TY - JOUR
T1 - The Cayley transform in complex, real and graded K -theory
AU - Bourne, Chris
AU - Kellendonk, Johannes
AU - Rennie, Adam
N1 - Funding Information:
We would like to thank Alex Kumjian, David Pask, and Aidan Sims for pointing out some graded issues that were a bit odd. We also thank Christopher Max for sharing the results of [1] with us, and Matthias Lesch for helpful conversations. Finally, Bram Mesland has once again provided timely and useful advice. This work was supported by World Premier International Research Center Initiative (WPI), MEXT, Japan. AR was partially supported by the BFS/TFS project Pure Mathematics in Norway and CB is supported by a JSPS Grant-in-Aid for Early-Career Scientists (No. 19K14548). All authors thank the Erwin Schrödinger Institute program Bivariant K-Theory in Geometry and Physics for hospitality and support during the production of this work.
Publisher Copyright:
© 2020 World Scientific Publishing Company.
PY - 2020/8/1
Y1 - 2020/8/1
N2 - We use the Cayley transform to provide an explicit isomorphism at the level of cycles from van Daele K-theory to KK-theory for graded C-algebras with a real structure. Isomorphisms between KK-theory and complex or real K-theory for ungraded C-algebras are a special case of this map. In all cases, our map is compatible with the computational techniques required in physical and geometrical applications, in particular, index pairings and Kasparov products. We provide applications to real K-theory and topological phases of matter.
AB - We use the Cayley transform to provide an explicit isomorphism at the level of cycles from van Daele K-theory to KK-theory for graded C-algebras with a real structure. Isomorphisms between KK-theory and complex or real K-theory for ungraded C-algebras are a special case of this map. In all cases, our map is compatible with the computational techniques required in physical and geometrical applications, in particular, index pairings and Kasparov products. We provide applications to real K-theory and topological phases of matter.
KW - K -theory
KW - operator algebras
KW - topological phases
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U2 - 10.1142/S0129167X20500743
DO - 10.1142/S0129167X20500743
M3 - Article
AN - SCOPUS:85091976034
SN - 0129-167X
VL - 31
JO - International Journal of Mathematics
JF - International Journal of Mathematics
IS - 9
M1 - 2050074
ER -