Remarks on C*-discrete inclusions

We show under certain constraints that $A$-valued semicircular systems give rise to C*-discrete inclusions, and thus are crossed products by an action of a tensor category. Along the way, we show the set of single algebraic generators of a dualizable bimodule forms an open subset. Furthermore, we obtain a Galois correspondence between the lattice of intermediate C*-discrete subalgebras intermediate to a given irreducible C*-discrete inclusion, and characterize these as targets of compatible expectations. Finally, we define ``freeness'' for actions of tensor categories on C*-algebras, and show simplicity is preserved under taking reduced crossed products.
Comment: This is a split part of the article Discrete Inclusions of C*-algebras [2305.05072v1] by the same authors, which was partitioned and replaced by [2305.05072v2] to better lay out the results therein