Isometry groups of twisted reduced group C∗-algebras

Abstract

An isometry of a unital [Formula: see text]-algebra with respect to a spectral triple is a [Formula: see text]-automorphism of the [Formula: see text]-algebra given by the conjugation by a unitary operator which commutes with the Dirac operator. We give a semidirect product topological characterization on the isometry group of a twisted reduced group [Formula: see text]-algebra of a discrete group with respect to the standard spectral triple induced by a length function on the group. We prove that this isometry group is compact in the point-norm topology, and in particular, for a finitely generated discrete group, this isometry group is a compact Lie group in the point-norm topology. We also extend this result to a unital [Formula: see text]-algebra with a filtration, and prove that its isometry group is a compact topological group in the point-norm topology.

On isometries of spectral triples associated to $AF$-algebras and crossed products

“…Park introduced in [14] the concept of isometry in the non-commutative setting (see also [13]). He showed that in the case of the spectral triple given by the continuous functions on a compact Riemannian oriented manifold, the Hilbert space of complex L 2 -forms and the de Rham operator as Dirac, this concept coincides with the ordinary notion of isometry.…”

“…the paragraph before Proposition 2.8 in [10]). Denoting by Aut l pGq the subgroup of AutpGq given by the automorphisms of G which preserve l, the map Φ gives a bijection between Char G ˆIsopA, H, Dq ˆAut l pGq and IsopA, H, Dq ˆIsopC r pGq, M l , 2 pGqq (see [13]). This suggests that in general the product of Iso-groups should embed inside the Iso of some tensor product of spectral triples.…”

“…From the point of view of non-commutative geometry, a natural problem is to understand what is the right notion of isometry for a non-commutative manifold. Such topic was investigated in [14,15,16,13,6] and at the time being the two definitions of noncommutative isometry appearing in these manuscripts are the only natural candidates known to the authors. More precisely, these are the automorphisms implemented by unitaries commuting with the Dirac operator and the automorphisms leading to preservation of the Connes distance on the state space, giving rise to the two groups Iso and ISO, respectively.…”

Isometry groups of inductive limits of metric spectral triples and Gromov-Hausdorff convergence

“…This work requires us to first prove that the group of isometries of a quantum compact metric space is compact. For the Riemannian manifold case, noncommutative tori and other examples, Iso was studied by Park in [31,32], for spectral triples over twisted reduced group C * -algebras associated to a length function as defined in [9] by Long-Wu in [30], for Goffeng-Mesland [15] spectral triples on Cuntz algebras by Conti-Rossi in [11], and for the Christensen-Ivan [6] spectral triples of AF-algebras by Bassi-Conti in [2]. Conti-Farsi consider both Iso and ISO for Kellendonk-Savinien spectral triples in [10].…”

Groups of isometries of the Cuntz algebras

“…Problems of this sort have rarely been addressed before. A recent case in point, though, is the paper [LW17], where suitable spectral triples on twisted reduced C * -algebras of discrete groups are looked at and the corresponding isometry groups are described completely. This is very much in line with the direction established by E. Park in [Eft95], where, to our knowledge, the notion of isometry with respect to a given spectral triple was defined for the first time.…”