EPSRC Reference: 
EP/N002377/1 
Title: 
W*bundle techniques and the structure of simple C*algebras 
Principal Investigator: 
Tikuisis, Professor AP 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Mathematical Sciences 
Organisation: 
University of Aberdeen 
Scheme: 
Overseas Travel Grants (OTGS) 
Starts: 
05 July 2015 
Ends: 
04 October 2015 
Value (£): 
4,034

EPSRC Research Topic Classifications: 

EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


Related Grants: 

Panel History: 

Summary on Grant Application Form 
C*algebras are mathematical objects that arose from the rigourisation of quantum mechanics. Each C*algebra is a set of continuous linear maps from a Hilbert space to itself, closed under a few natural algebraic and analytic operations. Upon their inception, it was quickly realised that C*algebras can be created in canonical ways from many other mathematical objects, modelling such things as symmetries, timeevolving systems, and large data sets. Time and again, interesting relationships have manifested between properties of the object being input and those of the resulting C*algebra.
For some time, it has been quite clear that different constructions can produce the same C*algebra; this is interesting externally, where it may imply a profound relationship between the differing input data, and internally, where it allows single C*algebras to be studied using the different techniques available from each different way of constructing it. However, a thorough elucidation of what conditions on the input objects produce different C*algebra outputs has yet to be achieved. Achieving this goal amounts to classifying C*algebras: showing that suitable, computable invariants (primarily, Ktheory) are sufficiently sensitive to always distinguish different C*algebras.
It has recently become apparent that to classify C*algebras, one should study regularity properties of the C*algebras  certain properties of C*algebras that indicate they are less complex and more tractable. Regular C*algebras are ones that have low (topological) dimension  in a way that exactly generalises dimension of a space. Just as low dimensional spaces are easier to visualise, it is often easier to prove things about them, to the extent that certain things that are true of all lowdimensional spaces are no longer true in higher dimensions. This carries forward to C*algebras: more and better things can be proven about low dimensional C*algebras than high dimensional ones. Returning to classification, it has been shown in many cases that C*algebras whose invariants take the same value are automatically the same (or isomorphic), provided that the C*algebras have low dimension.
I have been involved in research concerning regularity, and have found that a certain recent tool called W*bundles shows tremendous promise, although its fundamental theory has yet to be developed. From a C*algebra, one produces a W*bundle, and uses this as a tool.
This works because:
(i) the W*bundle has more structure, and it seems that it should be easier to prove things about it than about the C*algebra;
(ii) the W*bundle has a very special relationship to the C*algebra  it contains it in a special way  so that facts about the W*bundle can have important implications for the C*algebra.
The aim of this project is to further our understanding of structure and classification of C*algebras, by developing the theory of W*bundles.

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Organisation Website: 
http://www.abdn.ac.uk 