Let $X$ be a reflexive Banach space, and let $B(X)$ denote the set of all bounded linear operators $X\to X$.
Does there exist a subalgebra $A\subseteq B(X)$ with the following properties?
 1. A is unital.
 2. A is reflexive as a Banach space.
 3. *Every* closed maximal left ideal has *infinite* codimension.