2014
DOI: 10.1016/j.jfa.2014.08.024
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C-algebras from planar algebras II: The Guionnet–Jones–Shlyakhtenko C-algebras

Abstract: We study the C * -algebras arising in the construction of Guionnet-Jones-Shlyakhtenko (GJS) for a planar algebra. In particular, we show they are pairwise strongly Morita equivalent, we compute their K-groups, and we prove many properties, such as simplicity, unique trace, and stable rank 1. Interestingly, we see a K-theoretic obstruction to the GJS C * -algebra analog of Goldman-type theorems for II 1 -subfactors. This is the second article in a series studying canonical C * -algebras associated to a planar a… Show more

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Cited by 9 publications
(8 citation statements)
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“…Note that the distinguished vertex has minimal weight 1, so Assumption 3.3 holds. We have the following lemma from [HP14] which connects the GJS C * -algebra to the free loop algebras discussed in Section 3.1. 4.2.…”
Section: Application To Subfactor Theorymentioning
confidence: 99%
See 2 more Smart Citations
“…Note that the distinguished vertex has minimal weight 1, so Assumption 3.3 holds. We have the following lemma from [HP14] which connects the GJS C * -algebra to the free loop algebras discussed in Section 3.1. 4.2.…”
Section: Application To Subfactor Theorymentioning
confidence: 99%
“…Application to subfactor theory. The original motivation in our two articles [HP17,HP14] was to develop a connection between subfactor theory and C * -algebras with a view toward connections to Connes' non-commutative geometry [Con94]. The standard invariant of a finite index subfactor forms a shaded subfactor planar algebra [Jon99].…”
Section: Introductionmentioning
confidence: 99%
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“…Let Γ = (V, E) be as in Subsection 1.4. In contrast with the tracial setting in which one typically equips Γ with a vertex weightings (see [Har13,HP14,Har17,HN18b]), we will equip Γ with an edge weighting: a function µ : E → R + satisfying µ(e op ) = µ(e) −1 . Note that this is more general since any vertex weighting µ 0 : V → R + defines an edge weighting µ by µ(e) := µ 0 (t(e)) µ 0 (s(e)) .…”
Section: Graph Algebras With Edge Weightingsmentioning
confidence: 99%
“…As is [HP14,Har17], we set S(Γ, µ) to be the C*-algebra generated by A together with {Y e : e ∈ E}. The arguments used in [HP14,Theorem 5.19] can be used to show the following:…”
Section: Graph Algebras With Edge Weightingsmentioning
confidence: 99%