$C(X)$的保直径线性双射
Diameter preserving linear bijections of $C(X)$
摘要：
本文的目的是解决函数代数$C(X)$上的一个线性映射问题。证明了在$X$是第一可数紧Hausdorff空间的情况下，每个线性双射$ϕ:C(X)\to C(X)$都具有$diam(ϕ(f)(X))=diam(f(X))$ $(f\in C(X))$的形式为\[ ϕ(f)=τ\cdot f\circ φ+t(f)1 \qquad (f\in C(X)) \]，其中$τ$是模为1的复数，$φ:X\to X$是同胚，$t$是$C(X)$上的线性泛函.
英文摘要：
The aim of this paper is to solve a linear preserver problem on the function algebra $C(X)$. We show that in case $X$ is a first countable compact Hausdorff space, every linear bijection $ϕ:C(X)\to C(X)$ having the property that $diam(ϕ(f)(X))=diam(f(X))$ $(f\in C(X))$ is of the form \[ ϕ(f)=τ\cdot f\circ φ+t(f)1 \qquad (f\in C(X)) \] where $τ$ is a complex number of modulus 1, $φ:X\to X$ is a homeomorphism and $t$ is a linear functional on $C(X)$.
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