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 ${C_0(\G)^*}_\sharp$.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:703348 |
| Date | January 2016 |
| Creators | Trotter, Steven |
| Contributors | Daws, Matthew ; Kisil, Vladimir |
| Publisher | University of Leeds |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://etheses.whiterose.ac.uk/16168/ |
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