[Paper Review] On Type II noncommutative geometry and the JLO character

Alan Lai|arXiv (Cornell University)|Mar 22, 2010

Advanced Operator Algebra Research21 references3 citations

TL;DR

This paper extends the JLO character—a key tool in noncommutative geometry—from Type I to Type II noncommutative geometry by defining it for unbounded Breuer-Fredholm modules affiliated with a Type II von Neumann algebra. It proves that the JLO character in this setting defines the same entire cyclic cohomology class as the Connes-Chern character, generalizing the Type I result to infinite-dimensional, non-tracial geometries with real-valued indices.

ABSTRACT

The Jaffe-Lesniewski-Osterwalder (JLO) character is a homomorphism from K-homology to entire cyclic cohomology. This paper extends the domain of the JLO character to include Type II noncommutative geometry, the geometry represented by unbounded $θ$-summable Breuer-Fredholm modules; and shows that the JLO character coincides with the Chern-Connes character as a class in entire cyclic cohomolgoy.

Motivation & Objective

• To extend the domain of the JLO character to Type II noncommutative geometry, where the underlying algebra is a Type II von Neumann algebra.
• To define the JLO character for unbounded Breuer-Fredholm modules, which generalize unbounded Fredholm modules in the Type II setting.
• To establish that the JLO character class in entire cyclic cohomology coincides with the Connes-Chern character class in this extended framework.
• To generalize the index formula of Atiyah-Singer and the JLO-Comnes-Moscovici equivalence to the Type II setting.
• To provide a cohomological framework for the Breuer-Fredholm index, which takes values in ℝ rather than ℤ, in infinite-dimensional noncommutative geometries.

Proposed method

• The paper defines unbounded Breuer-Fredholm modules over a C*-algebra A, with a Dirac operator D affiliated with a Type II von Neumann algebra N and satisfying axioms analogous to Connes' unbounded Fredholm modules.
• It constructs the JLO character using the entire cyclic cohomology of A, based on the heat kernel trace of D, generalizing the Type I construction to the Type II setting.
• The JLO character is shown to be homotopy invariant under suitable deformations of the unbounded Breuer-Fredholm module.
• The paper establishes a correspondence between unbounded and bounded Breuer-Fredholm modules via functional calculus on D, extending the standard passage from unbounded to bounded Fredholm modules.
• It proves that the JLO character for p-summable unbounded Breuer-Fredholm modules is cohomologous to the Connes-Chern character of the associated bounded module.
• The proof relies on the theory of generalized singular numbers and τ-measurable operators in Type II von Neumann algebras, using trace ideals L_N^p and the τ-compactness condition.

Experimental results

Research questions

• RQ1 Can the JLO character be extended from Type I to Type II noncommutative geometry, where the Fredholm index takes values in ℝ rather than ℤ?
• RQ2 Does the JLO character for unbounded Breuer-Fredholm modules in the Type II setting define the same entire cyclic cohomology class as the Connes-Chern character?
• RQ3 How does the homotopy invariance of the JLO class behave in the Type II setting, particularly under deformations of the Dirac operator?
• RQ4 What is the relationship between the JLO character and the Connes-Chern character when passing from unbounded to bounded Breuer-Fredholm modules in Type II geometry?
• RQ5 To what extent do the standard tools of entire cyclic cohomology and trace ideals extend to the setting of τ-measurable operators in Type II von Neumann algebras?

Key findings

• The JLO character is well-defined for unbounded Breuer-Fredholm modules in Type II noncommutative geometry, extending its domain beyond the Type I case.
• The JLO character defines the same cohomology class in entire cyclic cohomology as the Connes-Chern character for the associated bounded Breuer-Fredholm module.
• The JLO character is homotopy invariant under continuous deformations of the unbounded Breuer-Fredholm module, preserving the cohomology class.
• For p-summable unbounded Breuer-Fredholm modules, the JLO character is cohomologous to the Connes-Chern character, generalizing the Type I result of Connes-Moscovici.
• The construction relies on the theory of τ-measurable operators and generalized singular numbers, ensuring compatibility with the trace structure of Type II von Neumann algebras.
• The index pairing in cohomology via the JLO character reproduces the Breuer-Fredholm index, which is real-valued, in contrast to the integer-valued index in Type I geometry.