Algebras of block Toeplitz matrices with commuting entries

Abstract: The maximal algebras of scalar Toeplitz matrices are known to be formed by generalized circulants. The identification of algebras consisting of block Toeplitz matrices is a harder problem, that has received little attention up to now. We consider the case when the block entries of the matrices belong to a commutative algebra $ \mathcal{A} $. After obtaining some general results, we classify all the maximal algebras for certain particular cases of $ \mathcal{A}$.