The spectral localizer for semifinite spectral triples

Schulz-Baldes, Hermann; Stoiber, Tom

doi:10.1090/proc/15230

The spectral localizer for semifinite spectral triples
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by Hermann Schulz-Baldes and Tom Stoiber

Proc. Amer. Math. Soc. 149 (2021), 121-134

DOI: https://doi.org/10.1090/proc/15230

Published electronically: October 20, 2020

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Abstract:

The notion of a spectral localizer is extended to pairings with semifinite spectral triples. By a spectral flow argument, any semifinite index pairing is shown to be equal to the signature of the spectral localizer. As an application, a formula for the weak invariants of topological insulators is derived. This provides a new approach to their numerical evaluation.

References

Moulay-Tahar Benameur, Alan L. Carey, John Phillips, Adam Rennie, Fyodor A. Sukochev, and Krzysztof P. Wojciechowski, An analytic approach to spectral flow in von Neumann algebras, Analysis, geometry and topology of elliptic operators, World Sci. Publ., Hackensack, NJ, 2006, pp. 297–352. MR 2246773
Chris Bourne and Emil Prodan, Non-commutative Chern numbers for generic aperiodic discrete systems, J. Phys. A 51 (2018), no. 23, 235202, 38. MR 3810727, DOI 10.1088/1751-8121/aac093
Chris Bourne and Hermann Schulz-Baldes, Application of semifinite index theory to weak topological phases, 2016 MATRIX annals, MATRIX Book Ser., vol. 1, Springer, Cham, 2018, pp. 203–227. MR 3792522
Manfred Breuer, Fredholm theories in von Neumann algebras. I, Math. Ann. 178 (1968), 243–254. MR 234294, DOI 10.1007/BF01350663
A. L. Carey, V. Gayral, J. Phillips, A. Rennie, and F. A. Sukochev, Spectral flow for nonunital spectral triples, Canad. J. Math. 67 (2015), no. 4, 759–794. MR 3361012, DOI 10.4153/CJM-2014-042-x
A. L. Carey, V. Gayral, A. Rennie, and F. A. Sukochev, Index theory for locally compact noncommutative geometries, Mem. Amer. Math. Soc. 231 (2014), no. 1085, vi+130. MR 3221983
Alan Carey and John Phillips, Unbounded Fredholm modules and spectral flow, Canad. J. Math. 50 (1998), no. 4, 673–718. MR 1638603, DOI 10.4153/CJM-1998-038-x
Alan L. Carey, John Phillips, Adam Rennie, and Fyodor A. Sukochev, The local index formula in semifinite von Neumann algebras. I. Spectral flow, Adv. Math. 202 (2006), no. 2, 451–516. MR 2222358, DOI 10.1016/j.aim.2005.03.011
Alan L. Carey, John Phillips, Adam Rennie, and Fyodor A. Sukochev, The local index formula in semifinite von Neumann algebras. II. The even case, Adv. Math. 202 (2006), no. 2, 517–554. MR 2222359, DOI 10.1016/j.aim.2005.03.010
Alain Connes and Henri Moscovici, Type III and spectral triples, Traces in number theory, geometry and quantum fields, Aspects Math., E38, Friedr. Vieweg, Wiesbaden, 2008, pp. 57–71. MR 2427588
J. Kaad, R. Nest, and A. Rennie, $KK$-theory and spectral flow in von Neumann algebras, J. K-Theory 10 (2012), no. 2, 241–277. MR 3004169, DOI 10.1017/is012003003jkt185
Terry A. Loring, $K$-theory and pseudospectra for topological insulators, Ann. Physics 356 (2015), 383–416. MR 3350651, DOI 10.1016/j.aop.2015.02.031
Terry A. Loring, Bulk spectrum and $K$-theory for infinite-area topological quasicrystals, J. Math. Phys. 60 (2019), no. 8, 081903, 16. MR 3994387, DOI 10.1063/1.5083051
Terry A. Loring and Hermann Schulz-Baldes, Finite volume calculation of $K$-theory invariants, New York J. Math. 23 (2017), 1111–1140. MR 3711272
Terry A. Loring and Hermann Schulz-Baldes, The spectral localizer for even index pairings, J. Noncommut. Geom. 14 (2020), no. 1, 1–23. MR 4107509, DOI 10.4171/JNCG/357
Terry A. Loring and Hermann Schulz-Baldes, Spectral flow argument localizing an odd index pairing, Canad. Math. Bull. 62 (2019), no. 2, 373–381. MR 3952525, DOI 10.4153/cmb-2018-013-x
Edgar Lozano Viesca, Jonas Schober, and Hermann Schulz-Baldes, Chern numbers as half-signature of the spectral localizer, J. Math. Phys. 60 (2019), no. 7, 072101, 13. MR 3982949, DOI 10.1063/1.5094300
John Phillips, Spectral flow in type I and II factors—a new approach, Cyclic cohomology and noncommutative geometry (Waterloo, ON, 1995) Fields Inst. Commun., vol. 17, Amer. Math. Soc., Providence, RI, 1997, pp. 137–153. MR 1478707
John Phillips and Iain Raeburn, An index theorem for Toeplitz operators with noncommutative symbol space, J. Funct. Anal. 120 (1994), no. 2, 239–263. MR 1266310, DOI 10.1006/jfan.1994.1032
Emil Prodan, A computational non-commutative geometry program for disordered topological insulators, SpringerBriefs in Mathematical Physics, vol. 23, Springer, Cham, 2017. MR 3618067, DOI 10.1007/978-3-319-55023-7
Emil Prodan and Hermann Schulz-Baldes, Bulk and boundary invariants for complex topological insulators, Mathematical Physics Studies, Springer, [Cham], 2016. From $K$-theory to physics. MR 3468838, DOI 10.1007/978-3-319-29351-6
Emil Prodan and Hermann Schulz-Baldes, Generalized Connes-Chern characters in $KK$-theory with an application to weak invariants of topological insulators, Rev. Math. Phys. 28 (2016), no. 10, 1650024, 76. MR 3572630, DOI 10.1142/S0129055X16500240