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Noncommutative geometry, quantum fields and motives
 Colloquium Publications, Vol.55, American Mathematical Society
, 2008
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Duality of compact groups and Hilbert C*systems for C*algebras with a nontrivial center
, 2004
"... In the present paper we prove a duality theory for compact groups in the case when the C*algebra A, the fixed point algebra of the corresponding Hilbert C*system (F, G), has a nontrivial center Z ⊃ C and the relative commutant satisfies the minimality condition A ′ ∩ F = Z, as well as a technical ..."
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Cited by 15 (2 self)
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In the present paper we prove a duality theory for compact groups in the case when the C*algebra A, the fixed point algebra of the corresponding Hilbert C*system (F, G), has a nontrivial center Z ⊃ C and the relative commutant satisfies the minimality condition A ′ ∩ F = Z, as well as a technical condition called regularity. The abstract characterization of the mentioned Hilbert C*system is expressed by means of an inclusion of C*categories TC < T, where TC is a suitable DRcategory and T a full subcategory of the category of endomorphisms of A. Both categories have the same objects and the arrows of T can be generated from the arrows of TC and the center Z. A crucial new element that appears in the present analysis is an abelian group C(G), which we call the chain group of G, and that can be constructed from certain equivalence relation defined on ̂ G, the dual object of G. The chain group, which is isomorphic to the character group of the center of G, determines the action of irreducible endomorphisms of A when restricted to Z. Moreover, C(G) encodes the possibility of defining a symmetry ǫ also for the
Superselection structures for C*algebras with nontrivial center
, 1996
"... We present and proof some results in the context of superselection theory with the following conditions fixed: the observable algebra A has a nontrivial center Z and its relative commutant w.r.t. the field algebra F coincides with Z , i.e. we have A 0 " F = Z oe C 1l . In this frame we study Hi ..."
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Cited by 14 (4 self)
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We present and proof some results in the context of superselection theory with the following conditions fixed: the observable algebra A has a nontrivial center Z and its relative commutant w.r.t. the field algebra F coincides with Z , i.e. we have A 0 " F = Z oe C 1l . In this frame we study Hilbert C*systems with a compact group. We propose a generalization of the notion of an irreducible endomorphism and study the influence of the sector structure on Z . Finally we give several characterizations of the stabilizer of A.
Quantum fields and motives
 J. Geom. Phys
, 2006
"... The main idea of renormalization is to correct the original Lagrangian of a quantum field theory by an infinite series of counterterms, labelled by the Feynman graphs that encode the combinatorics of the perturbative expansion of the theory. These counterterms have the effect of cancelling the ultra ..."
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Cited by 10 (2 self)
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The main idea of renormalization is to correct the original Lagrangian of a quantum field theory by an infinite series of counterterms, labelled by the Feynman graphs that encode the combinatorics of the perturbative expansion of the theory. These counterterms have the effect of cancelling the ultraviolet divergences. Thus, in the procedure of perturbative renormalization, one introduces a counterterm C(Γ) in the initial Lagrangian for every divergent one particle irreducible (1PI) Feynman diagram Γ. In the case of a renormalizable theory, all the necessary counterterms C(Γ) can be obtained by modifying the numerical parameters that appear in the original Lagrangian. It is possible to modify these parameters and replace them by (divergent) series, since they are not observable, unlike actual physical quantities that have to be finite. One of the fundamental difficulties with any renormalization procedure is a systematic treatment of nested and overlapping divergences in multiloop diagrams. Dimensional regularization and minimal subtraction. One of the most effective renormalization techniques in quantum field theory is
Towards a typed geometry of interaction
, 2005
"... We introduce a typed version of Girard’s Geometry of Interaction, called Multiobject GoI (MGoI) semantics. We give an MGoI interpretation for multiplicative linear logic (MLL) without units which applies to new kinds of models, including finite dimensional vector spaces. For MGoI (i) we develop a v ..."
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Cited by 4 (1 self)
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We introduce a typed version of Girard’s Geometry of Interaction, called Multiobject GoI (MGoI) semantics. We give an MGoI interpretation for multiplicative linear logic (MLL) without units which applies to new kinds of models, including finite dimensional vector spaces. For MGoI (i) we develop a version of partial traces and trace ideals (related to previous work of Abramsky, Blute, and Panangaden); (ii) we do not require the existence of a reflexive object for our interpretation (the original GoI 1 and 2 were untyped and hence involved a bureaucracy of domain equation isomorphisms); (iii) we introduce an abstract notion of orthogonality (related to work of Hyland and Schalk) and use this to develop a version of Girard’s theory of types, datum and algorithms in our setting, (iv) we prove appropriate Soundness and Completeness Theorems for our interpretations in partially traced categories with orthogonality; (v) we end with an application to completeness of (the original) untyped GoI in a unique decomposition category.
Abstract
, 2008
"... Dual group actions on C*–algebras and their description by Hilbert extensions ..."
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Dual group actions on C*–algebras and their description by Hilbert extensions