arXiv subjects
Publications and source records attributed to Ghadir Sadeghi.
In this paper, we explore the interplay between topological structures and phase retrieval in the context of projective Hilbert spaces. This work provides not only a deeper understanding and a new classification of the phase retrieval property in Hilbert spaces but also a way for further investigations into the topological underpinnings of quantum states.
A compact quantum metric space is a unital $C^*$-algebra equipped with a Lip-norm. Let $\{(A_n, L_n)\}$ be a sequence of compact quantum metric spaces, and let $ϕ_n:A_n\to A_{n+1}$ be a unital $^*$-homomorphism preserving Lipschitz elements for $n\geq 1$. We show that there exists a compact quantum metric space structure on the inductive limit $\varinjlim(A_n,ϕ_n)$ by means of the inverse limit of the state spaces $\{\mathcal{S}(A_n)\}$. We also give some sufficient conditions that two inductive limits of compact quantum metric spaces are Lipschitz isomorphic.
In this paper, we introduce the concept of $K$-fusion frames and propose the duality for such frames. The relation between the local frames of $K$-fusion frames with their dual is studied. The elements from the range of a bounded linear operator $K$ can be reconstructed by $K$-frames. Also, we establish $K$-fusion frame multipliers and investigate reconstruction of the range of $K$ by them.
Based on a maximal inequality type result of Cuculescu, we establish some noncommutative maximal inequalities such as Hajék--Penyi inequality and Etemadi inequality. In addition, we present a noncommutative Kolmogorov type inequality by showing that if $x_1, x_2, \ldots, x_n$ are successively independent self-adjoint random variables in a noncommutative probability space $(\mathfrak{M}, τ)$ such that $τ\left(x_k\right) = 0$ and $s_k s_{k-1} = s_{k-1} s_k$, where $s_k = \sum_{j=1}^k x_j$, then for any $λ> 0$ there exists a projection $e$ such that $$1 - \frac{(λ+ \max_{1 \leq k \leq n} \|x_k\|)^2}{\sum_{k=1}^n {\rm var}(x_k)}\leq τ(e)\leq \frac{τ(s_n^2)}{λ^2}.$$ As a result, we investigate the relation between convergence of a series of independent random variables and the corresponding series of their variances.
We establish a noncommutative Blackwell--Ross inequality for supermartingales under a suitable condition which generalize Khan's works to the noncommutative setting. We then employ it to deduce an Azuma-type inequality.
We establish an Azuma type inequality under a Lipshitz condition for martingales in the framework of noncommutative probability spaces and apply it to deduce a noncommutative Heoffding inequality as well as a noncommutative McDiarmid type inequality. We also provide a noncommutative Azuma inequality for noncommutative supermartingales in which instead of a fixed upper bound for the variance we assume that the variance is bounded above by a linear function of variables. We then employ it to deduce a noncommutative Bernstein inequality and an inequality involving $L_p$-norm of the sum of a martingale difference.
In this paper, we establish an extension of a noncommutative Bennett inequality with a parameter $1\leq r\leq2$ and use it together with some noncommutative techniques to establish a Rosenthal inequality. We also present a noncommutative Hoeffding inequality as follows: Let $(\mathfrak{M}, τ)$ be a noncommutative probability space, $\mathfrak{N}$ be a von Neumann subalgebra of $\mathfrak{M}$ with the corresponding conditional expectation $\mathcal{E}_{\mathfrak{N}}$ and let subalgebras $\mathfrak{N}\subseteq\mathfrak{A}_j\subseteq\mathfrak{M}\,\,(j=1, \cdots, n)$ be successively independent over $\mathfrak{N}$. Let $x_j\in\mathfrak{A}_j$ be self-adjoint such that $a_j\leq x_j\leq b_j$ for some real numbers $a_j o$ it holds that \begin{eqnarray*} {\rm Prob}\left(\left|\sum_{j=1}^n x_j-nμ\right|\geq t\right)\leq 2 \exp\left\{\frac{-2t^2}{\sum_{j=1}^n(b_j-a_j)^2}\right\}. \end{eqnarray*}
The classical Mazur--Ulam theorem which states that every surjective isometry between real normed spaces is affine is not valid for non-Archimedean normed spaces. In this paper, we establish a Mazur--Ulam theorem in the non-Archimedean strictly convex normed spaces.