K-Theory for Real C*-algebras via Unitary Elements with Symmetries

Abstract

We prove that all eight KO groups for a real C*-algebra can be constructed from homotopy classes of unitary matrices that respect a variety of symmetries. In this manifestation of the KO groups, all eight boundary maps in the 24-term exact sequence associated to an ideal in a real C*-algebra can be computed as exponential or index maps with formulas that are nearly identical to the complex case.

79 pages. Reorganization of sections and notational clarifications since previous edition

References in corpus (3)

• K-Theory and Pseudospectra for Topological Insulators
• Classification of "Quaternionic" Bloch-bundles: Topological Quantum Systems of type AII
• A Projective C*-Algebra Related to K-Theory

Cited by in corpus (13)

• K-Theory and Pseudospectra for Topological Insulators
• Controlled topological phases and bulk-edge correspondence
• An operator-based approach to topological photonics
• Quadratic pseudospectrum for identifying localized states
• Bulk Spectrum and K-theory for Infinite-Area Topological Quasicrystal
• Finite volume calculation of $K$-theory invariants
• On the C*-algebraic approach to topological phases for insulators
• The Cayley transform in complex, real and graded $K$-theory
• Cyclic cohomology for graded $C^{\ast ,r}$-algebras and its pairings with van Daele $K$-theory
• 'Real' gerbes and Dirac cones of topological insulators
• $K$-theory for real $k$-graph $C^{\ast }$-algebras
• Topological insulators from the perspective of non-commutative geometry and index theory
• Exact sequence between real and complex bivariant K theories and application to the Z2 pairing