Spaces of Operators, the ψ-Daugavet Property, and Numerical Indices

Spaces of Operators, the ψ-Daugavet Property, and Numerical Indices Suppose ψ : [0, ∞) → [1, ∞) is a strictly increasing function. A Banach space X is said to have the ψ-Daugavet Property if the inequality $$\|I_X\,{+}\,T\|\,{\geq}\, \psi(\|T\|)$$ holds for every compact operator T : X →  X. We show that, if 1 < p < ∞ and K(ℓp)↪ X ↪ B(ℓp), then X has the ψ-Daugavet Property with $$\psi(t)\,{=}\,(1\,{+}\,c_p t^q)^{1/q}$$ (here $$q\,{=}\,\max\{2,p\}$$ and c p is an absolute constant). We also prove that a C *-algebra A is commutative if and only if $$1\,{+}\,\|T\|\,{=}\,\sup\{\|I_A\,{+}\,\omega T\|\,||\omega| \,{=}\, 1\}$$ for any $$T: A \,{\rightarrow}\, A$$ . Together, these results allow us to distinguish between some types of von Neumann algebras by considering spaces of operators on them. Positivity Springer Journals

Spaces of Operators, the ψ-Daugavet Property, and Numerical Indices

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Kluwer Academic Publishers
Copyright © 2005 by Springer
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
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