In this paper we investigate finite rank operators in the Jacobson radical ${ℛ}_{𝒩\otimes ℳ}$ of $\mathrm{A}lg(𝒩\otimes ℳ)$, where $𝒩$, $ℳ$ are nests. Based on the concrete characterizations of rank one operators in $\mathrm{A}lg(𝒩\otimes ℳ)$ and ${ℛ}_{𝒩\otimes ℳ}$, we obtain that each finite rank operator in ${ℛ}_{𝒩\otimes ℳ}$ can be written as a finite sum of rank one operators in ${ℛ}_{𝒩\otimes ℳ}$ and the weak closure of ${ℛ}_{𝒩\otimes ℳ}$ equals $\mathrm{A}lg\left(𝒩\otimes ℳ\right)$ if and only if at least one of $𝒩$, $ℳ$ is continuous.

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 begin a program of generalizing basic elements of the theory of comparison, equivalence, and subequivalence, of elements in C*-algebras, to the setting of more general algebras. In particular, we follow the recent lead of Lin, Ortega, Rørdam, and Thiel of studying these equivalences, etc., in terms of open projections or module isomorphisms. We also define and characterize a new class of inner ideals in operator algebras, and develop a matching theory of open partial isometries in operator ideals...

We generalize some aspects of the theory of compact projections relative to a C*-algebra, to the setting of more general algebras. Our main result is that compact projections are the decreasing limits of 'peak projections', and in the separable case compact projections are just the peak projections. We also establish new forms of the noncommutative Urysohn lemma relative to an operator algebra, and we show that a projection is compact iff the associated face in the state space of the algebra is...

There are applied power mappings in algebras with logarithms induced by a given linear operator D in order to study particular properties of powers of logarithms. Main results of this paper will be concerned with the case when an algebra under consideration is commutative and has a unit and the operator D satisfies the Leibniz condition, i.e. D(xy) = xDy + yDx for x, y ∈ dom D. Note that in the Number Theory there are well-known several formulae expressed by means of some combinations of powers...

We prove that the spaces of (α,β)-derivations on certain operator algebras are topologically reflexive in the weak operator topology. Characterizations of automorphisms and (α,β)-derivations on reflexive algebras are also given.