Let $X$ be a (topological) space and let ${\mathscr I}$ be an ideal in $X$, that is, a collection of subsets of $X$ which contains all subsets of its elements and is closed under finite unions. The elements of ${\mathscr I}$ are called null. The space $X$ is locally null if each $x$ in $X$ has a null neighborhood in $X$. Let $C_b(X)$ denote the normed algebra of all continuous bounded real-valued mappings on $X$ equipped with the supremum norm, $C_0(X)$ denote the subalgebra of $C_b(X)$ consisting of elements vanishing at infinity and $C_{00}(X)$ the subalgebra of $C_b(X)$ consisting of elements with compact support.
We study the normed subalgebra $C^{\mathscr I}_{00}(X)$ of $C_b(X)$ consisting of all $f$ in $C_b(X)$ whose support has a null neighborhood in $X$, and the Banach subalgebra $C^{\mathscr I}_0(X)$ of $C_b(X)$ consisting of all $f$ in $C_b(X)$ such that $|f|^{-1}([1/n,\infty))$ has a null neighborhood in $X$ for all positive integer $n$. We prove that if $X$ is a normal locally null space then $C^{\mathscr I}_{00}(X)$ and $C^{\mathscr I}_0(X)$ are respectively isometrically isomorphic to $C_{00}(Y)$ and $C_0(Y)$ for a unique locally compact Hausdorff space $Y$; furthermore, $C^{\mathscr I}_{00}(X)$ is dense in $C^{\mathscr I}_0(X)$. We construct $Y$ explicitly as a subspace of the Stone--Čech compactification $βX$ of $X$. The space $Y$ is locally compact (and countably compact, in certain cases), contains $X$ densely, and in specific cases turns out to be familiar subspaces of $βX$. The known topological structure of $Y$ enables us to establish several commutative Gelfand--Naimark type theorems and derive results not generally expected to be deducible from the standard Gelfand theory.
Uniform approximation of continuous mappings by smooth mappings with no critical points on Hilbert manifolds
证明了从可分无限维Hilbert空间$X$到$\mathbb{R}^{m}$的每一个连续映射都能用$C^\infty$光滑映射{ \em with no临界点}一致逼近.这类结果可以看作是Morse-Sard定理的一种很强的近似形式。主要定理的一些推论如下。$X$的每两个不相交的闭子集可以被一个一维光滑流形分离，该流形是一个没有临界点的光滑函数的水平集;这一事实可以被看作是Hahn-Banach定理的几何版本的非线性模拟。特别地，$X$中的每一个闭集都可以被开集一致逼近，所述开集的边界是$X$的$C^\infty$光滑的一维余维子流形。最后，由于每个希尔伯特流形都是希尔伯特空间的一个开子集的同构，所以如果用任何在$X$上建模的光滑流形$M$替换希尔伯特空间$X$，所有这些结果仍然成立。
