CMS/SMC
Canadian Mathematical Society
www.cms.math.ca
Canadian Mathematical Society
  location:  PublicationsjournalsCJM
Abstract view

Convolution of Trace Class Operators over Locally Compact Quantum Groups

  Published:2012-09-10
 Printed: Oct 2013
  • Zhiguo Hu,
    Department of Mathematics and Statistics, University of Windsor, Windsor, Ontario, N9B 3P4
  • Matthias Neufang,
    School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, K1S 5B6
  • Zhong-Jin Ruan,
    Department of Mathematics, University of Illinois, Urbana, IL 61801, USA
Features coming soon:
Citations   (via CrossRef) Tools: Search Google Scholar:
Format:   LaTeX   MathJax   PDF  

Abstract

We study locally compact quantum groups $\mathbb{G}$ through the convolution algebras $L_1(\mathbb{G})$ and $(T(L_2(\mathbb{G})), \triangleright)$. We prove that the reduced quantum group $C^*$-algebra $C_0(\mathbb{G})$ can be recovered from the convolution $\triangleright$ by showing that the right $T(L_2(\mathbb{G}))$-module $\langle K(L_2(\mathbb{G}) \triangleright T(L_2(\mathbb{G}))\rangle$ is equal to $C_0(\mathbb{G})$. On the other hand, we show that the left $T(L_2(\mathbb{G}))$-module $\langle T(L_2(\mathbb{G}))\triangleright K(L_2(\mathbb{G})\rangle$ is isomorphic to the reduced crossed product $C_0(\widehat{\mathbb{G}}) \,_r\!\ltimes C_0(\mathbb{G})$, and hence is a much larger $C^*$-subalgebra of $B(L_2(\mathbb{G}))$. We establish a natural isomorphism between the completely bounded right multiplier algebras of $L_1(\mathbb{G})$ and $(T(L_2(\mathbb{G})), \triangleright)$, and settle two invariance problems associated with the representation theorem of Junge-Neufang-Ruan (2009). We characterize regularity and discreteness of the quantum group $\mathbb{G}$ in terms of continuity properties of the convolution $\triangleright$ on $T(L_2(\mathbb{G}))$. We prove that if $\mathbb{G}$ is semi-regular, then the space $\langle T(L_2(\mathbb{G}))\triangleright B(L_2(\mathbb{G}))\rangle$ of right $\mathbb{G}$-continuous operators on $L_2(\mathbb{G})$, which was introduced by Bekka (1990) for $L_{\infty}(G)$, is a unital $C^*$-subalgebra of $B(L_2(\mathbb{G}))$. In the representation framework formulated by Neufang-Ruan-Spronk (2008) and Junge-Neufang-Ruan, we show that the dual properties of compactness and discreteness can be characterized simultaneously via automatic normality of quantum group bimodule maps on $B(L_2(\mathbb{G}))$. We also characterize some commutation relations of completely bounded multipliers of $(T(L_2(\mathbb{G})), \triangleright)$ over $B(L_2(\mathbb{G}))$.
Keywords: locally compact quantum groups and associated Banach algebras locally compact quantum groups and associated Banach algebras
MSC Classifications: 22D15, 43A30, 46H05 show english descriptions Group algebras of locally compact groups
Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.
General theory of topological algebras
22D15 - Group algebras of locally compact groups
43A30 - Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.
46H05 - General theory of topological algebras
 

© Canadian Mathematical Society, 2014 : https://cms.math.ca/