The dynamical structure of partial group algebras with relations, with applications to subshift algebras

In this paper, we introduce partial group algebras with relations in a purely algebraic framework. Given a group and a set of relations, we define an algebraic partial action and prove that the resulting partial skew group ring is isomorphic to the associated partial group algebra with relations. Under suitable conditions — which always holds if the base ring is a field — we demonstrate that the partial skew group ring can also be described using a topological partial action. Furthermore, we show how subshift algebras can be realized as partial group algebras with relations. Using the topological partial action, we describe simplicity of subshift algebras in terms of the underlying dynamics of the subshift.