The algebra of supernatural matrices

The algebra of supernatural matrices is a key example in the theory of locally finite central simple algebras, which we developed in a previous paper [T. Bar-On, Sh. Gilat, E. Matzri and U. Vishne, Locally finite central simple algebras, Algebras Represent. Theory 26(2) (2023) 553–607]. This algebra has appeared under various names before, and deserves further study. Supernatural matrices are a minimal solution to the equation of unital algebras Mn(X) ^∼= X, which we compare to several similar conditions involving cancellation of matrices. Viewing a natural representation of this algebra, we show that supernatural matrices generalize both McCrimmon’s deep matrices algebra and m-petal Leavitt path algebra. We also study their simple representations.