On discrete twisted C*-dynamical systems, Hilbert C*-modules and regularity

We first give an overview of the basic theory for discrete unital twisted C*-dynamical systems and their covariant representations on Hilbert C*-modules. After introducing the notion of equivariant representations of such systems and their product with covariant representations, we prove a kind of Fell absorption principle saying that the product of an induced regular equivariant representation with a covariant faithful representation is weakly equivalent to an induced regular covariant representation. This principle is the key to our main result, namely that a certain property, formally weaker than Exel's approximation property, ensures that the system is regular, i.e., the associated full and reduced C*-crossed products are canonically isomorphic.Comment: Final version, to appear in Muenster J. Math. A permanence result for the weak approximation property, some corollaries of it and two examples have been added to Section 5. Some side results in Section 4 have been removed and will be included in a subsequent paper. The Introduction has also been partly rewritte

Scaling limit for subsystems and Doplicher-Roberts reconstruction

Given an inclusion $B \subset F$ of (graded) local nets, we analyse the structure of the corresponding inclusion of scaling limit nets $B_0 \subset F_0$, giving conditions, fulfilled in free field theory, under which the unicity of the scaling limit of $F$ implies that of the scaling limit of $B$. As a byproduct, we compute explicitly the (unique) scaling limit of the fixpoint nets of scalar free field theories. In the particular case of an inclusion $A \subset B$ of local nets with the same canonical field net $F$, we find sufficient conditions which entail the equality of the canonical field nets of $A_0$ and $B_0$.Comment: 31 page

Automorphisms of the UHF algebra that do not extend to the Cuntz algebra

Automorphisms of the canonical core UHF-subalgebra F_n of the Cuntz algebra O_n do not necessarily extend to automorphisms of O_n. Simple examples are discussed within the family of infinite tensor products of (inner) automorphisms of the matrix algebras M_n. In that case, necessary and sufficient conditions for the extension property are presented. It is also addressed the problem of extending to O_n the automorphisms of the diagonal D_n, which is a regular MASA with Cantor spectrum. In particular, it is shown the existence of product-type automorphisms of D_n that are not extensible to (possibly proper) endomorphisms of O_n

Modular Theory, Non-Commutative Geometry and Quantum Gravity

This paper contains the first written exposition of some ideas (announced in a previous survey) on an approach to quantum gravity based on Tomita-Takesaki modular theory and A. Connes non-commutative geometry aiming at the reconstruction of spectral geometries from an operational formalism of states and categories of observables in a covariant theory. Care has been taken to provide a coverage of the relevant background on modular theory, its applications in non-commutative geometry and physics and to the detailed discussion of the main foundational issues raised by the proposal.Comment: Special Issue "Noncommutative Spaces and Fields

Conformal nets and KK-theory

Given a completely rational conformal net A on the circle, its fusion ring acts faithfully on the K_0-group of a certain universal C*-algebra associated to A, as shown in a previous paper. We prove here that this action can actually be identified with a Kasparov product, thus paving the way for a fruitful interplay between conformal field theory and KK-theory

Diagonal automorphisms of the 22-adic ring C∗C^*-algebra

The $2$-adic ring $C^*$-algebra $\mathcal{Q}_2$ naturally contains a copy of the Cuntz algebra $\mathcal{O}_2$ and, a fortiori, also of its diagonal subalgebra $\mathcal{D}_2$ with Cantor spectrum. This paper is aimed at studying the group ${\rm Aut}_{\mathcal{D}_2}(\mathcal{Q}_2)$ of the automorphisms of $\mathcal{Q}_2$ fixing $\mathcal{D}_2$ pointwise. It turns out that any such automorphism leaves $\mathcal{O}_2$ globally invariant. Furthermore, the subgroup ${\rm Aut}_{\mathcal{D}_2}(\mathcal{Q}_2)$ is shown to be maximal abelian in ${\rm Aut}(\mathcal{Q}_2)$. Saying exactly what the group is amounts to understanding when an automorphism of $\mathcal{O}_2$ that fixes $\mathcal{D}_2$ pointwise extends to $\mathcal{Q}_2$. A complete answer is given for all localized automorphisms: these will extend if and only if they are the composition of a localized inner automorphism with a gauge automorphism.Comment: Improved exposition and corrected some typos and inaccuracie