Searcharxiv⌕ Search

arXiv subjects

Mohammad B. Asadi

Publications and source records attributed to Mohammad B. Asadi.

8 recordsLinked to original sources
arXiv2020

The radius of comparison of the tensor product of a C*-algebra with $C (X)$

Let $X$ be a compact metric space, let $A$ be a unital AH algebra with large matrix sizes, and let $B$ be a stably finite unital C*-algebra. Then we give a lower bound for the radius of comparison of $C(X) \otimes B$ and prove that the dimension-rank ratio satisfies $\operatorname{drr} (A) = \operatorname{drr} \left(C(X)\otimes A\right)$. We also give a class of unital AH algebras $A$ with $\operatorname{rc} \left(C(X) \otimes A\right) = \operatorname{rc} (A)$. We further give a class of stably finite exact $\mathcal{Z}$-stable unital C*-algebras with nonzero radius of comparison.

math.OA↗
arXiv2019

On the ternary domain of a completely positive map on a Hilbert C*-module

We associate to an operator valued completely positive linear map $φ$ on a $C^{\ast }$-algebra $A$ and a Hilbert $C^{\ast }$-module $X$ over $A$ a subset $X_{φ}$ of $X,$ called '\textit{ternary domain}' of $φ$ on $X,$ which is a Hilbert $C^{\ast }$-module over the multiplicative domain of $φ$ and every $φ$-map (i.e., associated quaternary map with $φ$) acts on it as a ternary map. We also provide several characterizations for this set. The ternary domain \ of $φ$ on $A\ $ is a closed two-sided $\ast $-ideal $T_{φ}$ of the multiplicative domain of $φ$. We show that $XT_{φ}=X_{φ}$ and give several characterizations of the set $X_{φ}.$ Furthermore, we establish some relationships between $X_{φ}$ and minimal Stinespring dilation triples associate to $φ$. Finally, we show that every operator valued completely positive linear map $φ$ on a $C^{\ast }$ -algebra $A$ induces a unique (in a some sense) completely positive linear map on the linking algebra of $X$ and we determine its multiplicative domain in terms of the multiplicative domain of $φ$ and the ternary domain of $φ$ on $X$.

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↗
arXiv2014

Spectral radius, numerical radius, and the product of operators

Let $σ(A)$, $ρ(A)$ and $r(A)$ denote the spectrum, spectral radius and numerical radius of a bounded linear operator $A$ on a Hilbert space $H$, respectively. We show that a linear operator $A$ satisfying $$ρ(AB)\le r(A)r(B) \quad\text{ for all bounded linear operators } B$$ if and only if there is a unique $μ\in σ(A)$ satisfying $|μ| = ρ(A)$ and $A = \frac{μ(I + L)}{2}$ for a contraction $L$ with $1\inσ(L)$. One can get the same conclusion on $A$ if $ρ(AB) \le r(A)r(B)$ for all rank one operators $B$. If $H$ is of finite dimension, we can further decompose $L$ as a direct sum of $C \oplus 0$ under a suitable choice of orthonormal basis so that $Re(C^{-1}x,x) \ge 1$ for all unit vector $x$.

math.FA↗
arXiv2013

Which multiplier algebras are $W^*$-algebras?

We consider the question of when the multiplier algebra $M(\mathcal{A})$ of a $C^*$-algebra $\mathcal{A}$ is a $ W^*$-algebra, and show that it holds for a stable $C^*$-algebra exactly when it is a $C^*$-algebra of compact operators. This implies that if for every Hilbert $C^*$-module $E$ over a $C^*$-algebra $\mathcal{A}$, the algebra $B(E)$ of adjointable operators on $E$ is a $ W^*$-algebra, then $\mathcal{A}$ is a $C^*$-algebra of compact operators. Also we show that a unital $C^*$-algebra $\mathcal{A}$ which is Morita equivalent to a $ W^*$-algebra must be a $ W^*$-algebra.

math.OA↗