Remarks on C*-discrete inclusions

TL;DR

This paper explores C*-discrete inclusions arising from semicircular systems, establishing a Galois correspondence for subalgebras, and defining freeness for tensor category actions, with implications for simplicity of crossed products.

Contribution

It introduces a framework connecting semicircular systems to C*-discrete inclusions and characterizes intermediate subalgebras via a Galois correspondence, also defining freeness for tensor category actions.

Findings

C*-discrete inclusions can be generated by semicircular systems under certain conditions.

A Galois correspondence between intermediate subalgebras and expectations is established.

Freeness of tensor category actions preserves simplicity in crossed products.

Abstract

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.