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Quantum isometry groups of spectral triples associated with approximately finite-dimensional C*-algebras are shown to arise as inductive limits of quantum symmetry groups of corresponding truncated Bratteli diagrams. This is used to determine explicitly the quantum isometry group of the natural spectral triple on the algebra of continuous functions on the middlethird Cantor set. It is also shown that the quantum symmetry groups of finite graphs or metric spaces coincide with the quantum isometry groups of the corresponding classical objects equipped with natural Laplacians.
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For $mu in (0,1), c> 0,$ we identify the quantum group $SO_mu(3)$ as the universal object in the category of compact quantum groups acting by `orientation and volume preserving isometries in the sense of cite{goswami2} on the natural spectral triple on the Podles sphere $S^2_{mu, c}$ constructed by Dabrowski, DAndrea, Landi and Wagner in cite{{Dabrowski_et_al}}.
Suppose that a compact quantum group $clq$ acts faithfully on a smooth, compact, connected manifold $M$, i.e. has a $C^*$ (co)-action $alpha$ on $C(M)$, such that the action $alpha$ is isometric in the sense of cite{Goswami} for some Riemannian structure on $M$. We prove that $clq$ must be commutative as a $C^{ast}$ algebra i.e. $clqcong C(G)$ for some compact group $G$ acting smoothly on $M$. In particular, the quantum isometry group of $M$ (in the sense of cite{Goswami}) coincides with $C(ISO(M))$.
We give local upper and lower bounds for the eigenvalues of the modular operator associated to an ergodic action of a compact quantum group on a unital C*-algebra. They involve the modular theory of the quantum group and the growth rate of quantum dimensions of its representations and they become sharp if other integral invariants grow subexponentially. For compact groups, this reduces to the finiteness theorem of Hoegh-Krohn, Landstad and Stormer. Consequently, compact quantum groups of Kac type admitting an ergodic action with a non-tracial invariant state must have representations whose dimensions grow exponentially. In particular, S_{-1}U(d) acts ergodically only on tracial C*-algebras. For quantum groups with non-involutive coinverse, we derive a lower bound for the parameters 0<lambda<1 of factors of type III_lambda that can possibly arise from the GNS representation of the invariant state of an ergodic action with a factorial centralizer.
Suppose that a compact quantum group ${mathcal Q}$ acts faithfully on a smooth, compact, connected manifold $M$, i.e. has a $C^{ast}$ (co)-action $alpha$ on $C(M)$, such that $alpha(C^infty(M)) subseteq C^infty(M, {mathcal Q})$ and the linear span of $alpha(C^infty(M))(1 otimes {mathcal Q})$ is dense in $C^infty(M, {mathcal Q})$ with respect to the Frechet topology. It was conjectured by the author quite a few years ago that ${mathcal Q}$ must be commutative as a $C^{ast}$ algebra i.e. ${mathcal Q} cong C(G)$ for some compact group $G$ acting smoothly on $M$. The goal of this paper is to prove the truth of this conjecture. A remarkable aspect of the proof is the use of probabilistic techniques involving Brownian stopping time.
A general form of contractive idempotent functionals on coamenable locally compact quantum groups is obtained, generalising the result of Greenleaf on contractive measures on locally compact groups. The image of a convolution operator associated to a contractive idempotent is shown to be a ternary ring of operators. As a consequence a one-to-one correspondence between contractive idempotents and a certain class of ternary rings of operators is established.
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