I am currently learning a lot about functional calculi and also attempts to axiomize them. Thus I got interested in infinite dimensional (commutative) $K$-algebras, i.e. algebras over some field $K$ (but esspecially $K \in \{\mathbb R, \mathbb C\}$) and their structure theory. The commutative case is more relevant for me but I would be also interested in the non-commutative case (where Lie-brackets might come into play). Further the finite dimensional case or cases where $\operatorname{char}(K) \neq 0$ isn't of too much interest for me.
But I really struggle to find books or other references who are dealing in depth with these kinds of objects and googling is very hard since the word "algebra" yields all sorts of references that deal with other topics in algebra. So I hope to get a good reference here.
Certain classes of simple Lie algebras in infinite dimension have been classified, e.g. Finitary simple Lie algebras. The result is given in Theorem $1.1$ and Corollary $1.2$:
Corollary 1.2 Let $F$ be an algebraically closed field of characteristic $0$. Then any infinite dimensional finitary simple Lie algebra over $F$ is isomorphic to one of the following: (1) a finitary special linear algebra ; (2) a finitary orthogonal algebra; (3) a finitary symplectic algebra.
A general classification of infinite-dimensional algebras is hopeless. There is a large literature on infinite-dimensional Lie algebras, and many books about it, dealing with structure theory, e.g., Kac's Infinite Dimensional Lie Algebras.