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Set Theory, Topology and Banach Spaces (Second International Topology Conference in Kielce)
July 7-11, 2008
Institute of Mathematics, Jan Kochanowski University in Kielce; co-organized by Technical University of Lodz
Kielce, Poland

Organizers
Taras Banakh, Piotr Koszmider, Wieslaw Kubis

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Extremely non-complex C(K) spaces
by
Miguel Martin
Universidad de Granada (Spain)
Coauthors: Piotr Koszmider and Javier Meri

We show that there exist infinite-dimensional extremely non-complex Banach spaces, i.e. spaces X such that the norm equality ∥Id + T2∥=1 + ∥T2∥ holds for every bounded linear operator T:X→ X. This answers in the positive Question 4.11 of [Kadets, Martín, Merí, Norm equalities for operators, Indiana U. Math.\ J. 56 (2007), 2385-2411]. More concretely, we show that this is the case of some C(K) spaces with few operators constructed in [Koszmider, Banach spaces of continuous functions with few operators, Math. Ann. 330 (2004), 151-183] and [Plebanek, A construction of a Banach space C(K) with few operators, Topology Appl. 143 (2004), 217-239]. We also construct compact spaces K1 and K2 such that C(K1) and C(K2) are extremely non-complex, C(K1) contains a complemented copy of C(2w) and C(K2) contains a (1-complemented) isometric copy of l∞.

Date received: June 23, 2008

Copyright © 2008 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caxg-16.