This thesis combines the fields of functional analysis and topology. $C^\ast$algebras are analytic objects used in noncommutative geometry and in particular we consider an invariant of them, namely $E$theory. $E$theory is a sequence of abelian groups defined in terms of homotopy classes of morphisms of $C^\ast$algebras. It is a bivariant functor from the category where objects are $C^\ast$algebras and arrows are $\ast$homomorphisms to the category where objects are abelian groups and arrows are group homomorphisms. In particular, $E$theory is a cohomology theory in its first variable and a homology theory in its second variable. We prove in the case of real graded $C^\ast$algebras that $E$theory has $8$fold periodicity. Further we create a spectrum for $E$theory. More precisely, we use the notion of quasitopological spaces and form a quasispectrum, that is a sequence of based quasitopological spaces with specific structure maps. We consider actions of the orthogonal group and we obtain a orthogonal quasispectrum which we prove has a smash product structure using the categorical framework. Then we obtain stable homotopy groups which give us $E$theory. Finally, we combine these ideas and a relation between $E$theory and $K$theory to obtain connections of the $E$theory orthogonal quasispectrum to $K$theory and $K$homology orthogonal quasispectra.
