Extremely non-complex C(K) spaces

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dc.creatorKoszmider, Piotr
dc.creatorMartin, Miguel
dc.creatorMeri, Javier
dc.date2008-11-04
dc.date.accessioned2026-07-07T12:00:41Z
dc.date.available2026-07-07T12:00:41Z
dc.descriptionWe 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 $C(2^ω)$ and $C(K_2)$ contains a (1-complemented) isometric copy of $\ell_\infty$.
dc.descriptionto appear in J. Math. Anal. Appl
dc.identifierhttps://arxiv.org/abs/0811.0577
dc.identifierhttp://arxiv.org/abs/0811.0577
dc.identifierJ. Math. Anal. Appl. 350 (2009), 584-598.
dc.identifierdoi:10.1016/j.jmaa.2008.04.021
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/206868
dc.subjectFunctional Analysis
dc.subjectOperator Algebras
dc.subject46B20, 47A99
dc.titleExtremely non-complex C(K) spaces
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