Remarks on ${\mathrm{C}}^{\ast }$-discrete inclusions

Abstract

We obtain a Galois correspondence between the lattice of intermediate ${\mathrm{C}}^{\ast }$-discrete subalgebras intermediate to a given irreducible ${\mathrm{C}}^{\ast }$-discrete inclusion and characterize these as targets of compatible expectations under a traciality condition. Furthermore, we define freeness for actions of tensor categories on ${\mathrm{C}}^{\ast }$-algebras and show that simplicity is preserved under taking reduced crossed products. Finally, we show under certain constraints that $A$-valued semicircular systems give rise to ${\mathrm{C}}^{\ast }$-discrete inclusions and thus are crossed products by an action of a tensor category. Along the way, we show that the set of single algebraic generators of a dualizable bimodule forms an open subset.