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

TL;DR

This paper develops a spectral model for (Real) $K$-theory of graded $C^*$-algebras using homotopy theory, establishing foundational properties, comparing with Kasparov's model, and proving Bott periodicity.

Contribution

It introduces a new spectral model for (Real) $K$-theory and proves Bott periodicity via a Dirac-dual Dirac approach, connecting homotopy theory with operator algebra $K$-theory.

Findings

Established a spectral model for (Real) $K$-theory.

Proved Bott periodicity using a Dirac-dual Dirac method.

Compared the new model with Kasparov's $K$-theory framework.

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.