Matrix bispectrality of full rank one algebras

We consider a noncommutative version of the bispectral problem proposed originally by Grünbaum. The latter had a number of unexpected connections with soliton theory and special functions of interest in Mathematical Physics. We take an algebraic route to describe the symmetry structure of the noncommutative algebras of bispectral operators by using the notion of full rank one algebra. By using this concept, we provide a general framework and derive a method to verify if one such matrix polynomial sub-algebra is bispectral. We give two examples illustrating the method. In the first one, we consider the eigenvalue to be scalar-valued, whereas in the second one, we assume it to be matrix-valued. In the former example we put forth a Pierce decomposition of that algebra. As a byproduct, we answer positively a conjecture of Grünbaum concerning certain noncommutative matrix algebras associated to the bispectral problem.