Free Analysis and Planar Algebras

We study 2-cabled analogs of Voiculescu’s trace and free Gibbs states on Jones planar algebras. These states are traces on a tower of graded algebras associated to a Jones planar algebra. Among our results is that, with a suitable de nition, niteness of free Fisher information for planar algebra traces implies that the associated tower of von Neumann algebras consists of factors, and that the standard invariant of the associated inclusion is exactly the original planar algebra. We also give conditions that imply that the associated von Neumann algebras are non-Γ non-L rigid factors. This is the published version of arXiv:1411.0268. Introduction A recent series of papers [9, 10, 4, 2] investigated towers of von Neumann algebras associated to a Jones planar algebra. To such a planar algebra, one rst associates a sequence of algebras Grk. Next, a special trace (the Voiculescu trace τ ) is constructed on each of these algebras. It turns out that W (Grk, τ) are factors (in fact, interpolated free group factors), and the standard invariant of their Jones tower is the original planar algebra P (compare with Popa’s results [25]). A question dating back to [9] is whether there are other “nice” choices of traces on Grk; in particular, one is interested in questions such as factoriality of W (Grk, τ) for various choices of τ , as well as the computation of the standard invariant of the resulting tower of algebras. A step in studying more general traces was taken in [11], where it was shown that certain random matrix models lead to “free Gibbs states” on planar algebras. Such a free Gibbs state corresponds to a certain trace τk on each Grk, and these traces have interesting combinatorial properties related to questions of enumeration of planar maps. The question of the isomorphism class of algebras associated to some of these free Gibbs states has been recently settled by B. Nelson [15] using his non-tracial extension [14] of free monotone transport introduced in [13]. He showed that if the potential of the free Gibbs state is su ciently close to the Voiculescu trace (in the sense that the potential is su ciently close to the quadratic one), then the von Neumann algebras generated by Grk under the free Gibbs state and the Voiculescu trace are isomorphic (and are thus interpolated free group factors). In this paper we investigate a di erent collection of traces, among which a special role is played by a trace which turns out to be as canonical as Voiculescu’s trace. We call the trace the 2-cabled Voiculescu trace. The main di erence from the Voiculescu trace is the replacement of summation over all TemplerleyLieb diagrams with summations over 2-cabled Temperley-Lieb diagrams. The relationship between the 2-cabled Voiculescu trace and the usual Voiculescu is akin to the di erence between circular and semicircular systems. Not surprisingly, many of the results from [9, 10] can be reproved for the 2-cabled version. We sketch the proofs of several of these. Among these results is the existence of 2-cabled versions of free Gibbs states. ?: Research supported by an NSF postdoctoral fellowship and DARPA Award HR0011-12-1-0009. †: Partially supported by ANR grant NEUMANN. ‡: Research supported by NSF grant DMS-1161411 and DARPA Award HR0011-12-1-0009. 115 2014 Maui and 2015 Qinhuangdao conferences in honour of Vaughan F. R. Jones’ 60th birthday Volume 46 of the Proceedings of the Centre for Mathematics and its Applications Page 116 It turns out that the 2-cabled setup is nicely amenable to using tools from free probability theory. Voiculescu’s free analysis di erential calculus has a nice diagrammatic expression adapted to this situation. The diagrammatic calculus is technically better behaved than the one used in the 1-cabled situation [11, 15]. Using ideas of [7] (which relied on some techniques of J. Peterson [24]), we are able to prove that if the free Fisher information of a planar algebra trace is nite, then the von Neumann algebras generated by Grk are factors, and the standard invariant of the resulting Jones tower is again the planar algebra P . This is in particular the case for any (two-cabled) free Gibbs state (whether or not the potential is close to quadratic). A consequence of our work is that possible phase transitions phenomena arising when one changes the potential of a free Gibbs state away from quadratic potentials cannot be captured by a change in the standard invariant. The outline of the paper is as follows. In the following section we recall some background material on graded algebras associated to planar algebras from [9, 10, 4]. In Section 2 we develop some general theory for traces on the graded algebra of a planar algebra. In particular we introduce planar algebra cumulants, which generalize the free cumulants of Speicher, and construct some new examples of planar algebra traces which generalize the Voiculescu trace studied in [9, 10, 4]. In §3, we adapt Voiculescu’s free di erential calculus to the planar algebra setting. Our main result states that if the planar algebra version of Voiculescu’s free Fisher information of a planar algebra trace is nite, then the tower of algebras associated to this trace has the same standard invariant as the original planar algebra. This is in particular the case when the trace we consider is the free Gibbs state associated to a potential. In this case, we also show that the associated von Neumann algebras are non-Γ, non L-rigid and prime. In §4 we further analyze the 2-cabled Voiculescu trace, and establish the isomorphism classes of the associated von Neumann algebras. In §5 we consider free Gibbs states on planar algebras, which are 2-cabled versions of those studied in [11]. Finally, §6 constructs a free monotone transport between the 2-cabled Voiculescu trace and free Gibbs states with potential su ciently close to the quadratic potential 1 2 . The proof is essentially the same as that of B. Nelson [15]. In particular, we show that Grk under such free Gibbs states still generate free group factors. 1. Graded algebras associated to planar algebras 1.1. Background. We begin by brie y recalling some constructions from [9], [4]. Let P = (Pn)n≥0 be a subfactor planar algebra. For n, k ≥ 0 let Pn,k be a copy of Pn+k. Elements of Pn,k will be represented by diagrams k k 2n x . In this diagrams and in diagrams below thick lines represent several parallel strings; sometimes, we add a numerical label to indicate their number (if the numeral is absent, the number of lines is presumed to be arbitrary, or understood from context). Thus in this diagram, the thick lines to the left and right each represent k strings, and the thick line at top represents 2n strings. We will typically suppress the marked point ?, and take the convention that it occurs at the top-left corner which is adjacent to an unshaded region. De ne a product ∧k : Pn,k × Pm,k → Pn+m,k by