  We analyze the existence of vector spaces of large dimension inside the set
$\mathcal{C}(L, \K) \setminus \overline{\mathcal{A}}$, where $L$ is a compact
Hausdorff space and $\mathcal{A}$ is a self-adjoint subalgebra of $\mathcal
C(L, \K)$ that vanishes nowhere on $L$ and does not necessarily separate the
points of $L$. The results depend strongly on an equivalence relation that is
defined on the algebra $\mathcal{A}$, denoted by $\sim_{\mathcal{A}}$, and a
cardinal number that depends on $\sim_{\mathcal{A}}$ which we call the order of
$\sim_{\mathcal{A}}$. We then introduce two different cases, when the order of
$\sim_{\mathcal{A}}$ is finite or infinite. In the finite case, we show that
$\mathcal{C}(L, \K) \setminus \overline{\mathcal{A}}$ is $n$-lineable but not
$(n+1)$-lineable with $n$ being the order of $\sim_{\mathcal{A}}$. On the other
hand, when the order of $\sim_{\mathcal{A}}$ is infinite, we obtain general
results assuming, for instance, that the codimension of the closure of
$\mathcal{A}$ is infinite or when $L$ is sequentially compact. To be more
precise, we introduce the notion of the Stone-Weiestrass character of $L$ which
is closely related to the topological weight of $L$ and allows us to describe
the lineability of $\mathcal{C}(L, \K) \setminus \overline{\mathcal{A}}$ in
terms of the Stone-Weierstrass character of subsets of $\sim_{\mathcal A}$. We
also prove, in the classical case, that $(\mathcal{C}(\partial{D}, \C)
\setminus \overline{\mbox{Pol}(\partial{D})}) \cup \{0\}$ (where
$\mbox{Pol}(\partial{D})$ is the set of all complex polynomials in one variable
restricted to the boundary of the unit disk) contains an isometric copy of
$\text{Hol}(\partial{D})$ and is strongly $\mathfrak c$-algebrable, extending
previous results from the literature.
