Let $M$ be a n-dimensional manifold, $C(M,\mathbb{R})$ be the function space of continuous function from $M$ to $\mathbb{R}$. What kind of properties should $C(M,\mathbb{R})$ has, to reflect the manifold structure?

Denote a L-shape line be $X$, we know it is not a manifold, but what difference can we observe from the algebras of continuous function compared to circle? i.e. What properties does $C(M,\mathbb{R})$ has and $C(X,\mathbb{R})$ does not? In this case their topological dimension is the same, so they should have same stable rank in terms of $C^*$-algebras.