Given a countably generated rigid C * -tensor category C, we construct a planar algebra P • whose category of projections Pro is equivalent to C. From P • , we use methods of Guionnet-Jones-Shlyakhtenko-Walker to construct a rigid C * -tensor category Bim whose objects are bifinite bimodules over an interpolated free group factor, and we show Bim is equivalent to Pro. We use these constructions to show C is equivalent to a category of bifinite bimodules over L(F ∞ ). C 2012 American Institute of Physics.As in the subfactor case, Hayashi and Yamagami gave a positive result for amenable rigid C * -tensor categories 16 (amenability for C * -tensor categories was first studied by Hiai and Izumi 15 ). Moreover, given a rigid C * -tensor category C, Yamagami constructed a category of bifite bimodules C bim over an amalgamated free product I I 1 -factor such that C bim is equivalent to C. 36 However, one can show these factors have property , so they are not interpolated free group factors (we briefly sketch this in the Appendix).In this paper, we give a result analogous to Popa and Shlyakhtenko's results for L(F ∞ ) for countably generated rigid C * -tensor categories, which answers part of Question 9 in Sec. 6 of Ref. 26. Recall that a rigid C * -tensor category C is generated by a set of objects S if for every Y ∈ C, there are X 1 , . . . , X n ∈ S such thatTheorem 1.3: Every countably generated rigid C * -tensor category can be realized as a category of bifinite bimodules over L(F ∞ ). Remark 1.4:Note that when C is finitely generated, we can prove Theorem 1.3 using 34 by adapting the technique in Theorem 4.1 of Ref. 9. We provide a sketch of the proof in the Appendix, where we also point out some difficulties of using the results of Ref. 34 when C is not finitely generated (see also Sec. 4 of Ref. 36).Hence we choose to use planar algebra technology to prove Theorem 1.3 since it offers the following advantages. First, the same construction works for both the finitely and infinitely generated cases. Second, planar diagrams arise naturally in the study of tensor categories, and a reader familiar with the diagrams may benefit from a planar algebraic approach. Third, we get an elegant description of the bimodules over L(F ∞ ) directly from the planar algebra (see Secs. III D and III E).There are three steps to the proof of Theorem 1.3.(1) Given a countably generated C * -tensor category C, we get a factor planar algebra P • such that the C * -tensor category Pro of projections of P • is equivalent to C. A factor planar algebra (called a fantastic planar algebra in Ref. 24) is an unshaded, spherical, evaluable C * -planar algebra. This step is well known to experts; we give most of the details in Sec. II. (2) Given a factor planar algebra P • , we construct a I I 1 -factor M and two rigid C * -tensor categories of bifinite bimodules over M:• Bim, built entirely from P • and obviously equivalent to Pro, and • CF, formed using Connes' fusion and linear operators.These categories are defined in Definitions 3.23 and 3.25. W...

We summarize the known obstructions to subfactors with principal graphs which begin with a triple point. One is based on Jones's quadratic tangles techniques, although we apply it in a novel way. The other two are based on connections techniques; one due to Ocneanu, and the other previously unpublished, although likely known to Haagerup.We then apply these obstructions to the classification of subfactors with index below 5. In particular, we eliminate three of the five families of possible principal graphs called "weeds" in the classification from [MS10].

Using Jones' quadratic tangles formulas, we automate the construction of the 4442, 3333, 3311, and 2221 spoke subfactors by finding sets of 1-strand jellyfish generators. The 4442 spoke subfactor is new, and the 3333, 3311, and 2221 spoke subfactors were previously known.Comment: 44 pages, many figures, to appear in Transactions of the American Mathematical Societ

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.