Mathematics – Functional Analysis
Scientific paper
2008-11-04
J. Math. Anal. Appl. 350 (2009), 584-598.
Mathematics
Functional Analysis
to appear in J. Math. Anal. Appl
Scientific paper
10.1016/j.jmaa.2008.04.021
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 $\ell_\infty$.
Koszmider Piotr
Martin Miguel
Meri Javier
No associations
LandOfFree
Extremely non-complex C(K) spaces does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Extremely non-complex C(K) spaces, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Extremely non-complex C(K) spaces will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-184694