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1.
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. 相似文献
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A. M. Bikchentaev 《Russian Mathematics (Iz VUZ)》2013,57(12):66-69
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. 相似文献
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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.
相似文献4.
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 $\tilde {\mathcal {M}}$ . 相似文献
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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... 相似文献
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A.M. Bikchentaev 《Siberian Mathematical Journal》2010,51(6):971-977
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. 相似文献
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A. M. Bikchentaev 《Russian Mathematics (Iz VUZ)》2012,56(2):75-79
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. 相似文献
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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... 相似文献