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Plus-Minus Property as a Generalization of the Daugavet Property

Shepelska, Varvara — 2010

Serdica Mathematical Journal

2000 Mathematics Subject Classification: Primary 46B20. Secondary 47A99, 46B42. It was shown in [2] that the most natural equalities valid for every rank-one operator T in real Banach spaces lead either to the Daugavet equation ||I+T|| = 1 + ||T|| or to the equation ||I − T|| = ||I+T||. We study if the spaces where the latter condition is satisfied for every finite-rank operator inherit the properties of Daugavet spaces.

Weak amenability of weighted group algebras on some discrete groups

Varvara Shepelska — 2015

Studia Mathematica

Weak amenability of ℓ¹(G,ω) for commutative groups G was completely characterized by N. Gronbaek in 1989. In this paper, we study weak amenability of ℓ¹(G,ω) for two important non-commutative locally compact groups G: the free group ₂, which is non-amenable, and the amenable (ax + b)-group. We show that the condition that characterizes weak amenability of ℓ¹(G,ω) for commutative groups G remains necessary for the non-commutative case, but it is sufficient neither for ℓ¹(₂,ω) nor for ℓ¹((ax + b),ω)...

Quotients of Banach Spaces with the Daugavet Property

Vladimir KadetsVarvara ShepelskaDirk Werner — 2008

Bulletin of the Polish Academy of Sciences. Mathematics

We consider a general concept of Daugavet property with respect to a norming subspace. This concept covers both the usual Daugavet property and its weak* analogue. We introduce and study analogues of narrow operators and rich subspaces in this general setting and apply the results to show that a quotient of L₁[0,1] by an ℓ₁-subspace need not have the Daugavet property. The latter answers in the negative a question posed to us by A. Pełczyński.

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