Block projection operators in normed solid spaces of measurable operators

We prove a Hermitian analog of the well-known operator triangle inequality for vonNeumann algebras. In the semifinite case we show that a block projection operator is a linear positive contraction on a wide class of solid spaces of Segal measurable operators. We describe some applications of the results.     

Minimality of Convergence in Measure Topologies on Finite von Neumann Algebras

We prove that the natural embedding of the metric ideal space on a finite von Neumann algebra $\mathcal{M}$ into the *-algebra of measurable operators $\tilde {\mathcal {M}}$ endowed with the topology of convergence in measure is continuous. Using this fact, we prove that the topology of convergence in measure is a minimal one among all metrizable topologies consistent with the ring structure on $\tilde {\mathcal {M}}$ .     

Paranormal measurable operators affiliated with a semifinite von Neumann algebra. II

Positivity - Let $${{\mathcal {M}}}$$ be a von Neumann algebra of operators on a Hilbert space $${\mathcal {H}}$$ and $$\tau $$ be a faithful normal semifinite trace on $$\mathcal {M}$$ . Let...     

Convergence in Measure and τ-Compactness of τ-Measurable Operators,Affiliated with a Semifinite von Neumann Algebra

Let τ be a faithful normal semifinite trace on a von Neumann algebra. We establish the Leibniz criterion for sign-alternating series of τ-measurable operators and present an analogue of the criterion of series “sandwich” series for τ-measurable operators. We prove a refinement of this criterion for the τ-compact case. In terms of measure convergence topology, the criterion of τ-compactness of an arbitrary τ-measurable operator is established. We also give a sufficient condition of 1) τ-compactness of the commutator of a τ-measurable operator and a projection; 2) convergence of τ-measurable operator and projection commutator sequences to the zero operator in the measure τ.

     

Paranormal Elements in Normed Algebra

For a normed algebra A and natural numbers k we introduce and investigate the ∥ · ∥ closed classes P _k (A). We show that P₁(A) is a subset of P _k (A) for all k. If T in P₁(A), then Tⁿ lies in P₁(A) for all natural n. If A is unital, U, V ∈ A are such that ∥U∥ = ∥V∥ = 1, VU = I and T lies in P _k (A), then UTV lies in P _k (A) for all natural k. Let A be unital, then 1) if an element T in P₁(A) is right invertible, then any right inverse element T^?1 lies in P₁(A); 2) for ßßIßß = 1 the class P₁(A) consists of normaloid elements; 3) if the spectrum of an element T, T ∈ P₁(A) lies on the unit circle, then ∥TX∥ = ∥X∥ for all X ∈ A. If A = B(H), then the class P₁(A) coincides with the set of all paranormal operators on a Hilbert space H.     

Commutativity of projections and characterization of traces on Von Neumann algebras

We find new necessary and sufficient conditions for the commutativity of projections in terms of operator inequalities. We apply these inequalities to characterize a trace on von Neumann algebras in the class of all positive normal functionals. We obtain some characterization of a trace on von Neumann algebras in terms of the commutativity of products of projections under a weight.     

The Haagerup problem on subadditive weights on W*-algebras. II

In 1975 U. Haagerup has posed the following question: Whether every normal subadditive weight on a W*-algebra is σ-weakly lower semicontinuous? In 2011 the author has positively answered this question in the particular case of abelian W*-algebras and has presented a general form of normal subadditive weights on these algebras. Here we positively answer this question in the case of finite-dimensional W*-algebras. As a corollary, we give a positive answer for subadditive weights with some natural additional condition on atomic W*-algebras. We also obtain the general form of such normal subadditive weights and norms for wide class of normed solid spaces on atomic W*-algebras.     

Seminorms Associated with Subadditive Weights on C*-Algebras

Mathematical Notes - Let ϕ be a subadditive weight on a C* -algebra A, and let Mϕ+ be the set of all elements x in A+ with ϕ(x) < +00. A seminorm ‖ • ‖ is...