### A Banach principle for semifinite von Neumann algebras.

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It is shown that every monocompact submeasure on an orthomodular poset is order continuous. From this generalization of the classical Marczewski Theorem, several results of commutative Measure Theory are derived and unified.

Let ℳ be a type II₁ von Neumann algebra, τ a trace in ℳ, and L²(ℳ,τ) the GNS Hilbert space of τ. If L²(ℳ,τ)₊ is the completion of the set ${\mathcal{M}}_{sa}$ of selfadjoint elements, then each element ξ ∈ L²(ℳ,τ)₊ gives rise to a selfadjoint unbounded operator ${L}_{\xi}$ on L²(ℳ,τ). In this note we show that the exponential exp: L²(ℳ,τ)₊ → L²(ℳ,τ), $exp\left(\xi \right)={e}^{i{L}_{\xi}}$, is continuous but not differentiable. The same holds for the Cayley transform $C\left(\xi \right)=({L}_{\xi}-i){({L}_{\xi}+i)}^{-1}$. We also show that the unitary group ${U}_{\mathcal{M}}\subset L\xb2(\mathcal{M},\tau )$ with the strong operator topology is not an embedded submanifold...

Given a von Neumann algebra M we consider its central extension E(M). For type I von Neumann algebras, E(M) coincides with the algebra LS(M) of all locally measurable operators affiliated with M. In this case we show that an arbitrary automorphism T of E(M) can be decomposed as $T={T}_{a}\circ {T}_{\varphi}$, where ${T}_{a}\left(x\right)=ax{a}^{-1}$ is an inner automorphism implemented by an element a ∈ E(M), and ${T}_{\varphi}$ is a special automorphism generated by an automorphism ϕ of the center of E(M). In particular if M is of type ${I}_{\infty}$ then every band preserving automorphism...

We establish a non-commutative analog of the classical Banach Principle on the almost everywhere convergence of sequences of measurable functions. The result is stated in terms of quasi-uniform (or almost uniform) convergence of sequences of measurable (with respect to a trace) operators affiliated with a semifinite von Neumann algebra. Then we discuss possible applications of this result.

We study Banach-Saks properties in symmetric spaces of measurable operators. A principal result shows that if the symmetric Banach function space E on the positive semiaxis with the Fatou property has the Banach-Saks property then so also does the non-commutative space E(ℳ,τ) of τ-measurable operators affiliated with a given semifinite von Neumann algebra (ℳ,τ).

A normal Banach quasi *-algebra (,) has a distinguished Banach *-algebra ${}_{b}$ consisting of bounded elements of . The latter *-algebra is shown to coincide with the set of elements of having finite spectral radius. If the family () of bounded invariant positive sesquilinear forms on contains sufficiently many elements then the Banach *-algebra of bounded elements can be characterized via a C*-seminorm defined by the elements of ().