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On spectral flow for operator algebras

Ping Wong Ng, Arindam Sutradhar, Cangyuan Wang

Abstract

Spectral flow was first studied by Atiyah and Lusztig, and first appeared in print in the work of Atiyah-Patodi-Singer (APS). For a norm-continuous path of self-adjoint Fredholm operators in the multiplier algebra $\mathcal{M}(\mathcal{B})$ with $\mathcal{B}$ separable and stable, spectral flow roughly measures the ``net mass" of spectrum that passes through zero in the positive direction, as we move along the continuous path. As the index of a Fredholm operator has had many fruitful and important generalizations to general operator algebras, generalizing the spectral flow of a path of self-adjoint Fredholm operators would also be of great interest to operator theory. We develop a notion of spectral flow which works for arbitrary separable stable canonical ideals -- including stably projectionless C*-algebras (which depends on a quite general notion of essential codimension). We show that, under appropriate hypotheses, spectral flow induces a group isomorphism $π_1(Fred_{SA,\infty},pt)\cong K_0(\mathcal{B})$, generalizing a result of APS. We also provide an axiomatization of spectral flow.

On spectral flow for operator algebras

Abstract

Spectral flow was first studied by Atiyah and Lusztig, and first appeared in print in the work of Atiyah-Patodi-Singer (APS). For a norm-continuous path of self-adjoint Fredholm operators in the multiplier algebra with separable and stable, spectral flow roughly measures the ``net mass" of spectrum that passes through zero in the positive direction, as we move along the continuous path. As the index of a Fredholm operator has had many fruitful and important generalizations to general operator algebras, generalizing the spectral flow of a path of self-adjoint Fredholm operators would also be of great interest to operator theory. We develop a notion of spectral flow which works for arbitrary separable stable canonical ideals -- including stably projectionless C*-algebras (which depends on a quite general notion of essential codimension). We show that, under appropriate hypotheses, spectral flow induces a group isomorphism , generalizing a result of APS. We also provide an axiomatization of spectral flow.
Paper Structure (19 sections, 57 theorems, 275 equations)

This paper contains 19 sections, 57 theorems, 275 equations.

Table of Contents

  1. Introduction
  2. The definition of spectral flow
  3. The DZWLPW definition
  4. The lifting projections hypothesis
  5. The general definition
  6. Functorial properties of spectral flow
  7. The Pre Spectral Flow isomorphism
  8. Fredholm operators, topological groups of invertible operators, and Grassmannian spaces
  9. A certain fibration
  10. Completing the Pre Spectral Flow Isomorphism
  11. The Spectral Flow Isomorphism Theorem and the Axiomatization of Spectral Flow
  12. The Spectral Flow Isomorphism Theorem
  13. Axiomatization
  14. Appendix: Some miscellaneous and useful computations, and the DZWLP Spectral Flow
  15. Generalized Fredholm operators and Fredholm index
  16. ...and 4 more sections

Key Result

Proposition 2.2

Let $\mathcal{B}$ be a separable stable C*-algebra. Let $P, Q, R, P', Q' \in \mathcal{M}(\mathcal{B})$ be projections such that $P - Q, Q - R, P' - Q' \in \mathcal{B}$. Then the following statements are true:

Theorems & Definitions (151)

  • Definition 2.1
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Remark 2.7
  • Lemma 2.8
  • ...and 141 more