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37
Noncommutative geometry, quantum fields and motives
 Colloquium Publications, Vol.55, American Mathematical Society
, 2008
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OPERATOR INTEGRALS, SPECTRAL SHIFT AND SPECTRAL FLOW
, 2007
"... We present a new and simple approach to the theory of multiple operator integrals that applies to unbounded operators affiliated with general von Neumann algebras. For semifinite von Neumann algebras we give applications to the Fréchet differentiation of operator functions that sharpen existing re ..."
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Cited by 25 (5 self)
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We present a new and simple approach to the theory of multiple operator integrals that applies to unbounded operators affiliated with general von Neumann algebras. For semifinite von Neumann algebras we give applications to the Fréchet differentiation of operator functions that sharpen existing results, and establish the BirmanSolomyak representation of the spectral shift function of M.G. Krein in terms of an average of spectral measures in the type II setting. We also exhibit a surprising connection between the spectral shift function and spectral flow.
THE DIXMIER TRACE AND ASYMPTOTICS OF ZETA FUNCTIONS
, 2006
"... We obtain general theorems which enable the calculation of the Dixmier trace in terms of the asymptotics of the zeta function and of the trace of the heat semigroup. We prove our results in a general semifinite von Neumann algebra. We find for p> 1 that the asymptotics of the zeta function det ..."
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Cited by 21 (11 self)
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We obtain general theorems which enable the calculation of the Dixmier trace in terms of the asymptotics of the zeta function and of the trace of the heat semigroup. We prove our results in a general semifinite von Neumann algebra. We find for p> 1 that the asymptotics of the zeta function determines an ideal strictly larger than L p, ∞ on which the Dixmier trace may be defined. We also establish stronger versions of other results on Dixmier traces and zeta functions.
Twisted cyclic theory, equivariant KKtheory and KMS states
"... Recently, examples of an index theory for KMS states of circle actions were discovered, [9, 13]. We show that these examples are not isolated. Rather there is a general framework in which we use KMS states for circle actions on a C ∗algebra A to construct Kasparov modules and semifinite spectral tr ..."
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Cited by 20 (6 self)
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Recently, examples of an index theory for KMS states of circle actions were discovered, [9, 13]. We show that these examples are not isolated. Rather there is a general framework in which we use KMS states for circle actions on a C ∗algebra A to construct Kasparov modules and semifinite spectral triples. By using a residue construction analogous to that used in the semifinite local index formula we associate to these triples a twisted cyclic cocycle on a dense subalgebra of A. This cocycle pairs with the equivariant KKtheory of the mapping cone algebra for the inclusion of the fixed point algebra of the circle action in A. The pairing is expressed in terms of spectral flow between a pair of unbounded self adjoint operators that are Fredholm in the semifinite sense. A novel aspect of our work is the discovery of an eta cocycle that forms a part of our twisted residue cocycle. To illustrate our theorems we observe firstly that they incorporate the results in [9, 13] as special cases. Next we use the ArakiWoods IIIλ representations of the Fermion algebra to show that there are examples which are not CuntzKrieger systems. 1.
Spectral action on noncommutative torus
 J. Noncommut. Geom
"... Dedicated to Alain Connes on the occasion of his 60th birthday The spectral action on noncommutative torus is obtained, using a Chamseddine– Connes formula via computations of zeta functions. The importance of a Diophantine condition is outlined. Several results on holomorphic continuation of series ..."
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Cited by 16 (8 self)
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Dedicated to Alain Connes on the occasion of his 60th birthday The spectral action on noncommutative torus is obtained, using a Chamseddine– Connes formula via computations of zeta functions. The importance of a Diophantine condition is outlined. Several results on holomorphic continuation of series of holomorphic functions are obtained in this context.
Twisted cyclic theory and an index theory for the gauge invariant KMS state on Cuntz algebras
"... This paper presents, by example, an index theory appropriate to algebras without trace. Whilst we work exclusively with the Cuntz algebras the exposition is designed to indicate how to develop a general theory. Our main result is an index theorem (formulated in terms of spectral flow) using a twiste ..."
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Cited by 15 (2 self)
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This paper presents, by example, an index theory appropriate to algebras without trace. Whilst we work exclusively with the Cuntz algebras the exposition is designed to indicate how to develop a general theory. Our main result is an index theorem (formulated in terms of spectral flow) using a twisted cyclic cocycle where the twisting comes from the modular automorphism group for the canonical gauge action on the Cuntz algebra. We introduce a modified K1group of the Cuntz algebra so as to pair with this twisted cocycle. As a corollary we obtain a noncommutative geometry interpretation for Araki’s notion of relative entropy in this example. We also note the connection of this example to the theory of noncommutative manifolds. Contents
A WALK IN THE NONCOMMUTATIVE GARDEN
"... 2. Handling noncommutative spaces in the wild: basic tools 2 3. Phase spaces of microscopic systems 6 4. Noncommutative quotients 9 ..."
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2. Handling noncommutative spaces in the wild: basic tools 2 3. Phase spaces of microscopic systems 6 4. Noncommutative quotients 9
Spectral flow is the integral of one forms on the Banach manifold of self adjoint Fredholm operators
"... Abstract. The aim of this paper is to prove the statement which forms the title. One may trace the issue back to the 1974 Vancouver ICM address of I.M. Singer. Our main theorem gives an analytic formula for the spectral flow along a smooth path of selfadjoint bounded BreuerFredholm operators in a ..."
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Abstract. The aim of this paper is to prove the statement which forms the title. One may trace the issue back to the 1974 Vancouver ICM address of I.M. Singer. Our main theorem gives an analytic formula for the spectral flow along a smooth path of selfadjoint bounded BreuerFredholm operators in a semifinite von Neumann algebra. The formula has a geometric interpretation which derives from the proof. Namely we start with a one form on the Banach manifold of all bounded selfadjoint BreuerFredholm operators and show that by integrating it along a sufficiently smooth path one calculates the spectral flow along this path. The original context for this result concerned paths of unbounded selfadjoint Fredholm operators. We therefore prove an analogous formula for spectral flow in the unbounded case as well. The proof is a synthesis of key contributions by previous authors, whom we acknowledge in detail in the introduction, combined with an additional important recent advance in the differential calculus of functions of noncommuting operators. 1.
Modular Theory, NonCommutative Geometry and Quantum Gravity
, 2010
"... 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 noncommutative geometry aiming at the reconstruction of spectral geometries from an operational formalism of state ..."
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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 noncommutative 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 noncommutative geometry and physics and to the detailed discussion of the main foundational issues raised by the proposal.