Searching for linear structures in the failure of the Stone-Weierstrass theorem

We analyze the existence of vector spaces of large dimension inside the set C(L, ) Ā, where L is a compact Hausdorff space and A is a self-adjoint subalgebra of C(L, ) 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 A, denoted by ~A, and a cardinal number that depends on ~A which we call the order of ~A. We then introduce two different cases, when the order of ~A is finite or infinite. In the finite case, we show that C(L, ) Ā is n-lineable but not $(n+1)$-lineable with n being the order of ~A. On the other hand, when the order of ~A is infinite, we obtain general results assuming, for instance, that the codimension of the closure of 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 C(L, ) Ā in terms of the Stone-Weierstrass character of subsets of ~A. We also prove, in the classical case, that (C(∂D, ) Pol(∂D)) ∪ \0\ (where Pol(∂D) is the set of all complex polynomials in one variable restricted to the boundary of the unit disk) contains an isometric copy of Hol(∂D) and is strongly c-algebrable, extending previous results from the literature.