Banach–Saks properties in symmetric spaces of measurable operators

Abstract

Abstract. 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 ).

Banach–Saks property in Marcinkiewicz spaces

“…These results complement those in [25] and [14] (see also [2,Theorem 6], where it is established that L p (0, 1), 1 p = 2 < ∞, does not embed isomorphically into C l p ).…”

Strict K-monotonicity and K-order continuity in symmetric spaces

“…This notion was discussed for the first time in papers [12] (see Theorem 4.2) for a separable symmetric space with the Fatou property. Next, the problem was improved in paper [11], where authors have established K -order continuity in the more general setting of symmetric spaces of measurable operators under additional assumption. We start our investigation with an equivalent condition for K -order continuity in a symmetric space E. Moreover, we show a necessary and sufficient condition for K -order continuity in symmetric spaces using a simple direct proof.…”

The Dixmier approximation theorem in algebras of measurable operators

“…Given x ∈ E (τ ), it follows from Theorem 9 that there exists a sequence {π n } ∞ n=1 in D such that π n x T m → 0 as n → ∞. Since π n x ≺≺ x for all n and E × (0, ∞) ⊆ S 0 (0, ∞), it now follows from Proposition 2.2 in [4] that π n x E(τ ) → 0 as n → ∞. This shows that 0 ∈ co E(τ ) U (x).…”