Spectral flow and Dixmier traces

By Alan L. Carey, J. Phillips, Fedor Sukochev. 2002

111 citations recorded by the source corpus.

Abstract We obtain general theorems which enable the calculation of the Dixmier trace in terms of the asymptotics of the zeta function and of the heat operator in a general semi-finite von Neumann algebra. Our results have several applications. We deduce a formula for the Chern character of an odd L (1,∞) -summable Breuer–Fredholm module in terms of a Hochschild 1-cycle. We explain how to derive a Wodzicki residue for pseudo-differential operators along the orbits of an ergodic R n action on a compact space X. Finally, we give a short proof of an index theorem of Lesch for generalised Toeplitz operators.

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/9aa8401c06e36f768f1bc2c0ecbbbc55808b4069

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In Algebra and Number Theory

1. Alan L. Carey
2. A. Rennie
3. Fedor Sukochev
4. David Pask
5. E. Semenov
6. J. Phillips
7. V. Gayral
8. R. Nest

In Mathematical Physics

1. Fedor Sukochev
2. A. Rennie
3. Alan L. Carey
4. David Pask
5. E. Semenov
6. J. Phillips
7. V. Gayral
8. R. Nest

Citations in the corpus

• Twisted cyclic theory, equivariant KK-theory and KMS states (2008, methodology): “These index pairings have not been seen before because they use in an essential way the semifinite index theory from [7, 11, 12].”
• Dixmier traces as singular symmetric functionals and applications to measurable operators (2005, unknown): “2 completes the identification suggested by the results of [1].”
• Dixmier traces and some applications in non-commutative geometry (2006, methodology): “Our exposition is based on the approach articulated in [31], [29], which considers such traces as a special class of continuous linear functionals on the corresponding operator ideals.”
• Invariant Banach limits and applications (2010, methodology): “In [5,2] similar results for the dilation operator σn, n ∈ N, and the Cesàro operator C (see relevant definitions below) were obtained through an approach based on the Schauder fixed point principle.”
• The Hochschild class of the Chern character for semifinite spectral triples (2003, methodology): “is trace class for all t40: Then we have Theorem 7 (Carey et al. [ 6 ]).”
• Moyal Planes are Spectral Triples (2003, result): “For this, recent results of Carey and coworkers [10] extending Connes’ trace theorem (see [16] and [45, Chap.”
• Spectral flow in Fredholm modules, eta invariants and the JLO cocycle (2003, methodology): “In a later paper [ 16 ], this result is crucial in proving Connes’ Dixmier-trace formula for the odd index in the general setting of (1, ∞) Breuer–Fredholm modules.”
• The noncommutative geometry of graph C*-algebras I: The index theorem (2005, methodology): “u∈ A unitary the pairing of [u] ∈ K1(A) with (A,H,D) is given by h[u],(A,H,D)i = ress=0τ(u[D,u∗](1+D2) −1/2 s). In particular, the residue on the right exists. For more information on this result see [4, 6, 7, 9]. 3. GraphC∗-Algebraswith Semifinite GraphTraces This section considers the existence of”
• The Dixmier trace and asymptotics of zeta functions (2006, unknown): “Our results follow primarily from (strengthened versions of) deep facts from [6] and recent advances in the study of singular traces, some of which seem not to be well known.”