Use this URL to cite or link to this record in EThOS:  http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.718839 
Title:  Etheory spectra  
Author:  Browne, Sarah Louise 
ISNI:
0000 0004 6349 1046


Awarding Body:  University of Sheffield  
Current Institution:  University of Sheffield  
Date of Award:  2017  
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Abstract:  
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.


Supervisor:  Mitchener, Paul David  Sponsor:  Not available  
Qualification Name:  Thesis (Ph.D.)  Qualification Level:  Doctoral  
EThOS ID:  uk.bl.ethos.718839  DOI:  Not available  
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