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Abstract
The set of non-commutative rational functions on n indeterminates, called the free skew field and denoted ℂ(<x1, ..., xn>) [Ami66], can sometimes be evaluated on a tuple of random variables in a tracial von Neumann algebra X1, ..., Xn ∈ (M, τ), resulting in a set in Aff(M) of non-commutative rational expressions in the random variables X1, ..., Xn, 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 xi’s, and a column vector [CR99].
The graded algebra construction Grk(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.
