Results in Quantum Field Theory.

Local thermal properties of the vacuum [BGLo1,GuLo4,GuLo5].
- The Unruh effect says that accelerated observers feel the vacuum as a thermal state. For conformally covariant nets of von Neumann algebras, it is shown that the vacuum is a local KMS state for observers moving along a suitable conformal flow (Bisognano-Wichmann property).
- KMS thermal equilibrium states at Hawking temperature 1/β for the Killing evolution associated with a class of stationary quantum black holes are characterized in terms of a boundedness property, namely the localized state vectors should have energy density levels increasing β-subexponentially. In particular, for a Poincaré covariant net of C^*-algebras on the Minkowski spacetime, the boundedness property for the boosts (in the Rindler wedge) is shown to be equivalent to the Bisognano-Wichmann property. In general, the boundedness condition is equivalent to a holomorphic property closely related to the Ruelle property for non-equilibrium steady states.
- For rapidly expanding universes, as the de Sitter spacetime, the Gibbons-Hawking effect states that the vacuum temperature is positive. It is shown that the vacuum can be dethermalized if the original time-evolution is replaced by a suitable quantum evolution, namely a one-parameter unitary group having only weak localization properties.
The modular reconstruction of the symmetry group [BGLo1,BGLo2,Guid1]. It is shown that the assumption of a geometric action of the modular group associated with the vacuum and the algebras of wedge regions gives rise to a canonical representation of the group of spacetime symmetries. This result apply to Minkowski and de Sitter spaces.
Modular PCT theorems [GuLo1,BGLo1,GuLo2]. It is shown that the geometric action of the modular conjugation J associated with the von Neumann algebra of a wedge region and the vacuum implies the PCT theorem for superselection sectors of a net of local algebras. Indeed, the antiunitary involution JR(π), which implements a full space-time reflection, intertwines any sector (with finite statistics) with its conjugate. It is shown that the geometric action of the modular conjugation, hence the PCT theorem, is a consequence of the geometric action of the modular group.
Spin and statistics theorems [GuLo2,Guid1,GuLo3,GLRV1].
- The proof of the spin and statistics connection is given for conformal nets on a circle, namely it is shown that the statistical phase of an irreducible sector with finite Jones' index coincides with the value of the corresponding group representation for the rotation of 2π .
- The spin and statistics connection is proved for covariant sectors of nets of local algebras on rotationally invariant spacetimes, under the assumption of the geometric action of the modular conjugation relative to suitable regions.
- For spacetimes with bifurcate Killing horizon, as is the case in the presence of black holes, the restriction of the observables to the Killing horizon together with geodesic KMS property for the Killing flow yields a conformally covariant quantum field theory on the circle and a conformal spin-statistics theorem for charged sectors localizable on the Killing horizon.
Superselection sectors [GuLo1,GLRV1].
- It is shown that superselection sectors with finite statistics in the sense of Doplicher, Haag and Roberts are Poincaré covariant if split property for space-like cones and duality for contractible causally complete regions are assumed. The same holds for topological charges, namely sectors localized in space-like cones.
- The Doplicher-Haag-Roberts theory of superselection sectors is extended to quantum field theory on arbitrary globally hyperbolic spacetimes. The statistics of a superselection sector may be defined as in flat spacetime and each charge has a conjugate charge when the spacetime possesses non-compact Cauchy surfaces. In this case, the field net and the gauge group can be constructed.
Results in Operator Algebras and Noncommutative Geometry.
Singular traces in semifinite von Neumann algebras and C^*-algebras [AGPS1, AGPS2, AGPS3, GuIs1, AGPS4,GuIs4, GuIs12, GuIs9]. Singular traces on semifinite von Neumann algebras or C^*-algebras (w.r.t. a given normal, or semicontinuous, trace tr) are those positive trace functionals vanishing on projections with finite trace tr.
- A characterization of singular traceability is given for the case of factors, namely those (tr-compact) operators T for which there exists a singular trace τ s.t. τ(T)=1 are determined in terms of a suitable asymptotic scale-invariance of their eigenvalue function, a condition generalizing the one given by Dixmier. In particular it is shown that some trace-class operators are singularly traceable. The mentioned condition is sufficient in the general case (non-trivial center).
- Type II₁ singular traces associated with ``continuous'' traces are studied, as traces defined on bimodules of measurable operators. A theory of Riemann integration on a C^*-algebra with a trace is developed, showing that type II₁ singular traces can be defined on bimodules of Riemann-measurable operators on the C^*-algebra of bounded Riemann-measurable operators.
- A characterization of the set of the exponents of singular traceability of an operator is given, showing that such set is an interval and giving explicit formulas for its endpoints.
Hausdorff measure theory in noncommutative geometry [CiGS1, GuIs3, GuIs5, GuIs8, GuIs9, GuIs11]. The concepts of Hausdorff functionals and dimension are introduced for Connes' spectral triples in noncommutative geometry, the Dixmier trace reconstructing the Hausdorff functionals (traces). It is shown that general singular traces produce Hausdorff-Besicovitch functionals (traces). For a given spectral triple they are labelled by the dimension interval, namely the interval of singular traceability exponents for the inverse of the Dirac operator. In the regular case such interval consists only of the Hausdorff dimension. In the general case it may be any subinterval of (0,+∞). Such notions are tested on fractals, showing the correspondence between classical and noncommutative concepts. In particular, tangent sets are defined for general closed subsets of Rⁿ, and it is shown that, for some class of fractals, the endpoints of the dimension interval are tangential dimensions, namely are the supremum, resp infimum, of the box dimensions of the tangent sets of the given fractal.
Asymptotic invariants for open manifolds [GuIs3,GuIs4,GuIs5,GuIs6,GuIs7]. Novikov-Shubin numbers α_p are introduced for open manifolds with bounded geometry and a given amenable exhaustion. α_p is the spectral asymptotic dimension of the p-th Laplacian measured via a semicontinuous trace on the C^*-algebra of quasi local operators acting on the L^p-bundle of p-forms. It is shown that such numbers are asymptotic invariants of the manifold, the invariance being under quasi-isometries. In the case p=0, the dimensional interpretation is strengthen by showing the coincidence of α₀ with a metric asymptotic dimension. Making use of the noncommutative Riemann integration, it is shown that such numbers are asymptotic noncommutative Hausdorff dimensions, and give rise to non trivial type II₁ singular traces.

Results in Quantum Field Theory.

Results in Operator Algebras and Noncommutative Geometry.