Free products of hyperfinite von Neumann algebras and free
              dimension

Dykema, Ken

doi:10.1215/s0012-7094-93-06905-0

Cited by 97 publications

(157 citation statements)

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“…Since rN 2 r contains a diffuse abelian von Neumann subalgebra with identity r, it follows from [9] that W * (qC q) contains a diffuse abelian von Neumann algebra with identity q. From this, one obtains the desired diffuse self-adjoint element in W * (qC q), and hence a unitary v ∈ qC q of trace zero.…”

Section: Form the Reduced Amalgamated Free Productmentioning

confidence: 94%

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

Hartglass

Penneys

2014

Journal of Functional Analysis

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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 algebra. This is the published version of arXiv:1401.2486.

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Section: Form the Reduced Amalgamated Free Productmentioning

confidence: 94%

“…(See[9,10].) Suppose that A, B, and C are tracial C * -algebras withD = p A ⊕B * C,and D is endowed with the canonical free product trace.…”

mentioning

confidence: 99%

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

Hartglass

Penneys

2014

Journal of Functional Analysis

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“…where r γ ≤ p γ and τ (r γ ) = µ(γ) − α∼γ n α,β µ(α). Moreover, the parameter, t, can be computed using Dykema's "free dimension" formulas [Dyk93,DR13]. In particular, M(Γ, µ) is a factor if and only if V > is empty.…”

Section: Introductionmentioning

confidence: 99%

Free transport for interpolated free group factors

Hartglass

Nelson

2018

Journal of Functional Analysis

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In this article, we study a form of free transport for the interpolated free group factors, extending the work of Guionnet and Shlyakhtenko for the usual free group factors [GS14]. Our model for the interpolated free group factors comes from a canonical finite von Neumann algebra M(Γ, µ) associated to a finite, connected, weighted graph (Γ, V, E, µ) [Har13, Har15]. With this model, we use an operator-valued version of Voiculescu's free difference quotient introduced in [Har15] to state a Schwinger-Dyson equation which is valid for the generators of M(Γ, µ). We construct free transport for appropriate perturbations of this equation. Also, M(Γ, µ) can be constructed using the machinery of Shlyakhtenko's operator-valued semicircular systems [Shl99]. arXiv:1702.08643v1 [math.OA] 28 Feb 2017 J * ([P + J f ] −1 ) = M #(x + f ) + (DW )(x + f ). Then, using P P +J f = P − J f P +J f and J * (P ) = M #x, we obtain J * J f P + J f + M #f + (DW )(X + f ) = 0

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“…In later work [Har13,BHP12], the author developed a canonical free-product von Neumann algebra, M(Γ, µ) that one can associate to an arbitrary undirected, weighted, graph with weighting µ : V (Γ) → R + . The key observation in [Har13] is that if (Γ, µ) is a subgraph of (Γ , µ), then there is a canonical, possibly nonunital, inclusion M(Γ, µ) → M(Γ , µ), compressions of which are a standard embedding in the sense of [Dyk93]. This observation showed that if a planar algebra P • is infinite depth, then in the construction in [GJS10], M k ∼ = L(F ∞ ) for all k.…”

Section: Introductionmentioning

confidence: 99%

Free Product C* -algebras Associated with Graphs, Free Differentials, and Laws of Loops

Hartglass¹

2017

Can. j. math.

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Abstract. We study a canonical C * -algebra, S(Γ, µ), that arises from a weighted graph (Γ, µ), speci c cases of which were previously studied in the context of planar algebras. We discuss necessary and su cient conditions of the weighting which ensure simplicity and uniqueness of trace of S(Γ, µ), and study the structure of its positive cone. We then study the * -algebra, A, generated by the generators of S(Γ, µ), and use a free di erential calculus and techniques of Charlesworth and Shlyakhtenko, as well as Mai, Speicher, and Weber to show that certain "loop" elements have no atoms in their spectral measure. A er modifying techniques of Shlyakhtenko and Skoufranis to show that self adjoint elements x ∈ M n (A) have algebraic Cauchy transform, we explore some applications to eigenvalues of polynomials in Wishart matrices and to diagrammatic elements in von Neumann algebras initially considered by Guionnet, Jones, and Shlyakhtenko.

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Free products of hyperfinite von Neumann algebras and free dimension

Cited by 97 publications

References 10 publications

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

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

Free transport for interpolated free group factors

Free Product C* -algebras Associated with Graphs, Free Differentials, and Laws of Loops

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