Strictly Convex Banach Algebras

We discuss two facets of the interaction between geometry and algebra in Banach algebras. In the class of unital Banach algebras, there is essentially one known example which is also strictly convex as a Banach space. We recall this example, which is finite-dimensional, and consider the open question of generalising it to infinite dimensions. In C*-algebras, we exhibit one striking example of the tighter relationship that exists between algebra and geometry there.