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

We use homotopy theoretic ideas to study the $K$-theory of (graded, Real) $C^{\ast }$-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.