In this paper, we give a comprehensive treatment of a “Clifford module flow” along paths in the skew-adjoint Fredholm operators on a real Hilbert space that takes values in KO∗(ℝ) via the Clifford index of Atiyah–Bott–Shapiro. We develop its properties for both bounded and unbounded skew-adjoint operators including an axiomatic characterization. Our constructions and approach are motivated by the principle that spectral flow = Fredholm index. That is, we show how the KO-valued spectral flow relates to a KO-valued index by proving a Robbin–Salamon type result. The Kasparov product is also used to establish a spectral flow = Fredholm index result at the level of bivariant K-theory. We explain how our results incorporate previous applications of ℤ/2ℤ-valued spectral flow in the study of topological phases of matter.

中文翻译：

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斜伴随 Fredholm 算子的 KO 值谱流

在这篇论文中，我们给出了一个“Clifford 模流”在一个真实的 Hilbert 空间上沿斜伴随 Fredholm 算子路径的综合处理，该空间取值KO*(ℝ)通过 Atiyah-Bott-Shapiro 的 Clifford 指数。我们为有界和无界斜伴随算子开发了它的属性，包括公理化表征。我们的结构和方法的动机是这样的原则 谱流 = 弗雷德霍尔姆指数. 也就是说，我们展示了如何KO值谱流与KO通过证明 Robbin–Salamon 类型结果的值索引。Kasparov 乘积也用于建立一个谱流 = 弗雷德霍尔姆指数双变量水平的结果ķ-理论。我们解释了我们的结果如何结合以前的应用ℤ/2ℤ物质拓扑相研究中的值谱流。