On the locally functionally countable subalgebra of C(X) on locally functionally countable subalgebra of C(X)

Abstract

Let $C_c(X)=\{f\in C(X) : |f(X)|\leq \aleph_0\}$, $C^F(X)=\{f\in C(X): |f(X)|<\infty\}$, and $L_c(X)=\{f\in C(X) : \overline{C_f}=X\}$, where $C_f$ is the union of all open subsets $U\subseteq X$ such that $|f(U)|\leq\aleph_0$, and $C_F(X)$ be the socle of $C(X)$ (i.e., the sum of minimal ideals of $C(X)$). It is shown that if $X$ is a locally compact space, then $L_c(X)=C(X)$ if and only if $X$ is locally scattered. We observe that $L_c(X)$ enjoys most of the important properties which are shared by $C(X)$ and $C_c(X)$. Spaces $X$ such that $L_c(X)$ is regular (von Neumann) are characterized. Similarly to $C(X)$ and $C_c(X)$, it is shown that $L_c(X)$ is a regular ring if and only if it is $\aleph_0$-selfinjective. We also determine spaces $X$ such that ${\rm Soc}{\big(}L_c(X){\big)}=C_F(X)$ (resp., ${\rm Soc}{\big(}L_c(X){\big)}={\rm Soc}{\big(}C_c(X){\big)}$). It is proved that if $C_F(X)$ is a maximal ideal in $L_c(X)$, then $C_c(X)=C^F(X)=L_c(X)\cong \prod\limits_{i=1}^n R_i$, where $R_i=\mathbb R$ for each $i$, and $X$ has a unique infinite clopen connected subset. The converse of the latter result is also given. The spaces $X$ for which $C_F(X)$ is a prime ideal in $L_c(X)$ are characterized and consequently for these spaces, we infer that $L_c(X)$ can not be isomorphic to any $C(Y)$.

Keywords

• functionally countable space
• socle, zero-dimensional space
• scattered space
• locally scattered space, $\aleph_0$-selfinjective.