Supercharacters of discrete algebra groups

The supercharacter theory of a finite group was introduced by Diaconis and Isaacs as an alternative to the usual irreducible character theory, and illustrated with a construction in the case of finite algebra groups. We extend this construction to arbitrary countable discrete algebra groups, where superclasses and indecomposable supercharacters play the role of conjugacy classes and indecomposable characters, respectively. However, we adopt an ergodic theoretical point of view. The theory is then illustrated with a characterization of standard supercharacters of the group of upper unitriangular matrices over an algebraically closed field of prime characteristic.