AI 中文总结
该研究确定了局部紧豪斯多夫空间上连续函数巴拿赫代数间范数环同构的具体形式，实域情形还将Gelfand-Kolmogoroff定理的范数版本推广到局部紧场景。
AI 中文摘要
设X和Y为局部紧豪斯多夫空间，𝕂属于{ℝ,𝔽}。我们称双射T:C₀(X,𝕂)→C₀(Y,𝕂)为范数环同构，若对任意f,g∈C₀(X,𝕂)，均满足‖T(f+g)‖=‖T(f)+T(g)‖且‖T(fg)‖=‖T(f)T(g)‖。我们确定这类映射的形式：当𝕂=𝔽时，在附加对任意f∈C₀(X,𝔽)满足‖T(overline{f})‖=‖T(f)‖的条件下，存在连续函数w:Y→{λ∈𝔽:|λ|=1}、同胚φ:Y→X及Y的开闭子集Y₀，使得对任意f∈C₀(X,𝔽)和y∈Y，T(f)(y)在y∈Y₀时为w(y)f(φ(y))，在y∈Y\backslash Y₀时为w(y)overline{f(φ(y))}；当𝕂=ℝ时，存在连续函数w:Y→{±1}及同胚φ:Y→X，使得T(f)(y)=w(y)f(φ(y))。特别地，实域情形将Gelfand-Kolmogoroff定理的范数版本推广到了局部紧情形。
英文摘要
Let $X$ and $Y$ be locally compact Hausdorff spaces and let $\mathbb{K}\in \{\mathbb{R},\mathbb{C}\}$. We say that a bijection $T\colon C_0(X,\mathbb{K})\to C_0(Y,\mathbb{K})$ is a ring isomorphism in norm if \[ \|T(f+g)\|=\|T(f)+T(g)\|,\qquad \|T(fg)\|=\|T(f)T(g)\| \] for every $f,g\in C_0(X,\mathbb{K})$. We determine the form of such maps. When $\mathbb{K}=\mathbb{C}$, under the additional assumption that $\|T(\overline f)\|=\|T(f)\|$ for every $f\in C_0(X,\mathbb{C})$, there exist a continuous function $w\colon Y\to\{λ\in\mathbb{C}:|λ|=1\}$, a homeomorphism $φ\colon Y\to X$, and a closed and open subset $Y_0\subset Y$ such that \[ T(f)(y)= \begin{cases} w(y)f(φ(y)),& y\in Y_0,\\ w(y)\overline{f(φ(y))},& y\in Y\setminus Y_0, \end{cases} \] for every $f\in C_0(X,\mathbb{C})$ and $y\in Y$. When $\mathbb{K}=\mathbb{R}$, there exist a continuous function $w\colon Y\to\{\pm1\}$ and a homeomorphism $φ\colon Y\to X$ such that \[ T(f)(y)=w(y)f(φ(y)) \] for every $f\in C_0(X,\mathbb{R})$ and $y\in Y$. In particular, the real case extends the norm version of the Gelfand--Kolmogoroff theorem to the locally compact setting.
