Spectral flow for multiplier algebras

Ping Wong Ng  (University of Louisiana at Lafayette)

10:00-11:00, December 9, 2023   Science Building A503

Abstract:

Let B be a separable stable C*-algebra. For a norm-continuous path of self-adjoint Fredholm operators in the multiplier algebra M(B), spectral flow roughly measures the ``net mass" of spectrum that passes through zero in the positive direction, as we move along the continuous path. Spectral flow was first studied by Atiyah and Lustig, and first appeared in print in the work of Atiyah-Patodi-Singer. Ever since then, the subject has exploded in many directions beyond the scope of this talk. We focus on the case of bounded operators, and our point of view is that just as the Fredholm index has led to many interesting results in operator algebras, spectral flow will also lead to interesting results in this context. 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), and we discuss a projection-lifting hypothesis which is present in all previous treatments of spectral flow. Our first main result is that for a separable stable C*-algebra B, spectral flow induces a group isomorphism pi_1(F_{SA, infinity}) \rightarrow K_0(B) where F_{SA, infinity} is a class of self-adjoint Fredholm operators in M(B) that satisfy a certain strong lifting projections hypothesis (or strong infinite condition). Under appropriate hypotheses, we also provide an axiomatization of spectral flow.

About the speaker:

Ping Wong Ng,路易斯安那大学拉法叶分校教授，是知名的算子代数专家，主要研究C*-代数延拓及分类理论。Ping Wong Ng已在国际高水品期刊发表论文57篇，参与国际会议、进行学术报告/讨论班近100次，并为Bulletin of the Canadian Mathematics Society, Bulletin of the Malaysian Mathematical Sciences Society, Canadian Journal of Mathematics, Illinois Journal of Mathematics等多个高水平学术刊物做评审工作。

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