C*-algebras from planar algebras I: Canonical C*-algebras associated to a planar algebra

Abstract

From a planar algebra, we give a functorial construction to produce numerous associated C ∗ ^* -algebras. Our main construction is a Hilbert C ∗ ^* -bimodule with a canonical real subspace which produces Pimsner-Toeplitz, Cuntz-Pimsner, and generalized free semicircular C ∗ ^* -algebras. By compressing this system, we obtain various canonical C ∗ ^* -algebras, including Doplicher-Roberts algebras, Guionnet-Jones-Shlyakhtenko algebras, universal (Toeplitz-) Cuntz-Krieger algebras, and the newly introduced free graph algebras. This is the first article in a series studying canonical C ∗ ^* -algebras associated to a planar algebra.

Free transport for interpolated free group factors

“…(by [HP14a]) these hypothesis are satisfied for the canonical map from B to A. Moreover, if the above is a strict inequality, Lemma 2.1 and an argument similar to Lemma 37 in [Dab14] tells us that this map is injective.…”

“…We call τ the free graph law corresponding to (Γ, µ). As shown in [HP14a], τ is a faithful tracial state on S(Γ, µ).…”

“…The algebras M(Γ, µ) were initially studied in [GJS11] in determining the isomorphism classes of von Neumann algebras arising from planar algebras. Slightly more general versions of M(Γ, µ) were studied by the first author in [Har13], and the C*-algebra counterparts S(Γ, µ) were studied in [HP14a,HP14b,Har15]. L(F t ) t = 6(a − a 2 ) a 1-a n a ∈ [ 1 3 , 1) n ≥ 1 L(F t ) t = (n − 6)a 2 + 6a…”

“…We denote S(Γ, µ) to be the C*-algebra generated by ∞ (V ) and (x ) ∈ E . From [Shl99,HP14a], there is a faithful conditional expectation E : S(Γ, µ) → ∞ (V ) given by…”

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

“…In an effort to study the diagrammatic C * -algebras (A 0 ⊂ A 1 ⊂ A 2 ⊂ · · · ) that arise in the construction in [GJS10], the author and Penneys defined the C * -algebra analogue of M(Γ, µ), called S(Γ, µ) [HP14a,HP14b]. Although this algebra was defined for an arbitrary weighting µ, most properties of S(Γ, µ) were studied only in the case of Γ being the principal graph of a planar algebra with µ the associated weighting.…”