# 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

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

, Volume 9 (4) – Jan 1, 2005
17 pages

Publisher
Springer Journals
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-005-2779-7
Publisher site
See Article on Publisher Site

### Abstract

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.

### Journal

PositivitySpringer Journals

Published: Jan 1, 2005

## You’re reading a free preview. Subscribe to read the entire article.

### DeepDyve is your personal research library

It’s your single place to instantly
that matters to you.

over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month ### Explore the DeepDyve Library ### Search Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly ### Organize Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place. ### Access Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals. ### Your journals are on DeepDyve Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more. All the latest content is available, no embargo periods. DeepDyve ### Freelancer DeepDyve ### Pro Price FREE$49/month
\$360/year

Save searches from
PubMed

Create lists to

Export lists, citations