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

Let $A$ be a real commutative Banach algebra with unity. Let $a_0in Asetminus{0}$. Let $mathbb Z a_0:={na_0}_{nin mathbb Z}$. Then, $mathbb Z a_0$ is a discrete subgroup of $A$. For any $nin mathbb Z$, the Frechet derivative of the mapping $$x , in , A mapsto x+na_0 , in , A$$ 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 $t_1in T$ and $t_2in T$ such that $t_1 ot=t_2$ 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 $mathbb R , subsetneq , C(T;mathbb R)$ 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.