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.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:718839 |
| Date | January 2017 |
| Creators | Browne, Sarah Louise |
| Contributors | Mitchener, Paul David |
| Publisher | University of Sheffield |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://etheses.whiterose.ac.uk/17812/ |
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