compact_quantum_group
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compact_quantum_group [2019/10/04 09:18] d.gromada |
compact_quantum_group [2021/11/23 11:56] (current) |
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| Let $\pi$ be the [[wp>Gelfand–Naimark–Segal_construction|GNS representation]] of $C(G)$ corresponding to the Haar state $h$. We denote by $L^2(G)$ the corresponding Hilbert space. It holds that the Haar state $h$ is //faithful// on $\Pol(G)$ (i.e. $h(aa^*)=0\Leftrightarrow a=0$). Hence, $\pi$ provides a faithful representation of $\Pol(G)$ on $L^2(G)$. | Let $\pi$ be the [[wp>Gelfand–Naimark–Segal_construction|GNS representation]] of $C(G)$ corresponding to the Haar state $h$. We denote by $L^2(G)$ the corresponding Hilbert space. It holds that the Haar state $h$ is //faithful// on $\Pol(G)$ (i.e. $h(aa^*)=0\Leftrightarrow a=0$). Hence, $\pi$ provides a faithful representation of $\Pol(G)$ on $L^2(G)$. | ||
| - | Conversely, for any Hopf $*$-algebra with a positive integral $A_0$, we can consider its universal C*-completion (see bellow) $A:=C^*(A_0)$, which defines a compact quantum group. This provides an alternative //algebraic// definition of compact quantum groups. | + | Conversely, for any Hopf $*$-algebra with a positive integral $A_0$, we can consider its universal C*-completion (see below) $A:=C^*(A_0)$, which defines a compact quantum group. This provides an alternative //algebraic// definition of compact quantum groups. |
| ==== The universal C*-algebra ==== | ==== The universal C*-algebra ==== | ||
compact_quantum_group.1570180726.txt.gz · Last modified: 2021/11/23 11:56 (external edit)
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