 there are contractions u, υ ∊ \( \mathcal{A} \) such that |x + y| ≦ u|x|u* + υ|y|υ* + ɛe.

An A-valued norm is defined on a Hilbert C*-module V over a commutative C*-algebra A in terms of a standard inner product on V. We introduce a 2-inner product on V and define an A-valued 2-norm there. For both C*-valued norms on V a number on inequalities (which are in fact order relations on A) are given.

The construction of the generalized continuous wavelet transform (GCWT) on Hilbert spaces is a special case of the coherent state transform construction, where the coherent state system arises as an orbit of an admissible vector under a strongly continuous unitary representation of a locally compact group. In this thesis we extend this construction to the setting of Hilbert C*-modules. In particular, we define a coherent state transform and a GCWT on Hilbert modules. This construction gives a reconstruction formula and a resolution of the identity formula analogous to those found in the Hilbert space setting. Moreover, the existing theory of standard normalized tight frames in finite countably generated Hilbert modules can be viewed as a discrete case of this construction We also show that the image space of the coherent state transform on Hilbert module is a reproducing kernel Hilbert module. We discuss the kernel and the intertwining property of the group coherent state transform.

We study a certain class of frames for a full Hilbert C*-module which characterize Hilbert C*-modules over the C*-algebra of compact operators. An example of such frames are orthonormal bases.

We give the necessary and sufficient conditions for the norm and the C*-valued triangle equalities in (pre)-Hilbert C*-modules.

We study orthogonality preserving and approximately orthogonality preserving mappings in the setting of inner product C ∗ -modules. In particular, if V and W are inner product C ∗ -modules over the C ∗ -algebra A , any scalar multiple of an A -linear isometry is an A -linear orthogonality preserving mapping. The converse does not hold in general, but it holds if A contains K ( H ) (the C ∗ -algebra of all compact operators on a Hilbert space H ). Furthermore, we give the estimate of ‖ 〈 T x , T y 〉 − ‖ T ‖ 2 〈 x , y 〉 ‖ for an A -linear approximately orthogonality preserving mapping T : V → W when V and W are inner product C ∗ -modules over a C ∗ -algebra containing K ( H ) . In the case A = K ( H ) and V , W are Hilbert, we also prove that an A -linear approximately orthogonality preserving mapping can be approximated by an A -linear orthogonality preserving mapping.

The paper is devoted to closed modular densely defined operators on a Hilbert C*-module W over the C*-algebra of compact operators on a Hilbert space. A bijective operation preserving correspondence of the set of all such operators and the set of all closed densely defined operators on a Hilbert space is obtained. The polar decomposition of a closed operator is generalized as well as some results concerning operators with compact resolvent, relative compact operators and the generalized inverse.

The operator-valued Schur class is defined to be the set of holomorphic functions S mapping the unit disk into the space of contraction operators between two Hilbert spaces. There are a number of alternate characterizations: the operator of multiplication by S defines a contraction operator between two Hardy Hilbert spaces, S satisfies a von Neumann inequality, a certain operator-valued kernel associated with S is positive-definite, and S can be realized as the transfer function of a dissipative (or even conservative) discrete-time linear input/state/output linear system. Various multivariable generalizations of this class have appeared recently, one of the most encompassing being that of Muhly and Solel where the unit disk is replaced by the strict unit ball of the elements of a dual correspondence E σ associated with a W*-correspondence E over a W*-algebra \( \mathcal{A} \) together with a *-representation σ of \( \mathcal{A} \). The main new point which we add here is the introduction of the notion of reproducing kernel Hilbert correspondence and identification of the Muhly-Solel Hardy spaces as reproducing kernel Hilbert correspondences associated with a completely positive analogue of the classical Szego kernel. In this way we are able to make the analogy between the Muhly-Solel Schur class and the classical Schur class more complete. We also illustrate the theory by specializing it to some well-studied special cases; in some instances there result new kinds of realization theorems.

In this paper, we study the erasures of submodules of a fusion frame in Hilbert [Formula: see text]-modules and we present some sufficient conditions under which a sequence remains a standard fusion frame after deletion of some submodules. We also introduce a new perturbation for g-frames and fusion frames in Hilbert [Formula: see text]-modules and it is shown that under some conditions g-frames and fusion frames are stable under this perturbation. We generalize some of the results obtained for perturbations of g-frames and fusion frames, especially we consider perturbations of direct sums, duals and approximate duals of g-frames.

We determine the condition for a morphism of / between extensions of Hilbert C*-modules to exist and give the description of a morphism out of an extension of a Hilbert C*-module in a general case.

In this paper we introduce frames of submodules for countably generated Hilbert K(H)-modules. We show the correspondence between frames of submodules for a Hilbert K(H)-module V and frames of subspaces for the Hilbert space V e V; where e 2 K(H) is an arbi- trary orthogonal one-dimensional projection. We also establish a relation between their frame operators.

Let us have a split extension W of a Hilbert C*-module V by a Hilbert C*-module Z. Like in the case of C*-algebras (well known theorem of T. A. Loring), every morphism out of W, more precisely from W to an arbitrary Hilbert C*-module U, can be described as a pair of morphisms from V and Z, respectively, into U that satisfies certain conditions. It turns out that besides the generalization of the Loring's condition, an additional condition has to be posed.

The subject of the paper is representations of Hilbert C*-modules. We introduce the concept of a nondegenerate, irreducible and cyclic representation and the cornmutant of a Hilbert C*-module. Relations between representations of a Hilbert C*-module, its linking algebra and underlying C*-algebras are discussed. We also study Hilbert CCR-and GCR-modules and extend some results from CCR-and GCR algebras. The results we obtain are similar to the results known for C*-algebras.

For a given $q in kom$ with $|q| le 1$, we study the $C$-numerical range of a Hilbert space operator where $C$ is an operator of the form [ left( begin{array}{ccc} qI_n & sqrt{1-|q|^2}I_n \ 0_n & 0_n end{array} right) oplus 0. ] Some known results on the $q$-numerical range are extended to this set.

The aim of this paper is to connect the results of D. Baki c and B. Gulja s about C -extensions of Hilbert C -modules with results of D.P. Blecher about Hilbert C -extensions of operator spaces. In the rst part, we give conditions on a completely bounded linear operator between Hilbert C -modules for the possibility of extending the operator to a "corner-preserving" C -morphism of the corresponding linking-algebras (or, equivalently, for the operator being a Hilbert C -morphism). The second part provides an order preserving bijection between the sets of C - extensions of a Hilbert C -module and its Hilbert C -extensions, the latter being a generalized version of Blecher's Hilbert C -extensions of operator spaces dened in (5).

The paper begins a study of extensions of Hilbert \cez-modules. An (essential) extension of a Hilbert \cez-module $V$ over a \cez-algebra \as is defined as a quadruple $(W, \bss, \Phi, \varphi)$ consisting of a Hilbert \bss-module $W$, an injective morphism of \cez-algebras $\varphi : \ass \rightarrow \bss$ such that $\mbox{; ; ; Im}; ; ; \, \varphi$ is an (essential) ideal in \bss, and a morphism of Hilbert \cez-modules $\Phi : V \rightarrow W$ such that $\mbox{; ; ; Im}; ; ; \, \Phi$ is an ideal submodule of $W$. This leads to the exact sequence $0 \rightarrow V \rightarrow W \rightarrow W/\mbox{; ; ; Im}; ; ; \, \Phi \rightarrow 0$ of Hilbert \cez-modules. It is proved that for each Hilbert \ass-module $V$ there exists the largest essential extension $(V_d, \ogrr(\ass), \Gamma, \gamma)$ such that for any other essential extension $(W, \bss, \Phi, \varphi, )$ of $V$ one can embed $W$ into $V_d$. It is also shown that the \cez-algebras of all adjointable operators acting on $V$ and $V_d$, respectively, are isomorphic.

We study irreducible representations of a Hilbert C*-module and construct the faithful representation which is a direct sum of irreducible representations.

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We describe the pullback construction in the category of Hilbert C -modules (with a suitable class of morphisms) in terms of pullbacks of underlying C -algebras. In the second section the Busby in- variant for extensions of Hilbert C -modules is introduced and it is proved that each extension is uniquely determined, up to isomorphism, by the corresponding Busby map. The induced extensions of the underlying C - algebras as well as of the corresponding linking algebras are also discussed. The paper ends with a Hilbert C -module version of a familiar result which states that a C -algebra is projective if and only if it is corona projective.

Let H be a complex Hilbert space, dim H>1, and let A⊆B(H) be a C*-algebra such that the ideal K(H) of all compact operators on H is contained in A. Let X be a Hilbert C*-module over A. We prove that any function F:X→X which preserves the absolute value of the A-valued inner product on X is of the form F(x)=φ(x)Ux, x∈X, where φ is a phase function and U is an A-linear isometry. The result generalizes Wigner’s classical unitary-antiunitary theorem and its extension to Hilbert K(H)-modules.

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A concept of an orthonormal basis for Hilbert \cez-modules is discussed. It is proved that each Hilbert \cez-module $W$ over an arbitrary \cez-algebra \as of (not necessarily all) compact operators on a Hilbert space possesses an orthonormal basis. The \cez-algebra of all adjointable operators on $W$ is naturally represented on a Hilbert space contained in $W$. Also, "compact" operators on $W$ are characterized.

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The paper describes some basic properties of a class of module maps of Hilbert C*-modules. In Section 1 ideal submodules are considered and the canonical Hilbert C*-module structure on the quotient of a Hilbert C*-module over an ideal submodule is described. Given a Hilbert C*-module V, an ideal submodule $V_{inss}$, and the quotient $V/V_{inss}$, canonical morphisms of the corresponding C*-algebras of adjointable operators are discussed. In the second part of the paper a class of module maps of Hilbert C*-modules is introduced.Given Hilbert C*-modules V and W and a morphism $varphi : ass rightarrow bss$ of the underlying cez-algebras, a map $Phi : V rightarrow W$ belongs to the class under consideration if it preserves inner products modulo $varphi$: $langle Phi(x), Phi(y) rangle = varphi(langle x,y rangle)$ for all $x,y in V$. It is shown that each morphism Φ of this kind is necessarily a contraction such that the kernel of Φ is an ideal submodule of V. A related class of morphisms of the corresponding linking algebras is also discussed.

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.

By a theorem of D. P. Blecher, a Hilbert C*-module V over a C*-algebra A (faithfully and nondegenerately represented on a Hilbert space H) is characterized by a certain Hilbert space HV, such that V can be embedded in the algebra B(H, HV) of bounded operators between H and HV. In this paper it is shown: 1. For a Hilbert C*-module over a C*-algebra of compact operators the Hilbert space HV conicides with a Hilbert subspace of the module, which characterizes all adjointable operators on the module; 2. For any Hilbert C*-module V, its strict completion can be realized in the algebra B(H, HV).

This article is a review of the basic results on tensor products of C*-algebras, operator spaces and Hilbert C*-modules.

Hilbert C*-modules are objects like Hilbert spaces, except that the inner product, instead of being complex valued, takes its values in a C*-algebra. The theory of these modules, together with their bounded and unbounded operators, is not only rich and attractive in its own right but forms an infrastructure for some of the most important research topics in operator algebras. This book is based on a series of lectures given by Professor Lance at a summer school at the University of Trondheim. It provides, for the first time, a clear and unified exposition of the main techniques and results in this area, including a substantial amount of new and unpublished material. It will be welcomed as an excellent resource for all graduate students and researchers working in operator algebras.

In this paper, we introduce the notion of (θ1, θ2, θ3, ϕ)-derivations on Hilbert C*-modules. Moreover, we investigate the generalized Hyers–Ulam–Rassias stability, Isac–Rassias type stability and superstability of (θ1, θ2, θ3, ϕ)-derivations on Hilbert C*-modules.

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.

We show that each right Hilbert C*-module can be equipped with a left Hilbert C*-module structure over the C*-algebra of generalized compact operators.

In a Hilbert space H, discrete families of vectors {hj} with the property that f=∑j〈hj‖ f〉hj for every f in H are considered. This expansion formula is obviously true if the family is an orthonormal basis of H, but also can hold in situations where the hj are not mutually orthogonal and are ‘‘overcomplete.’’ The two classes of examples studied here are (i) appropriate sets of Weyl–Heisenberg coherent states, based on certain (non‐Gaussian) fiducial vectors, and (ii) analogous families of affine coherent states. It is believed, that such ‘‘quasiorthogonal expansions’’ will be a useful tool in many areas of theoretical physics and applied mathematics.

This paper is an investigation of right modules over a B ∗ {B^\ast } -algebra B which posses a B-valued “inner product” respecting the module action. Elementary properties of these objects, including their normability and a characterization of the bounded module maps between two such, are established at the beginning of the exposition. The case in which B is a W ∗ {W^\ast } -algebra is of especial interest, since in this setting one finds an abundance of inner product modules which satisfy an analog of the self-duality property of Hilbert space. It is shown that such self-dual modules have important properties in common with both Hilbert spaces and W ∗ {W^\ast } -algebras. The extension of an inner product module over B by a B ∗ {B^\ast } -algebra A containing B as a ∗ ^\ast -subalgebra is treated briefly. An application of some of the theory described above to the representation and analysis of completely positive maps is given.