Given a unital C* algebra $A$, or more specifically, $A:=\mathcal L(H)$ be the bounded linear operators of a certain Hilbert space $H$.
How do its subalgebras look like if one only considers the pure algebra structure? It is possible that $\exists A_B\subset A \lor A_*\subset A$ (unital) subalgebras of $A$, such that $A_B$ is only a Banach algebra and not * algebra, or that $A_*$ is only a * algebra and no Banach algebra?
Also, maybe this belongs to another question, but I read very often that unital C* algebras can either be embedded (*-isometrically and isomorphically) into a subalgebra of $\mathcal L(H)$ for some Hilbert space $H$, or into $\mathcal C_0(X)$ of a locally compact Hausdorff space $X$. But doesn't that latter case (for commutative cases) also belong to the more general non-commutative case (hence the first case)?
Thanks in advance!