A few remarks on Pimsner–Popa bases and regular subfactors of depth 2

We prove that a finite index regular inclusion of $II_1$ -factors with commutative first relative commutant is always a crossed product subfactor with respect to a minimal action of a biconnected weak Kac algebra. Prior to this, we prove that every finite index inclusion of $II_1$ -factors which is of depth 2 and has simple first relative commutant (respectively, is regular and has commutative or simple first relative commutant) admits a two-sided Pimsner–Popa basis (respectively, a unitary orthonormal basis).

Strong Morita equivalence for inclusions of $C^*$-algebras induced by twisted actions of a countable discrete group

We consider two twisted actions of a countable discrete group on $\sigma$-unital $C^*$-algebras. Then by taking the reduced crossed products, we get two inclusions of $C^*$-algebras. We suppose that they are strongly Morita equivalent as inclusions of $C^*$-algebras. Also, we suppose that one of the inclusions of $C^*$-algebras is irreducible, that is, the relative commutant of one of the $\sigma$-unital $C^*$-algebra in the multiplier $C^*$-algebra of the reduced twisted crossed product is trivial. We show that the two actions are then strongly Morita equivalent up to some automorphism of the group.

Duality for compact group actions on operator algebras and applications: Irreducible inclusions and Galois correspondence

We consider compact group actions on C*- and W*-algebras. We prove results that relate the duality property of the action (as defined in the Introduction) with other relevant properties of the system such as the relative commutant of the fixed point algebras being trivial (called the irreducibility of the inclusion) and also to the Galois correspondence between invariant C*-subalgebras containing the fixed point algebra and the class of closed normal subgroups of the compact group.

On diagonal quasi-free automorphisms of simple Cuntz-Krieger algebras

We show that an outer action of a finite abelian group on a simple Cuntz-Krieger algebra is strongly approximately inner in the sense of Izumi if the action is given by diagonal quasi-free automorphisms and the associated matrix is aperiodic. This is achieved by an approximate cohomology vanishing-type argument for the canonical shift restricted to the relative commutant of the set of domain projections of the canonical generating isometries in the fixed point algebra.

The Relative Drinfeld Commutant of a Fusion Category and α-Induction

We establish a correspondence among simple objects of the relative commutant of a full fusion subcategory in a larger fusion category in the sense of Drinfeld, irreducible half-braidings of objects in the larger fusion category with respect to the fusion subcategory, and minimal central projections in the relative tube algebra. Based on this, we explicitly compute certain relative Drinfeld commutants of fusion categories arising from α-induction for braided subfactors. We present examples arising from chiral conformal field theory.

REALIZATION OF MINIMAL C*-DYNAMICAL SYSTEMS IN TERMS OF CUNTZ–PIMSNER ALGEBRAS

In the present article, we provide several constructions of C*-dynamical systems [Formula: see text] with a compact group [Formula: see text] in terms of Cuntz–Pimsner algebras. These systems have a minimal relative commutant of the fixed-point algebra [Formula: see text] in [Formula: see text], i.e. [Formula: see text], where [Formula: see text] is the center of [Formula: see text], which is assumed to be non-trivial. In addition, we show in our models that the group action [Formula: see text] has full spectrum, i.e. any unitary irreducible representation of [Formula: see text] is carried by a [Formula: see text]-invariant Hilbert space within [Formula: see text]. First, we give several constructions of minimal C*-dynamical systems in terms of a single Cuntz–Pimsner algebra [Formula: see text] associated to a suitable [Formula: see text]-bimodule ℌ. These examples are labelled by the action of a discrete Abelian group ℭ (which we call the chain group) on [Formula: see text] and by the choice of a suitable class of finite dimensional representations of [Formula: see text]. Second, we present a more elaborate contruction, where now the C*-algebra [Formula: see text] is generated by a family of Cuntz–Pimsner algebras. Here, the product of the elements in different algebras is twisted by the chain group action. We specify the various constructions of C*-dynamical systems for the group [Formula: see text], N ≥ 2.