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.

.

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.

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.

.

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.

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.

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.

.

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.