On spectral flow for operator algebras

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 $\pi_1(Fred_{SA,\infty},pt)\cong K_0(\mathcal{B})$, generalizing a result of APS. We also provide an axiomatization of spectral flow.