The spectral localizer for semifinite spectral triples

Schulz-Baldes, Hermann; Stoiber, Tom

doi:10.1090/proc/15230

The spectral localizer for semifinite spectral triples

Authors: Hermann Schulz-Baldes and Tom Stoiber
Journal: Proc. Amer. Math. Soc. 149 (2021), 121-134
MSC (2010): Primary 19K56, 46L80
DOI: https://doi.org/10.1090/proc/15230
Published electronically: October 20, 2020
MathSciNet review: 4172591
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/10.1017/is012003003jkt185
Terry A. Loring, $K$-theory and pseudospectra for topological insulators, Ann. Physics 356 (2015), 383–416. MR 3350651, DOI https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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
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
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 https://doi.org/10.1142/S0129055X16500240