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The spectral model of (Real) $K$-theory

Anupam Datta

Abstract

We use homotopy theoretic ideas to study the $K$-theory of (graded, Real) $C^*$-algebras in detail. After laying the foundations, and deriving the formal properties, the comparison of the model with the Kasparov picture of $K$-theory has been made, and Bott periodicity has been proven using a Dirac-dual Dirac method.

The spectral model of (Real) $K$-theory

Abstract

We use homotopy theoretic ideas to study the -theory of (graded, Real) -algebras in detail. After laying the foundations, and deriving the formal properties, the comparison of the model with the Kasparov picture of -theory has been made, and Bott periodicity has been proven using a Dirac-dual Dirac method.

Paper Structure

This paper contains 35 sections, 109 theorems, 162 equations.

Table of Contents

  1. Introduction
  2. The first definitions, examples, and properties
  3. Graded and Real C*-algberas
  4. Hom sets as spaces
  5. The Spectral K-theory space underlying a Hilbert space
  6. Two fundamental properties of spectral K-theory
  7. Stability
  8. Long exact sequence
  9. Two important graded Real *-homomorphisms
  10. The diagonal map
  11. The codiagonal map
  12. Addition and H-space structures
  13. Sums and naturality
  14. H-space structures
  15. Sums preserve H-space structures
  16. ...and 20 more sections

Key Result

Theorem 2.5

(See GOOD) There is an equivalence between the category of Real $C^*$-algebras with Real structure preserving $*$-homomorphisms, and the category of real $C^*$-algebras and $*$-homomorphisms. The correspondence goes as followsthe process of assigning a Real $C^*$-algebra to a real one as given here

Theorems & Definitions (292)

  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Example 2.4
  • Theorem 2.5
  • Definition 2.6
  • Example 2.7
  • Remark 1
  • Definition 2.8
  • Remark 2
  • ...and 282 more