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Real valued spectral flow. (English) Zbl 0831.46065
Curto, Raúl E. (ed.) et al., Multivariable operator theory. A joint summer research conference on multivariable operator theory, July 10-18, 1993, University of Washington, Seattle, WA, USA. Providence, RI: American Mathematical Society. Contemp. Math. 185, 307-318 (1995).
Summary: The notion of spectral flow as introduced by M. F. Atiyah, V. K. Patodi and I. M. Singer [Math. Proc. Cambr. Philos. Soc. 79, 71-99 (1976; Zbl 0325.58015)], provides an isomorphism between the fundamental group of the non-trivial path component $${\mathcal F}^{sa}_*$$, of selfadjoint Fredholm operators, $${\mathcal F}^{sa}$$ and the additive group of integers, $$\mathbb{Z}$$. In this paper, we generalize these ideas to type $$II_\infty$$ von Neumann algebras, $$N$$ and obtain a real valued spectral flow, $$sf$$ as an isomorphism between $$\pi_1({\mathcal F}^{sa}_{II, *})$$ and $$\mathbb{R}$$, where $${\mathcal F}^{sa}_{II,*}$$ is the corresponding non-trivial path component of the selfadjoint Breuer- Fredholm elements, $${\mathcal F}^{sa}_{II}$$ in $$N$$.
For the entire collection see [Zbl 0819.00022].

##### MSC:
 46L10 General theory of von Neumann algebras 46L80 $$K$$-theory and operator algebras (including cyclic theory) 46L55 Noncommutative dynamical systems 58J20 Index theory and related fixed-point theorems on manifolds