A remark on embedding of a cylinder on a real commutative Banach algebra

Abstract

Let $A$ be a real commutative Banach algebra with unity. Let $a_0\in A\setminus \{0\}$. Let $\mathbb{Z}{a}_0:=\{n{a}_0{\}}_{n\in \mathbb{Z}}$. Then, $\mathbb{Z}{a}_0$ is a discrete subgroup of $A$. For any $n\in \mathbb{Z}$, the Frechet derivative of the mapping $$x\in A\mapsto x+n{a}_0\in A$$ is the identity map on $A$ and, especially, an $A$-linear transformation on $A$. So, the quotient group $A/(\mathbb{Z}{a}_0)$ is a $1$-dimensional $A$-manifold and the covering projection $$x\in A\mapsto x+\mathbb{Z}{a}_0\in A/(\mathbb{Z}{a}_0)$$ is an $A$-map. We call $A/(\mathbb{Z}{a}_0)$ the $1$-dimensional $A$-cylinder by $a_0$. Let $T$ be a compact Hausdorff space. Suppose that there exist $t_1\in T$ and $t_2\in T$ such that $t_1\ne t_2$ holds. Then, the set $C(T;\mathbb{R})$ of all real-valued continuous functions on $T$ is a real commutative Banach algebra with unity and $\mathbb{R}⊊C(T;\mathbb{R})$ holds. In this paper, we show that there exists $a_0\in C(T;\mathbb{R})\setminus \mathbb{R}$ such that for any $k\in \mathbb{N}$, the $1$-dimensional $C(T;\mathbb{R})$-cylinder $(C(T;\mathbb{R}))/(\mathbb{Z}{a}_0)$ by $a_0$ cannot be embedded in the finite direct product space $(C(T;\mathbb{R}){)}^k$ as a $C(T;\mathbb{R})$-submanifold.