We study the non-autonomous stochastic Cauchy problem on a real Banach space E, $dU\left(t\right)=A\left(t\right)U\left(t\right)dt+B\left(t\right)d{W}_H\left(t\right)$, t ∈ [0,T], U(0) = u₀. Here, $W_H$ is a cylindrical Brownian motion on a real separable Hilbert space H, ${\left(B\left(t\right)\right)}_{t\in [0,T]}$ are closed and densely defined operators from a constant domain (B) ⊂ H into E, ${\left(A\left(t\right)\right)}_{t\in [0,T]}$ denotes the generator of an evolution family on E, and u₀ ∈ E. In the first part, we study existence of weak and mild solutions by methods of van Neerven and Weis. Then we use a well-known factorisation method in the setting of evolution...

Let ℳ be a hyperfinite finite von Nemann algebra and ${\left({ℳ}_k\right)}_{k\ge 1}$ be an increasing filtration of finite-dimensional von Neumann subalgebras of ℳ. We investigate abstract fractional integrals associated to the filtration ${\left({ℳ}_k\right)}_{k\ge 1}$. For a finite noncommutative martingale $x={\left(x_k\right)}_{1\le k\le n}\subseteq L₁\left(ℳ\right)$ adapted to ${\left({ℳ}_k\right)}_{k\ge 1}$ and 0 < α < 1, the fractional integral of x of order α is defined by setting $I^{\alpha }x={\sum }_{k=1}^n{\zeta }_k^{\alpha }d{x}_k$ for an appropriate sequence ${\left({\zeta }_k\right)}_{k\ge 1}$ of scalars. For the case of a noncommutative dyadic martingale in L₁() where is the type II₁ hyperfinite factor equipped...

We generalize, to the setting of Arveson’s maximal subdiagonal subalgebras of finite von Neumann algebras, the Szegő $L^p$-distance estimate and classical theorems of F. and M. Riesz, Gleason and Whitney, and Kolmogorov. As a byproduct, this completes the noncommutative analog of the famous cycle of theorems characterizing the function algebraic generalizations of $H^{\infty }$ from the 1960’s. A sample of our other results: we prove a Kaplansky density result for a large class of these algebras, and give a necessary...

We show that a real Banach algebra A such that ||a²|| = ||a||² for a ∈ A is a subalgebra of the algebra $C_{ℍ}\left(X\right)$ of continuous quaternion-valued functions on a compact set X.

We prove an existence theorem of solutions for a nonconvex sweeping process with nonconvex noncompact perturbation in Hilbert space. We do not assume that the values of the orient field are compact.

In this paper we deal with the Cauchy problem for differential inclusions governed by $m$-accretive operators in general Banach spaces. We are interested in finding the sufficient conditions for the existence of integral solutions of the problem $x^{\text{'}}\left(t\right)\in -Ax\left(t\right)+f(t,x\left(t\right))$, $x\left(0\right)=x_0$, where $A$ is an $m$-accretive operator, and $f$ is a continuous, but non-compact perturbation, satisfying some additional conditions.