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Register a new userExistence and examples of quantum isometry group for a class of compact metric spaces
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Authors
Debashish Goswami
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We formulate a definition of isometric action of a compact quantum group (CQG) on a compact metric space, generalizing Banicas definition for finite metric spaces. For metric spaces $(X,d)$ which can be isometrically embedded in some Euclidean space, we prove the existence of a universal object in the category of the compact quantum groups acting isometrically on $(X,d)$. In fact, our existence theorem applies to a larger class, namely for any compact metric space $(X,d)$ which admits a one-to-one continuous map $f : X raro IR^n$ for some $n$ such that $d_0(f(x),f(y))=phi(d(x,y))$ (where $d_0$ is the Euclidean metric) for some homeomorphism $phi$ of $IR^+$. As concrete examples, we obtain Wangs quantum permutation group $cls_n^+$ and also the free wreath product of $IZ_2$ by $cls_n^+$ as the quantum isometry groups for certain compact connected metric spaces constructed by taking topological joins of intervals in cite{huang1}.
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Starting with a vertex-weighted pointed graph $(Gamma,mu,v_0)$, we form the free loop algebra $mathcal{S}_0$ defined in Hartglass-Penneys article on canonical $rm C^*$-algebras associated to a planar algebra. Under mild conditions, $mathcal{S}_0$ is a non-nuclear simple $rm C^*$-algebra with unique tracial state. There is a canonical polynomial subalgebra $Asubset mathcal{S}_0$ together with a Dirac number operator $N$ such that $(A, L^2A,N)$ is a spectral triple. We prove the Haagerup-type bound of Ozawa-Rieffel to verify $(mathcal{S}_0, A, N)$ yields a compact quantum metric space in the sense of Rieffel. We give a weighted analog of Benjamini-Schramm convergence for vertex-weighted pointed graphs. As our $rm C^*$-algebras are non-nuclear, we adjust the Lip-norm coming from $N$ to utilize the finite dimensional filtration of $A$. We then prove that convergence of vertex-weighted pointed graphs leads to quantum Gromov-Hausdorff convergence of the associated adjusted compact quantum metric spaces. As an application, we apply our construction to the Guionnet-Jones-Shyakhtenko (GJS) $rm C^*$-algebra associated to a planar algebra. We conclude that the compact quantum metric spaces coming from the GJS $rm C^*$-algebras of many infinite families of planar algebras converge in quantum Gromov-Hausdorff distance.
We show that any quantum family of maps from a non commutative space to a compact quantum metric space has a canonical quantum semi metric structure.
We show that there is a one-to-one correspondence between compact quantum subgroups of a co-amenable locally compact quantum group $mathbb{G}$ and certain left invariant C*-subalgebras of $C_0(mathbb{G})$. We also prove that every compact quantum subgroup of a co-amenable quantum group is co-amenable. Moreover, there is a one-to-one correspondence between open subgroups of an amenable locally compact group $G$ and non-zero, invariant C*-subalgebras of the group C*-algebra $C^*(G)$.
Suppose that a compact quantum group Q acts faithfully and isomet- rically (in the sense of [10]) on a smooth compact, oriented, connected Riemannian manifold M . If the manifold is stably parallelizable then it is shown that the compact quantum group is necessarily commutative as a C ast algebra i.e. Q = C(G) for some compact group G. Using this, it is also proved that the quantum isometry group of Rieffel deformation of such manifold M must be a Rieffel-Wang deformation of C(ISO(M))
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.
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