This is an announcement for the paper "Extremely non-complex C(K) spaces" by Piotr Koszmider, Miguel Martin, and Javier Meri .

Abstract: We show that there exist infinite-dimensional extremely non-complex Banach spaces, i.e.\ spaces $X$ such that the norm equality $|Id + T^2|=1 + |T^2|$ holds for every bounded linear operator $T:X\longrightarrow X$. This answers in the positive Question 4.11 of [Kadets, Martin, Meri, Norm equalities for operators, \emph{Indiana U.\ Math.\ J.} \textbf{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, \emph{Math.\ Ann.} \textbf{330} (2004), 151--183] and [Plebanek, A construction of a Banach space $C(K)$ with few operators, \emph{Topology Appl.} \textbf{143} (2004), 217--239]. We also construct compact spaces $K_1$ and $K_2$ such that $C(K_1)$ and $C(K_2)$ are extremely non-complex, $C(K_1)$ contains a complemented copy of $C(2^\omega)$ and $C(K_2)$ contains a (1-complemented) isometric copy of $\ell_\infty$.

Archive classification: math.FA math.OA

Mathematics Subject Classification: 46B20, 47A99

Remarks: to appear in J. Math. Anal. Appl

The source file(s), JMAA-07-3370R1.tex: 65250 bytes, is(are) stored in gzipped form as 0811.0577.gz with size 20kb. The corresponding postcript file has gzipped size 135kb.

Submitted from: mmartins@ugr.es

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