Hancox, Jason and Belton, Alexander and Lindsay, Martin
(2018)
*Quantum stochastic flows on universal partial isometry matrix C*-algebras.*
PhD thesis, Lancaster University.

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## Abstract

We give a simple algebraic construction of quantum stochastic flows on universal C*-algebras generated by partial isometry matrix relations. This is a large class of C*-algebras that subsumes the family of graph C*-algebras and, more generally, Cuntz-Krieger algebras. The construction expands on the main results of the 2015 paper by Belton and Wills which builds quantum stochastic flows from a stochastic flow generator defined on a dense *-subalgebra, subject to a growth condition. We characterise such quantum stochastic flows on the Cuntz algebras and give examples of when the growth condition is achieved. As a specialisation of the above we then consider Levy processes on universal C*-bialgebras generated by partial isometry matrices. Similarly to the C*-algebra scenario this class of C*-bialgebras is a large class which includes all universal compact quantum groups. The added structure of the C*-bialgebra removes the necessity of the growth condition required for quantum stochastic flows on C*-algebras. We construct a new family of universal C*-bialgebras which we call the deformed biunitary C*-bialgebras. The class of deformed biunitary C*-bialgebras includes the universal unitary compact quantum groups of Van Daele and Wang. We consider a sub-family of the deformed biunitary C*-bialgebras, the isometry C*-bialgebras and scrutinise the structure of Levy processes on theses C*-bialgebras. Included in the isometry *-bialgebras is the Toeplitz algebra; we also examine this very closely. We investigate how Levy processes on the Toeplitz algebra act on its commutative sub-C*-algebras. To motivate the noncommutative setting we also consider classical Markov chains in terms of kernels. In so doing, we prove some characterisation results for bounded linear operators on some Banach spaces related to measurable and topological spaces in terms of kernels.