arXiv subjects
Publications and source records attributed to Gh. Sadeghi.
We employ some techniques involving projections in a von Neumann algebra to establish some maximal inequalities such as the strong and weak symmetrization, Levy, Levy-Skorohod, and Ottaviani inequalities in the realm of the quantum probability spaces.
Let $\mathfrak{M}$ be a semifinite von Neumann algebra on a Hilbert space equipped with a faithful normal semifinite trace $τ$. A closed densely defined operator $x$ affiliated with $\mathfrak{M}$ is called $τ$-measurable if there exists a number $λ\geq 0$ such that $τ\left(e^{|x|}(λ,\infty)\right)<\infty$. A number of useful inequalities, which are known for the trace on Hilbert space operators, are extended to trace on $τ$-measurable operators. In particular, these inequalities imply Clarkson inequalities for $n$-tuples of $τ$-measurable operators. A general parallelogram law for $τ$-measurable operators are given as well.
Let $\A$ be a $C^*$-algebra and $\B$ be a von Neumann algebra that both act on a Hilbert space $\Ha$. Let $\M$ and $\N$ be inner product modules over $\A$ and $\B$, respectively. Under certain assumptions we show that for each mapping $f\colon{\mathcal M} \to {\mathcal N}$ satisfying $$\||\ip{f(x)}{f(y)}|-|\ip{x}{y}| \|\leqϕ(x,y)\qquad (x,y\in{\mathcal M}),$$ where $ϕ$ is a control function, there exists a solution $I\colon{\mathcal M} \to {\mathcal N}$ of the Wigner equation $$|\ip{I(x)}{I(y)}|=|\ip{x}{y}|\qquad (x, y \in {\mathcal M})$$ such that $$\|f(x)-I(x)\|\leq\sqrt{ϕ(x,x)} \qquad (x\in {\mathcal M}).$$