相关论文: Spectral flow and Dixmier traces
We study the relationships between Dixmier traces, zeta-functions and traces of heat semigroups beyond the dual of the Macaev ideal and in the general context of semifinite von Neumann algebras. We show that the correct framework for this…
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…
In \cite{APSIII} Atiyah, Patodi and Singer introduced spectral flow for elliptic operators on odd dimensional compact manifolds. They argued that it could be computed from the Fredholm index of an elliptic operator on a manifold of one…
We give a comprehensive account of an analytic approach to spectral flow along paths of self-adjoint Breuer-Fredholm operators in a type $I_{\infty}$ or $II_\infty$ von Neumann algebra ${\mathcal N}$. The framework is that of {\it odd…
We develop the theory of modulated operators in general principal ideals of compact operators. For Laplacian modulated operators we establish Connes' trace formula in its local Euclidean model and a global version thereof. It expresses…
We define the notion of von Neumann spectral triple and consider the associated index problem. We compute the analytic Chern-Connes character of such a von Neumann spectral triple and prove the corresponding local formula for its Hochschild…
We present an ab initio approach to integration theory for nonunital spectral triples. This is done without reference to local units and in the full generality of semifinite noncommutative geometry. The main result is an equality between…
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…
Trace formulas appear in many forms in noncommutative geometry (NCG). In the first part of this thesis, we obtain results for asymptotic expansions of trace formulas like heat trace expansions by adapting the theory of Multiple Operator…
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 Fr\'echet…
We derive a formula for the regularized trace of operators with compact spectrum which act on the space of square integrable functions on the quotient of a semisimple Liegroup of real rank one by a convex-cocompact subgroup. The sum of…
We introduce the notion of a {\vartheta}-summable Fredholm module over a locally convex dg algebra {\Omega} and construct its Chern character as a cocycle on the entire cyclic complex of {\Omega}, extending the construction of Jaffe,…
We study extended zeta-function residues on principal ideals of compact operators and their connections with Dixmier traces. We establish a Lidskii-type formula for continuous singular traces on these ideals. Using this formula, we obtain a…
We revisit traces of holomorphic families of pseudodifferential operators on a closed manifold in view of geometric applications. We then transpose the corresponding analytic constructions to two different geometric frameworks; the…
We compute the Dixmier trace of pseudo-Toeplitz operators on the Fock space. As an application we find a formula for the Dixmier trace of the product of commutators of Toeplitz operators on the Hardy and weighted Bergman spaces on the unit…
Consider a selfadjoint unbounded operator D on a Hilbert space H and a one parameter norm continuous family of selfadjoint bounded operators {A(t)} parametrized by the real line. Then under certain conditions \cite{RS95} that include the…
We investigate the spectral asymptotic behavior of operator-valued classical pseudo-differential operators ($\Psi$DOs) for negative order with symbols taking values in a semifinite von Neumann algebran $\mathcal{M}$ equipped with a normal…
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…
We review previous work on spectral flow in connection with certain self-adjoint model operators $\{A(t)\}_{t\in \mathbb{R}}$ on a Hilbert space $\mathcal{H}$, joining endpoints $A_\pm$, and the index of the operator $D_{A}^{}= (d/d t) + A$…
In this paper we provide sharp results for the Dixmier traceability of discrete pseudo-differential operators on $\ell^2(\mathbb{Z}^n)$. In this setting, we introduce a suitable notion of a class of classical symbols which provide a class…