arXiv subjects

Hamed Nikpey

Publications and source records attributed to Hamed Nikpey.

5 recordsLinked to original sources

arXiv2019

$C^*$-Operator systems and crossed products

The purpose of this paper is to introduce a consistent notion of universal and reduced crossed products by actions and coactions of groups on operator systems and operator spaces. In particular we shall put emphasis to reveal the full power of the universal properties of the the universal crossed products. It turns out that to make things consistent, it seems useful to perform our constructions on some bigger categories which allow the right framework for studying the universal properties and which are stable under the construction of crossed products even for non-discrete groups. In the case of operator systems, this larger category is what we call a $C^*$-operator system, i.e., a selfadjoint subspace $X$ of some $\mathcal B(H)$ which contains a $C^*$-algebra $A$ such that $AX=X=XA$. In the case of operator spaces, the larger category is given by what we call $C^*$-operator bimodules. After we introduced the respective crossed products we show that the classical Imai-Takai and Katayama duality theorems for crossed products by group (co-)actions on $C^*$-algebras extend one-to-one to our notion of crossed products by $C^*$-operator systems and $C^*$-operator bimodules.

math.OA↗

arXiv2016

Operator Valued Maps on Hilbert $C^*$-Modules

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.

math.OA↗

arXiv2016

Completely semi-$φ$-maps

We introduce completely semi-$φ$-maps on Hilbert $C^*$-modules as a generalization of $φ$-maps. This class of maps provides examples of CP-extendable maps which are not CP-H-extendable, in Skeide-Sumesh's sense. Using the CP-extendability of completely semi-$φ$-maps, we give a representation theorem, similar to Stinespring's representation theorem, for this class of maps which can be considered as strengthened and generalized form of Asadi's and Bhat-Ramesh-Sumesh's analogues of Stinespring representation theorem for $φ$-maps. We also define an order relation on the set of all completely semi-$φ$-maps and establish a Radon-Nikodym type theorem for this class of maps in terms of their representations.

math.OA↗

arXiv2016

On the extendability of some classes of maps on Hilbert $C^*$-modules

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.

math.OA↗

arXiv2016

Crossed products of operator spaces

Let $V$ be an operator space and $\iso(V)$ be the group of all completely isometric bijective linear mappings on $V$. Let $G$ act on $V$ via a strongly continuous group homomorphism $α:G \to \iso (V)$. We define the full (and reduced) operator space crossed product $V\rtimes^{\op}_{α,(r)}G$ and show that for a $C^*$-algebra with its canonical operator space structure, it coincides with the corresponding $C^*$-algebra crossed product. Unfortunately, the proof of Theorem 4.3 of the paper contains a serious gap, which leads to the withdrawal of the paper.

math.OA↗