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 norm-closed 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.

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PS: The following are equivalent for $A$
 - Every closed maximal left ideal of $A$ has infinite codimension.
 - Every primitive ideal of $A$ has infinite codimension.
 - Every irreducible representation of $A$ is infinite dimensional.