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In this paper, we present results concerning orthogonality in Hilbert C ∗ -modules. Moreover, for a C ∗ -algebra A , we prove theorems concerning the multi- A -linearity and its preservation by A -linear operators. New version of solution of the orthogonality equation on Hilbert C ∗ -modules and mappings preserving orthogonality are also investigated. [ABSTRACT FROM AUTHOR]

In this paper, we give new characterizations of modular Riesz bases in Hilbert C ∗ -modules. We prove that modular Riesz bases share many properties with Riesz bases in Hilbert spaces. Moreover, we show that there are also important differences; for example, there exist exact frames that are not modular Riesz bases. [ABSTRACT FROM AUTHOR]

Let U be a Hilbert A-module and L(U) the set of all adjointable A-linear maps on U. Let K = {Λx ∈ L(U, Vx) : x ∈ X } and L = {Γx ∈ L(U, Vx) : x ∈ X } be two continuous g-frames for U, K is said to be similar with L if there exists an invertible operator J ∈ L(U) such that Γx = ΛxJ, for all x ∈ X . In this paper, we define the concepts of closeness and nearness between two continuous g-frames. In particular, we show that K and L are near, if and only if they are similar. [ABSTRACT FROM AUTHOR]

This paper considers generators of Heisenberg modules in the case of twisted group C ∗ -algebras of closed subgroups of locally compact abelian (LCA) groups and how the restriction and/or periodization of these generators yield generators for other Heisenberg modules. Since generators of Heisenberg modules are exactly the generators of (multi-window) Gabor frames, our methods are going to be from Gabor analysis. In the latter setting, the procedure of restriction and periodization of generators is well known. Our results extend this established part of Gabor analysis to the general setting of LCA groups. We give several concrete examples where we demonstrate some of the consequences of our results. Finally, we show that vector bundles over an irrational noncommutative torus may be approximated by vector bundles for finite-dimensional matrix algebras that converge to the irrational noncommutative torus with respect to the module norm of the generators, where the matrix algebras converge in the quantum Gromov–Hausdorff distance to the irrational noncommutative torus. [ABSTRACT FROM AUTHOR]

In this paper, we study a category whose objects are Hilbert C∗-modules and whose morphisms are completely semi-ϕ-maps. We give a characterization of injective objects in this category. In fact, we investigate extendability of completely semi-ϕ-maps on Hilbert C∗-modules, leading to an analog of the Arveson’s extension theorem for completely semi-ϕ-maps (in contrast with ϕ-maps). This theorem together with previous results suggest that the completely semi-ϕ-maps are proper generalizations of the completely positive maps. [ABSTRACT FROM AUTHOR]

The duality principle for Gabor frames is one of the pillars of Gabor analysis. We establish a far-reaching generalization to Morita equivalence bimodules with some extra properties. For certain twisted group C ∗ -algebras, the reformulation of the duality principle to the setting of Morita equivalence bimodules reduces to the well-known Gabor duality principle by localizing with respect to a trace. We may lift all results at the module level to matrix algebras and matrix modules, and in doing so, it is natural to introduce (n , d) -matrix Gabor frames, which generalize multi-window super Gabor frames. We are also able to establish density theorems for module frames on equivalence bimodules, and these localize to density theorems for (n , d) -matrix Gabor frames. [ABSTRACT FROM AUTHOR]

Paschke's version of Stinespring's theorem associates a Hilbert C ∗ -module along with a generating vector to every completely positive map. Building on this, to every quantum dynamical semigroup (QDS) on a C ∗ -algebra 𝒜 one may associate an inclusion system E = (E t) of Hilbert 𝒜 - 𝒜 -modules with a generating unit ξ = (ξ t). Suppose ℬ is a von Neumann algebra, consider M 2 (ℬ) , the von Neumann algebra of 2 × 2 matrices with entries from ℬ. Suppose (Φ t) t ≥ 0 with Φ t = ϕ t 1 ψ t ψ t ∗ ϕ t 2 , is a QDS on M 2 (ℬ) which acts block-wise and let (E t i) t ≥ 0 be the inclusion system associated to the diagonal QDS (ϕ t i) t ≥ 0 with the generating unit (ξ t i) t ≥ 0 , i = 1 , 2. It is shown that there is a contractive (bilinear) morphism T = (T t) t ≥ 0 from (E t 2) t ≥ 0 to (E t 1) t ≥ 0 such that ψ t (a) = 〈 ξ t 1 , T t a ξ t 2 〉 for all a ∈ ℬ. We also prove that any contractive morphism between inclusion systems of von Neumann ℬ - ℬ -modules can be lifted as a morphism between the product systems generated by them. We observe that the E 0 -dilation of a block quantum Markov semigroup (QMS) on a unital C ∗ -algebra is again a semigroup of block maps. [ABSTRACT FROM AUTHOR]