极非复C（K）空间
Extremely non-complex C(K) spaces
摘要：
我们证明了存在无限维极非复Banach空间，即空间$X$使得范数等式$\|Id + T^2\|=1 + \|T^2\|$对每个有界线性算子$T:X\longrightarrow X$成立.这在[Kadets，Martin，Meri，Norm equalities for operators，\emph{Indiana U. Math. J.} \textbf{56}（2007），2385--2411]的肯定问题4.11中得到了回答。更具体地说，我们证明了这是在[Koszmider，具有少量运算符的连续函数的Banach空间，\emph{Math. Ann.} \textbf{330}（2004），151- 183]和[Plebanek，具有少量运算符的Banach空间$C(K)$的构造，\emph{Topology Appl.} \textbf{143}（2004），217- 239]中构造的某些具有少量运算符的$C(K)$空间的情况。我们还构造了紧致空间$K_1$和$K_2$，使得$C(K_1)$和$C(K_2)$是非常非复的，$C(K_1)$包含$C(2^ω)$的一个补拷贝，$C(K_2)$包含$\ell_\infty$的一个（1-补）等距拷贝.
英文摘要：
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 $C(2^ω)$ and $C(K_2)$ contains a (1-complemented) isometric copy of $\ell_\infty$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
