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Real $K$-Theory for $C^*$-Algebras: Just the Facts

Jeff Boersema, Claude Schochet

TL;DR

The paper develops KO-theory for real C*-algebras and its counterpart KO^* for spaces, emphasizing connections to complex K-theory $K_*$, KR, Clifford algebras, and the Ten-Fold Way in physics. It introduces KO_*-theory via projections and unitary groups, establishes homology and cofibre frameworks, and proves Bott periodicity with an 8-fold cycle, including external/internal products and a real Künneth theorem. It then builds a geometric-topological bridge through topological spaces, classifying spaces, and vector bundles, culminating in a Homotopy Road Map that links Bott periodicity, Clifford algebras, symmetric spaces, and physical symmetries. The work also provides explicit low-dimensional homotopy computations for the Ten-Fold Way and outlines real-vector-bundle classifications and Swan-type theorems, offering a practical foundation for applying KO-theory in mathematics and physics. Overall, it supplies a comprehensive, implementation-friendly treatment of KO-theory for real C*-algebras and real spaces with substantial links to geometry, topology, and physics.

Abstract

This work is intended to present the basic properties of $KO$-theory for real $C^*$-algebras and to explain its relationship with complex $K$-theory and with $KR$- theory. Whenever possible we will rely upon proofs in printed literature, particularly the work of Karoubi, Wood, Schröder, and more recent work of Boersema and J. M. Rosenberg. In addition, we shall explain how $KO$-theory is related to the Ten-Fold Way in physics and point out how some deeper features of $KO$-theory for operator algebras may provide powerful new tools there. Commutative real $C^*$-algebras not of the form $C(X, R)$ will play a special role. Unfortunately, there is no single reference for $KO$-theory for operator algebras that begins to compare with Blackadar's wonderful exposition of complex $K$-theory. This work is intended to provide a platform upon which mathematicians and mathematical physicists can rely in order to use these new tools in their research. As we are writing for a diverse audience of functional analysts, topologists, and physicists, we often present material well-known to one group of people and unfamiliar to another.

Real $K$-Theory for $C^*$-Algebras: Just the Facts

TL;DR

The paper develops KO-theory for real C*-algebras and its counterpart KO^* for spaces, emphasizing connections to complex K-theory , KR, Clifford algebras, and the Ten-Fold Way in physics. It introduces KO_*-theory via projections and unitary groups, establishes homology and cofibre frameworks, and proves Bott periodicity with an 8-fold cycle, including external/internal products and a real Künneth theorem. It then builds a geometric-topological bridge through topological spaces, classifying spaces, and vector bundles, culminating in a Homotopy Road Map that links Bott periodicity, Clifford algebras, symmetric spaces, and physical symmetries. The work also provides explicit low-dimensional homotopy computations for the Ten-Fold Way and outlines real-vector-bundle classifications and Swan-type theorems, offering a practical foundation for applying KO-theory in mathematics and physics. Overall, it supplies a comprehensive, implementation-friendly treatment of KO-theory for real C*-algebras and real spaces with substantial links to geometry, topology, and physics.

Abstract

This work is intended to present the basic properties of -theory for real -algebras and to explain its relationship with complex -theory and with - theory. Whenever possible we will rely upon proofs in printed literature, particularly the work of Karoubi, Wood, Schröder, and more recent work of Boersema and J. M. Rosenberg. In addition, we shall explain how -theory is related to the Ten-Fold Way in physics and point out how some deeper features of -theory for operator algebras may provide powerful new tools there. Commutative real -algebras not of the form will play a special role. Unfortunately, there is no single reference for -theory for operator algebras that begins to compare with Blackadar's wonderful exposition of complex -theory. This work is intended to provide a platform upon which mathematicians and mathematical physicists can rely in order to use these new tools in their research. As we are writing for a diverse audience of functional analysts, topologists, and physicists, we often present material well-known to one group of people and unfamiliar to another.
Paper Structure (41 sections, 91 theorems, 515 equations, 1 table)

This paper contains 41 sections, 91 theorems, 515 equations, 1 table.

Table of Contents

  1. Introduction
  2. $KO_*$-Theory for Real $C^*$-Algebras
  3. Real $C^*$-algebras
  4. New $C^*$-algebras from Old Ones
  5. Homotopy Constructions
  6. Homology Theories on Real $C^*$-algebras
  7. $KO_*$-theory for real $C^*$-Algebras: Definitions and Basic Properties
  8. External Products, the Künneth Theorem, and Internal Products
  9. Bott Periodicity
  10. $KO^*$-theory for Spaces
  11. The Basics: Topological Spaces
  12. The Basics: Fibrations, Fibre Bundles, Principal Bundles, and $\mathbf{BO}$
  13. Bott Periodicity Revisited; First View of the Homotopy Road Map
  14. Next Contact with Physics: Low Homotopy Groups of the Ten-Fold Way
  15. 1. The Groups $U_n$ and $SU_n$
  16. ...and 26 more sections

Key Result

Theorem 2.2

(Arens-Kaplansky AK, Thm. 9.1) Suppose that $A$ is a commutative real $C^*$-algebra. Then:

Theorems & Definitions (231)

  • Definition 2.1
  • Theorem 2.2
  • Remark 2.3
  • Definition 3.1
  • Example 3.3
  • Definition 3.4
  • Definition 3.5
  • Definition 3.6
  • Definition 3.7
  • Proposition 3.8
  • ...and 221 more