An invariant for homogeneous spaces of compact quantum groups (Q305716)

The authors present a program producing a dimensional invariant of an ergodic \(C^*\)-dynamical system and work out its explicit computation for some examples, including all type A compact quantum groups.NEWLINENEWLINEMore precisely, let \(G\) be a compact quantum group acting on a \(C^*\)-algebra \(A\) via a \(*\)-homo\-mor\-phism \(\tau:A\rightarrow A\otimes C( G) \) such that the span of \(( I\otimes b) \tau( a) \) with \(a\in A\) and \(b\in C( G) \) is dense in \(A\otimes C( G) \). We call \(( A,G,\tau) \) an ergodic \(C^*\)-dynamical system when \(A\) is a homogeneous space of \(G\) in the sense that the fixed point algebra \(\{ a\in A:\tau( a) =a\otimes I\} \) is \(\mathbb{C}I\) and hence \(A\) has a unique invariant state \(\rho\) satisfying \(( \rho\otimes\mathrm{id}) \tau( a) =\rho( a) I\) for all \(a\in A\) and related to the Haar state \(h\) on \(G\) by \(( \mathrm{id}\otimes h) \tau( a) =\rho( a) I\) for \(a\in A\).NEWLINENEWLINEFor a given ergodic \(C^*\)-dynamical system \(( A,G,\tau) \), it is natural to consider the GNS representation \(( \mathcal{H}_{\rho} ,\pi_{\rho},\eta_{\rho}) \) associated with \(\rho\), where the Hilbert space \(\mathcal{H}_{\rho}\) is a completion of a quotient linear space of \(A\) and carries a unitary action of \(G\), described by a unitary element \(u_{\tau}\) of the multiplier algebra \(M( \mathcal{K}( \mathcal{H}_{\rho }) \otimes C( G) ) \), canonically arising from the action \(\tau\) of \(G\) on \(A\) such that \(( \pi_{\rho},u_{\tau}) \) is a covariant representation of \(( A,G,\tau) \) in the sense that \(( \pi_{\rho}\otimes\mathrm{id}) \tau( a) =u_{\tau }( \pi_{\rho}( a) \otimes I) u_{\tau}^{\ast}\) for all \(a\in A\). Furthermore, there is an associated dense \(*\)-subalgebra of \(A\) defined as the algebra \(\mathcal{A}\) of all \(a\in A\) with \(\tau( a) \) in the algebraic tensor product of \(A\) and \(A( G) \), where \(A( G) \) is the dense \(*\)-subalgebra of \(C( G) \) generated by the matrix entries of irreducible unitary representations of \(G\). Let \(\mathcal{E}\) be the class of spectral triples \(( \mathcal{H}_{\rho },\pi_{\rho},D) \) for \(\mathcal{A}\), equivariant with respect to \(( \pi_{\rho},u_{\tau}) \) in the sense that \(D\otimes I\) commutes with \(u_{\tau}\). Then the authors introduce the definition of the spectral dimension \(\mathcal{S}\dim( A,G,\tau) \) of \(( A,G,\tau) \) as the infimum of \(p>0\) such that \(D\) is \(p\)-summable for some spectral triple \(( \mathcal{H}_{\rho},\pi_{\rho},D) \in\mathcal{E}\).NEWLINENEWLINEThe spectral dimension \(\mathcal{S}\dim\) is then computed for concrete examples, including the canonical cases of \(SU( 2) \) acting on \(C( SU( 2) ) \), \(\mathbb{T}^{n}\) acting on a noncommutative torus \(A_{\Theta}\), \(SU_{q}( \ell+1) \) acting on \(C( SU_{q}( \ell+1) ) \), \(SU_{q}( \ell+1) \) acting on \(C( \mathbb{S}_{q}^{2\ell+1}) \), \(SU_{q}( 2) \) acting on the Podleś sphere \(C( S_{q0}^{2}) \), and \(A_{u}( Q) \) acting on Cuntz algebras.