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Summability For Nonunital Spectral Triples, Adam C. Rennie Sep 2015

Summability For Nonunital Spectral Triples, Adam C. Rennie

Associate Professor Adam Rennie

This paper examines the issue of summability for spectral triples for the class of nonunital algebras. For the case of (p, -) summability, we prove that the Dixmier trace can be used to define a (semifinite) trace on the algebra of the spectral triple. We show this trace is well-behaved, and provide a criteria for measurability of an operator in terms of zeta functions. We also show that all our hypotheses are satisfied by spectral triples arising from eodesically complete Riemannian manifolds. In addition, we indicate how the Local Index Theorem of Connes-Moscovici extends to our nonunital setting.


Smoothness And Locality For Nonunital Spectral Triples, Adam C. Rennie Sep 2015

Smoothness And Locality For Nonunital Spectral Triples, Adam C. Rennie

Associate Professor Adam Rennie

To deal with technical issues in noncommuntative geometry for nonunital algebras, we introduce a useful class of algebras and their modules. Thes algebras and modules allo us to extend all of the smoothness results for spectral triples to the nonunital case. In addition, we show that smooth spectral tiples are closed under the C- functional calculus of self-adjoint elements. In the final section we show that our algebras allow the formulation of Poincare Duality and that the algebras of smooth spectral triples are H-unital.


Index Theory For Locally Compact Noncommutative Geometries, Alan Carey, V Gayral, Adam Rennie, F Sukochev Sep 2015

Index Theory For Locally Compact Noncommutative Geometries, Alan Carey, V Gayral, Adam Rennie, F Sukochev

Associate Professor Adam Rennie

Spectral triples for nonunital algebras model locally compact spaces in noncommutative geometry. In the present text, we prove the local index formula for spectral triples over nonunital algebras, without the assumption of local units in our algebra.


A Residue Formula For The Fundamental Hochschild 3-Cocycle For Suq(2), Ulrich Krahmer, Adam Rennie, Roger Senior Sep 2015

A Residue Formula For The Fundamental Hochschild 3-Cocycle For Suq(2), Ulrich Krahmer, Adam Rennie, Roger Senior

Associate Professor Adam Rennie

An analogue of a spectral triple over SUq(2) is constructed for which the usual assumption of bounded commutators with the Dirac operator fails. An analytic expression analogous to that for the Hochschild class of the Chern character for spectral triples yields a non-trivial twisted Hochschild 3-cocycle. The problems arising from the unbounded commutators are overcome by defining a residue functional using projections to cut down the Hilbert space.


The Resolvent Cocycle In Twisted Cyclic Cohomology And A Local Index Formula For The Podle's Sphere, Adam Rennie, Roger Senior Sep 2015

The Resolvent Cocycle In Twisted Cyclic Cohomology And A Local Index Formula For The Podle's Sphere, Adam Rennie, Roger Senior

Associate Professor Adam Rennie

We continue the investigation of twisted homology theories in the context of dimension drop phenomena. This work unifies previous equivariant index calculations in twisted cyclic cohomology. We do this by proving the existence of the resolvent cocycle, a finitely summable analogue of the JLO cocycle, under weaker smoothness hypotheses and in the more general setting of 'modular' spectral triples. As an application we show that using our twisted resolvent cocycle, we can obtain a local index formula for the Podles sphere. The resulting twisted cyclic cocycle has non-vanishing Hochschild class which is in dimension 2.


The K-Theory Of Heegaard Quantum Lens Spaces, Piotr M. Hajac, Adam Rennie, Bartosz Zielinski Sep 2015

The K-Theory Of Heegaard Quantum Lens Spaces, Piotr M. Hajac, Adam Rennie, Bartosz Zielinski

Associate Professor Adam Rennie

Representing Z/NZ as roots of unity, we restrict a natural U(1)-action on the Heegaard quantum sphere to Z/NZ, and call the quotient spaces Heegaard quantum lens spaces. Then we use this representation of Z/NZ to construct an associated complex loine bundle. This paper proves the stable non-triviality of these line bundles over any of the quantum lens spaces we consider. We use the pullback structure of the C*-algebra of the lens space to compute its K-theory via the Mayer-Vietoris sequence, and an explicit form of the Bass connecting homomorphism to prove the stable non-triviality ...


Dense Domains, Symmetric Operators And Spectral Triples, Iain Forsyth, B Mesland, Adam Rennie Sep 2015

Dense Domains, Symmetric Operators And Spectral Triples, Iain Forsyth, B Mesland, Adam Rennie

Associate Professor Adam Rennie

This article is about erroneous attempts to weaken the standard definition of unbounded Kasparov module (or spectral triple). This issue has been addressed previously, but here we present concrete counterexamples to claims in the literature that Fredholm modules can be obtained from these weaker variations of spectral triple. Our counterexamples are constructed using self-adjoint extensions of symmetric operators.


The Local Index Formula In Noncommutative Geometry Revisited, Alan Carey, John Phillips, Adam Rennie, Fyodor Sukochev Nov 2013

The Local Index Formula In Noncommutative Geometry Revisited, Alan Carey, John Phillips, Adam Rennie, Fyodor Sukochev

Associate Professor Adam Rennie

In this review we discuss the local index formula in noncommutative geomety from the viewpoint of two new proofs are partly inspired by the approach of Higson especially that in but they differ in several fundamental aspedcts, in particular they apply to semifinite spectral triples for a *s-subalgebra A of a general semifinite von Neumann algebra. Our proofs are novel even in the setting of the original theorem and reduce the hypotheses of the theorem to those necessary for its statement. These proofs rely on the introduction of a function valued cocycle which is 'almost' a (b, B)-cocycle in ...


Semi-Finite Noncommutative Geometry And Some Applications, Alan L. Carey, Adam C. Rennie, John Phillips Nov 2013

Semi-Finite Noncommutative Geometry And Some Applications, Alan L. Carey, Adam C. Rennie, John Phillips

Associate Professor Adam Rennie

These notes are a summary of talks given in Shonan, Japan in February 2008 with modifications from a later series of talks at the Hausdorff Institute for Mathematics in Bonn in July 2008 and at the Erwin Schr¨odinger Institute in October 2008. The intention is to give a short discussion of recent results in noncommutative geometry (NCG) where one extends the usual point of view of [22] by replacing the bounded operators B(H) on a Hilbert space H by certain sub-algebras; namely semi-finite von Neumann algebras. These are weakly closed subalgebras of the bounded operators on a Hilbert ...


The Local Index Formula In Semifinite Von Neumann Algebras Ii: The Even Case, Alan Carey, John Phillips, Adam Rennie, F Sukochev Oct 2013

The Local Index Formula In Semifinite Von Neumann Algebras Ii: The Even Case, Alan Carey, John Phillips, Adam Rennie, F Sukochev

Associate Professor Adam Rennie

We generalise the even local index formula of Connes and Moscovici to the case of spectral triples for a ∗-subalgebra A of a general semifinite von Neumann algebra. The proof is a variant of that for the odd case which appears in Part I. To allow for algebras with a non-trivial centre we have to establish a theory of unbounded Fredholm operators in a general semifinite von Neumann algebra and in particular prove a generalised McKean-Singer formula.


Riemannian Manifolds In Noncommutative Geometry, Steven Lord, Adam Rennie, Joseph C. Varilly Oct 2013

Riemannian Manifolds In Noncommutative Geometry, Steven Lord, Adam Rennie, Joseph C. Varilly

Associate Professor Adam Rennie

We present a definition of Riemannian manifold in noncommutative geometry. Using products of unbounded Kasparov modules, we show one can obtain such Riemannian manifolds from noncommutative spinc manifolds; and conversely, in the presence of a spinc structure. We also show how to obtain an analogue of Kasparov's fundamental class for a Riemannian manifold, and the associated notion of Poincaré duality. Along the way we clarify the bimodule and first-order conditions for spectral triples.


Modular Index Invariants Of Mumford Curves, Adam C. Rennie, Alan L. Carey, M Marcolli Oct 2013

Modular Index Invariants Of Mumford Curves, Adam C. Rennie, Alan L. Carey, M Marcolli

Associate Professor Adam Rennie

We continue an investigation initiated by Consani–Marcolli of the relation between the algebraic geometry of p-adic Mumford curves and the noncommutative geometry of graph C∗-algebras associated to the action of the uniformizing p-adic Schottky group on the Bruhat–Tits tree. We reconstruct invariants of Mumford curves related to valuations of generators of the associated Schottky group, by developing a graphical theory for KMS weights on the associated graph C∗-algebra, and using modular index theory for KMS weights. We give explicit examples of the construction of graph weights for low genus Mumford curves. We then show that the ...


The Chern Character Of Semifinite Spectral Triples, Alan L. Carey, John Phillips, Adam C. Rennie, Fyodor A. Sukochev Oct 2013

The Chern Character Of Semifinite Spectral Triples, Alan L. Carey, John Phillips, Adam C. Rennie, Fyodor A. Sukochev

Associate Professor Adam Rennie

In previous work we generalised both the odd and even local index formula of Connes and Moscovici to the case of spectral triples for a ∗-subalgebra A of a general semifinite von Neumann algebra. Our proofs are novel even in the setting of the original theorem and rely on the introduction of a function valued cocycle (called the resolvent cocycle) which is 'almost' a (b,B)-cocycle in the cyclic cohomology of A. In this paper we show that this resolvent cocycle 'almost' represents the Chern character, and assuming analytic continuation properties for zeta functions, we show that the associated ...


Pseudo-Riemannian Spectral Triples And The Harmonic Oscillator, Koen Van Den Dungen, Mario Paschke, Adam C. Rennie Oct 2013

Pseudo-Riemannian Spectral Triples And The Harmonic Oscillator, Koen Van Den Dungen, Mario Paschke, Adam C. Rennie

Associate Professor Adam Rennie

We define pseudo-Riemannian spectral triples, an analytic context broad enough to encompass a spectral description of a wide class of pseudo-Riemannian manifolds, as well as their noncommutative generalisations. Our main theorem shows that to each pseudo-Riemannian spectral triple we can associate a genuine spectral triple, and so a K-homology class. With some additional assumptions we can then apply the local index theorem. We give a range of examples and some applications. The example of the harmonic oscillator in particular shows that our main theorem applies to much more than just classical pseudo-Riemannian manifolds. 2013 Elsevier B.V.


Kk-Theory And Spectral Flow In Von Neumann Algebras, J Kaad, R Nest, Adam C. Rennie Oct 2013

Kk-Theory And Spectral Flow In Von Neumann Algebras, J Kaad, R Nest, Adam C. Rennie

Associate Professor Adam Rennie

We present a definition of spectral flow for any norm closed ideal J in any von Neumann algebra N. Given a path of selfadjoint operators in N which are invertible in N/J, the spectral flow produces a class in Ko(J).Given a semifinite spectral triple (A, H, D) relative to (N, t) with A separable, we construct a class [D] ? KK1(A, K(N)). For a unitary u ? A, the von Neumann spectral flow between D and u*Du is equal to the Kasparov product [u]A[D], and is simply related to the numerical spectral flow, and ...


The Local Index Formula In Semifinite Von Neumann Algebras I: Spectral Flow, Alan Carey, John Phillips, Adam Rennie, Fyodor Sukochev Oct 2013

The Local Index Formula In Semifinite Von Neumann Algebras I: Spectral Flow, Alan Carey, John Phillips, Adam Rennie, Fyodor Sukochev

Associate Professor Adam Rennie

We generalise the local index formula of Connes and Moscovici to the case of spectral triples for a ∗-subalgebra A of a general semifinite von Neumann algebra. In this setting it gives a formula for spectral flow along a path joining an unbounded self-adjoint Breuer-Fredholm operator, affiliated to the von Neumann algebra, to a unitarily equivalent operator. Our proof is novel even in the setting of the original theorem and relies on the introduction of a function valued cocycle which is 'almost' a (b,B)-cocycle in the cyclic cohomology of A.


Noncommutative Atiyah-Patodi-Singer Boundary Conditions And Index Pairings In Kk-Theory, Alan L. Carey, John Phillips, Adam C. Rennie Oct 2013

Noncommutative Atiyah-Patodi-Singer Boundary Conditions And Index Pairings In Kk-Theory, Alan L. Carey, John Phillips, Adam C. Rennie

Associate Professor Adam Rennie

We investigate an extension of ideas of Atiyah-Patodi-Singer (APS) to a noncommutative geometry setting framed in terms of Kasparov modules. We use a mapping cone construction to relate odd index pairings to even index pairings with APS boundary conditions in the setting of KK-theory, generalising the commutative theory. We find that Cuntz-Krieger systems provide a natural class of examples for our construction and the index pairings coming from APS boundary conditions yield complete K-theoretic information about certain graph C*-algebras


An Analytic Approach To Spectral Flow In Von Neumann Algebras, M-T Benameur, Alan Carey, John Phillips, Adam Rennie, Fyodor Sukochev, K Wojciechowski Oct 2013

An Analytic Approach To Spectral Flow In Von Neumann Algebras, M-T Benameur, Alan Carey, John Phillips, Adam Rennie, Fyodor Sukochev, K Wojciechowski

Associate Professor Adam Rennie

The analytic approach to spectral flow is about ten years old. In that time it has evolved to cover an ever wider range of examples. The most critical extension was to replace Fredholm operators in the classical sense by Breuer-Fredholm operators in a semifinite von Neumann algebra. The latter have continuous spectrum so that the notion of spectral flow turns out to be rather more difficult to deal with. However quite remarkably there is a uniform approach in which the proofs do not depend on discreteness of the spectrum of the operators in question. The first part of this paper ...


Twisted Cyclic Cohomology And Modular Fredholm Modules, Adam Rennie, Andrzej Sitarz, Makoto Yamashita Oct 2013

Twisted Cyclic Cohomology And Modular Fredholm Modules, Adam Rennie, Andrzej Sitarz, Makoto Yamashita

Associate Professor Adam Rennie

Connes and Cuntz showed in [Comm. Math. Phys. 114 (1988), 515-526] that suitable cyclic cocycles can be represented as Chern characters of finitely summable semifinite Fredholm modules. We show an analogous result in twisted cyclic cohomology using Chern characters of modular Fredholm modules. We present examples of modular Fredholm modules arising from Podles sphereś and from SUq (2).


Spectral Flow Invariants And Twisted Cyclic Theory For The Haar State On Suq(2), A L. Carey, A Rennie, K Tong Oct 2013

Spectral Flow Invariants And Twisted Cyclic Theory For The Haar State On Suq(2), A L. Carey, A Rennie, K Tong

Associate Professor Adam Rennie

In [A.L. Carey, J. Phillips, A. Rennie, Twisted cyclic theory and an index theory for the gauge invariant KMS state on Cuntz algebras. arXiv:0801.4605], we presented a K-theoretic approach to finding invariants of algebras with no non-trivial traces. This paper presents a new example that is more typical of the generic situation. This is the case of an algebra that admits only non-faithful traces, namely SUq.2/ and also KMS states. Our main results are index theorems (which calculate spectral flow), one using ordinary cyclic cohomology and the other using twisted cyclic cohomology, where the twisting comes ...


Spectral Triples: Examples And Index Theory, Alan L. Carey, John Phillips, Adam C. Rennie Oct 2013

Spectral Triples: Examples And Index Theory, Alan L. Carey, John Phillips, Adam C. Rennie

Associate Professor Adam Rennie

The main objective of these notes is to give some intuition about spectral triples and the role they play in index theory. The notes are basically a road map, with much detail omitted. To give a complete account of all the topics covered would require at least a book, so we have opted for a sketch.


The Dixmier Trace And Asymptotics Of Zeta Functions, Alan L. Carey, Adam C. Rennie, Aleksandr Sedaev, Fyodor A. Sukochev Oct 2013

The Dixmier Trace And Asymptotics Of Zeta Functions, Alan L. Carey, Adam C. Rennie, Aleksandr Sedaev, Fyodor A. Sukochev

Associate Professor Adam Rennie

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 semi-finite von Neumann algebra. We find for p > 1 that the asymptotics of the zeta function determines an ideal strictly larger than Lp,∞ 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 And An Index Theory For The Gauge Invariant Kms State On Cuntz Algebras On, Alan L. Carey, John Phillips, Adam C. Rennie Oct 2013

Twisted Cyclic Theory And An Index Theory For The Gauge Invariant Kms State On Cuntz Algebras On, Alan L. Carey, John Phillips, Adam C. Rennie

Associate Professor Adam Rennie

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 K1-group 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 ...


Semifinite Spectral Triples Associated With Graph C*-Algebras, Alan L. Carey, John Phillips, Adam Rennie Oct 2013

Semifinite Spectral Triples Associated With Graph C*-Algebras, Alan L. Carey, John Phillips, Adam Rennie

Associate Professor Adam Rennie

We review the recent construction of semifinite spectral triples for graph C^*-algebras. These examples have inspired many other developments and we review some of these such as the relation between the semifinite index and the Kasparov product, examples of noncommutative manifolds, and an index theorem in twisted cyclic theory using a KMS state.


Twisted Cyclic Theory, Equivariant Kk-Theory And Kms States, Alan L. Carey, Sergey Neshveyev, R Nest, Adam Rennie Oct 2013

Twisted Cyclic Theory, Equivariant Kk-Theory And Kms States, Alan L. Carey, Sergey Neshveyev, R Nest, Adam Rennie

Associate Professor Adam Rennie

Given a C-algebra A with a KMS weight for a circle action, we construct and compute a secondary invariant on the equivariant K-theory of the mapping cone of AT ,! A, both in terms of equivariant KK-theory and in terms of a semifinite spectral flow. This in particular puts the previously considered examples of Cuntz algebras [10] and SUqð2Þ [14] in a general framework. As a new example we consider the Araki-Woods IIIl representations of the Fermion algebra.


Families Of Type Iii Kms States On A Class Of C-Algebras Containing On And Qn, A L. Carey, J Phillips, I F. Putnam, A Rennie Oct 2013

Families Of Type Iii Kms States On A Class Of C-Algebras Containing On And Qn, A L. Carey, J Phillips, I F. Putnam, A Rennie

Associate Professor Adam Rennie

We construct a family of purely infinite C¤-algebras, Q¸ for ¸ 2 (0, 1) that are classified by their K-groups. There is an action of the circle T with a unique KMS state à on each Q¸. For ¸ = 1/n, Q1/n »= On, with its usual T action and KMS state. For ¸ = p/q, rational in lowest terms, Q¸ »= On (n = q − p + 1) with UHF fixed point algebra of type (pq)1. For any n > 1, Q¸ »= On for infinitely many ¸ with distinct KMS states and UHF fixed-point algebras. For any ¸ 2 (0, 1), Q¸ 6= O1. For ¸ irrational ...


Orbifolds Are Not Commutative Geometries, Adam C. Rennie, Joseph C. Varilly Oct 2013

Orbifolds Are Not Commutative Geometries, Adam C. Rennie, Joseph C. Varilly

Associate Professor Adam Rennie

In this note we show that the crucial orientation condition for commutative geometries fails for the natural commutative spectral triple of an orbifold M/G.