Semifinite K-theory index pairing - Imported Knowlpedia Content

Definition

Let $(\mathcal A,\mathcal N,D,\tau)$ be a semifinite spectral triple and put $P=1_{[0,\infty)}(D)$ . Its semifinite K-theory index pairing is the homomorphism obtained by replacing the ordinary Fredholm index with the $\tau$ -Breuer index. In odd parity, a unitary $u\in M_n(\mathcal A)$ represents a class in $K_1(\mathcal A)$ , and

\langle [u],[D]\rangle_\tau=\operatorname{Ind}_\tau(PuP),

where $PuP:P(H^n)\to P(H^n)$ is Breuer-Fredholm. In even parity, a projection $e\in M_n(\mathcal A)$ is paired with the Breuer index of the compression $eD^+e:eH_+^n\to eH_-^n$ . The value is real in general because $\tau$ -dimensions need not be integers.

Spectral-flow realization

For the odd pairing, the Breuer index can equivalently be computed as the semifinite spectral flow from $D$ to $uDu^*$ , with the sign determined by the orientation of the chosen path. This equality is the analytic bridge between K-theory and the trace-valued crossing count Carey and Phillips, Theorems 1.9 and 2.17.

The pairing depends only on the K-theory class and the homotopy class of the semifinite Fredholm module, provided the usual matrix amplification and unitization conventions are used.

Examples and scope

When $\mathcal N=B(H)$ and $\tau$ is the ordinary operator trace, the Breuer index is the integer Fredholm index, so the construction reduces to the usual spectral-triple pairing. In a type $\mathrm{II}_\infty$ factor, finite projections can have arbitrary nonnegative real trace, and the pairing can therefore take nonintegral real values.

References

A. L. Carey and J. Phillips, “Unbounded Fredholm Modules and Spectral Flow,” Canadian Journal of Mathematics 50 (1998), 673–718. DOI record. Relevant: §§1–2 on the Breuer index and spectral-flow formula.
A. L. Carey, J. Phillips, A. Rennie, and F. A. Sukochev, “The Local Index Formula in Semifinite von Neumann Algebras I: Spectral Flow,” Advances in Mathematics 202 (2006), 451–516. DOI record. Relevant: §§2–4 on semifinite triples, spectral flow, and the odd pairing.