$\newcommand{\Z}{\mathbb Z}$ By an operator on $\ell^\infty(\Z)$, I mean a bounded linear map $\ell^\infty(\Z)\to\ell^\infty(\Z)$. (Note that I am not assuming weak-star continuity.) By shift-invariance, I mean that such a map $T$ should commute with the shift operator $S:\ell^\infty(\Z)\to\ell^\infty(\Z)$ that is defined by $(S\xi)(n) = \xi(n-1)$.
Let $B$ denote the algebra of all shift-invariant operators on $\ell^\infty(\Z)$. It is known, as a special case of general Banach-algebraic results, that $B$ may be identified with $\ell^1(\Z)^{**}$ equipped with the first Arens product; in particular, $B$ is highly non-commutative, since it contains a copy of the semigroup $\beta\Z$. (See this question and its answer for some related background.)
Question 1. Does there exist an infinite-dimensional Banach space $X$ and a continuous representation $\pi: B \to {\cal B}(X)$ which is irreducible, in the strict algebraic sense (i.e. the only non-zero $\pi(B)$-invariant linear subspace of $X$ is $X$ itself)?
Question 2. As in Question 1, but with "irreducible" weakened to "topologically irreducible".
