Use this URL to cite or link to this record in EThOS: https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.703348
Title: Involutive algebras and locally compact quantum groups
Author: Trotter, Steven
ISNI:       0000 0004 6061 2244
Awarding Body: University of Leeds
Current Institution: University of Leeds
Date of Award: 2016
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Abstract:
In this thesis we will be concerned with some questions regarding involutions on dual and predual spaces of certain algebras arising from locally compact quantum groups. In particular we have the $L^1(\G)$ predual of a von Neumann algebraic quantum group $(L^\infty(\G), \Delta)$. This is a Banach algebra (where the product is given by the pre-adjoint of the coproduct $\Delta$), however in general we cannot make this into a Banach $*$-algebra in such a way that the regular representation is a $*$-homomorphism. We can however find a dense $*$-subalgebra $L^1_\sharp(\G)$ that satisfies this property and is a Banach algebra under a new norm. This was originally considered by Kustermans and Vaes when defining the universal C$^*$-algebraic quantum group, however little else has been studied regarding this algebra in general. In this thesis we study the $L^1_\sharp$-algebra of a locally compact quantum group in this thesis. In particular we show how this has a (not necessarily unique) operator space structure such that this forms a completely contractive Banach algebra, we study some properties for compact quantum groups, we study the object for the compact quantum group $\mathrm{SU}_q(2)$ and we study the operator biprojectivity of the $L^1_\sharp$-algebra. In addition we also briefly study some related properties of $C_0(\G)^*$ and its $*$-subalgebra ${C_0(\G)^*}_\sharp$.