UCLA

The set of non-commutative rational functions on n indeterminates, called the free skew field and denoted C () [Ami66], can sometimes be evaluated on a tuple of random variables in a tracial von Neumann algebra X_1, ..., X_n ∈ (M, τ), resulting in a set in Aff(M) of non-commutative rational expressions in the random variables X_1, ..., X_n, where Aff(M) ⊇ M is the algebra of affiliated operators [MSY23].

Constructing the free skew field can be done with linearization, a technique for representing non-commutative rational functions as products of a row vector, the inverse of a matrix that is linear in the x_i’s, and a column vector [CR99].

The graded algebra construction Gr_k(P) associated to a planar algebra can be thought of as generalizing the set non-commutative polynomials and matrices over those polynomials [GJS10]. We prove an analog of a linearization result in the context of planar algebras, a step on the path towards the construction of a planar algebra analog of the free skew fieldin n indeterminates.