A Banach lattice $X$ is said to be an AM-space if $\|x \vee y\|=\max\{\|x\|, \|y\|\}$ for every $x,y \in X_+$. A classical theorem of Kakutani establishes that AM-spaces are precisely the closed sublattices of $C(K)$ spaces. This result provides an intrinsic characterization of the closed sublattices of $C(K)$. However, since $C(K)$ naturally carries a Banach algebra structure as well, a natural question arises: Can one characterize the closed subspaces of $C(K)$ that are simultaneously sublattices and subalgebras? In this talk, we demonstrate that such a characterization is indeed possible by adding a simple algebraic constraint to the AM-space condition. Introducing this interplay between the AM-space and algebraic structures will require a new characterization of the AM-space property.
(This is joint work with Pedro Tradacete.)