Almost sure convergence of iterates of contractions in noncommutative L2-spaces.
Automorphisms of the algebra of operators in preserving conditioning
Let α be an isometric automorphism of the algebra of bounded linear operators in (p ≥ 1). Then α transforms conditional expectations into conditional expectations if and only if α is induced by a measure preserving isomorphism of [0, 1].
Pointwise limit theorem for a class of unbounded operators in -spaces
We distinguish a class of unbounded operators in , r ≥ 1, related to the self-adjoint operators in ². For these operators we prove a kind of individual ergodic theorem, replacing the classical Cesàro averages by Borel summability. The result is equivalent to a version of Gaposhkin’s criterion for the a.e. convergence of operators. In the proof, the theory of martingales and interpolation in -spaces are applied.
Borel methods of summability and ergodic theorems
Passing from Cesàro means to Borel-type methods of summability we prove some ergodic theorem for operators (acting in a Banach space) with spectrum contained in ℂ∖(1,∞).
On some versions of Jensen's inequality on operator algebras
Pointwise Convergence Theorems in L2 over a von Neumann Algebra.
Convergence of orthogonal series of projections in Banach spaces
For a sequence of mutually orthogonal projections in a Banach space, we discuss all possible limits of the sums in a “strong” sense. Those limits turn out to be some special idempotent operators (unbounded, in general). In the case of X = L₂(Ω,μ), an arbitrary unbounded closed and densely defined operator A in X may be the μ-almost sure limit of (i.e. μ-a.e. for all f ∈ (A)).
Almost sure approximation of unbounded operators in
The possibilities of almost sure approximation of unbounded operators in by multiples of projections or unitary operators are examined.
Gleason measures and some inner products on the algebra of bounded operators on a Hilbert space
The bundle convergence in von Neumann algebras and their -spaces
A stronger version of almost uniform convergence in von Neumann algebras is introduced. This "bundle convergence" is additive and the limit is unique. Some extensions of classical limit theorems are obtained.
The unconditional pointwise convergence of orthogonal seriesin over a von Neumann algebra
The paper is devoted to some problems concerning a convergence of pointwise type in the -space over a von Neumann algebra M with a faithful normal state Φ [3]. Here is the completion of M under the norm .
On summability methods generated by R-integrable functions
Dilation theorems for completely positive maps and map-valued measures
The Stinespring theorem is reformulated in terms of conditional expectations in a von Neumann algebra. A generalisation for map-valued measures is obtained.
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