E-theory spectra

This thesis combines the fields of functional analysis and topology.

$C^\ast$-algebras are analytic objects used in non-commutative 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 quasi-topological spaces and form a quasi-spectrum, that is a sequence of based quasi-topological spaces with specific structure maps. We consider actions of the orthogonal group and we obtain a orthogonal quasi-spectrum 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 quasi-spectrum to $K$-theory and $K$-homology orthogonal quasi-spectra.