<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msup></mml:math>-algebras from planar algebras II: The Guionnet–Jones–Shlyakhtenko <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mo>⁎</mml:mo></mml:mrow></mml:msup></mml:math>-algebras

Hartglass, Michael; Penneys, David

doi:10.1016/j.jfa.2014.08.024

Cited by 9 publications

(8 citation statements)

References 32 publications

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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%

“…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%

“…In this case, the cutdown S 0 of S(Γ, µ) at v 0 = is isomorphic to the Guionnet-Jones-Shlyakhtenko (GJS) C * -algebra [HP17,HP14]. This algebra is the C * -completion of the graded algebra Gr 0 arising from their diagrammatic reproof [GJS10] of Popa's celebrated subfactor reconstruction theorem [Pop95].…”

Section: Introductionmentioning

confidence: 99%

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Compact Quantum Metric Spaces from Free Graph Algebras

Aguilar¹,

Hartglass²,

Penneys³

2021

Preprint

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Starting with a vertex-weighted pointed graph (Γ, µ, v0), we form the free loop algebra S0 defined in Hartglass-Penneys' article on canonical C * -algebras associated to a planar algebra. Under mild conditions, S0 is a non-nuclear simple C * -algebra with unique tracial state. There is a canonical polynomial subalgebra A ⊂ S0 together with a Dirac number operator N such that (A, L 2 A, N ) is a spectral triple. We prove the Haagerup-type bound of Ozawa-Rieffel to verify (S0, 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 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) C * -algebra associated to a planar algebra. We conclude that the compact quantum metric spaces coming from the GJS C * -algebras of many infinite families of planar algebras converge in quantum Gromov-Hausdorff distance.

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Section: Application To Subfactor Theorymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Compact Quantum Metric Spaces from Free Graph Algebras

Aguilar¹,

Hartglass²,

Penneys³

2021

Preprint

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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%

Non-tracial free graph von Neumann algebras

Hartglass¹,

Nelson²

2018

Preprint

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Given a finite, directed, connected graph Γ equipped with a weighting µ on its edges, we provide a construction of a von Neumann algebra equipped with a faithful, normal, positive linear functional (M(Γ, µ), ϕ). When the weighting µ is instead on the vertices of Γ, the first author showed the isomorphism class of (M(Γ, µ), ϕ) depends only on the data (Γ, µ) and is an interpolated free group factor equipped with a scaling of its unique trace (possibly direct sum copies of C). Moreover, the free dimension of the interpolated free group factor is easily computed from µ. In this paper, we show for a weighting µ on the edges of Γ that the isomorphism class of (M(Γ, µ), ϕ) depends only on the data (Γ, µ), and is either as in the vertex weighting case or is a free Araki-Woods factor equipped with a scaling of its free quasi-free state (possibly direct sum copies of C). The latter occurs when the subgroup of R + generated by µ(e 1 ) • • • µ(e n ) for loops e 1 • • • e n in Γ is non-trivial, and in this case the point spectrum of the free quasi-free state will be precisely this subgroup. As an application, we give the isomorphism type of some infinite index subfactors considered previously by Jones and Penneys.

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Non-tracial Free Graph von Neumann Algebras

Hartglass

Nelson

2020

Commun. Math. Phys.

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C⁎-algebras from planar algebras II: The Guionnet–Jones–Shlyakhtenko C⁎-algebras

Cited by 9 publications

References 32 publications

Compact Quantum Metric Spaces from Free Graph Algebras

Compact Quantum Metric Spaces from Free Graph Algebras

Non-tracial free graph von Neumann algebras

Non-tracial Free Graph von Neumann Algebras

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