A semi-finite algebra associated to a subfactor planar algebra

作者：Guionnet A; Jones V; Shlyakhtenko D^*

来源：Journal of Functional Analysis, 2011, 261(5): 1345-1360.

DOI：10.1016/j.jfa.2011.05.004

• Von Neumann algebras
• Free probability
• Subfactors

We canonically associate to any planar algebra two type II(infinity) factors m(+/-). The subfactors constructed previously by the authors in Guionnet et al. (2010) [6] are isomorphic to compressions of m(+/-) to finite projections. We show that each m(+/-) is isomorphic to an amalgamated free product of type 1 von Neumann algebras with amalgamation over a fixed discrete type I von Neumann subalgebra. In the finite-depth case, existing results in the literature imply that m(+/-) congruent to m(-) is the amplification a free group factor on a finite number of generators. As an application, we show that the factors M(j) constructed in Guionnet et al. (in press) [6] are isomorphic to interpolated free group factors L(F(r(j))),r (j) = 1 + 2 delta(-2j) (delta - 1)I, where delta(2) is the index of the planar algebra and / is its global index. Other applications include computations of laws of Jones Wenzl projections.

• 出版日期2011-9-1