[Paper Review] An orthogonal approach to the subfactor of a planar algebra

Vaughan F. R. Jones, Dimitri Shlyakhtenko+1 more|ArXiv.org|Jul 25, 2008

Advanced Operator Algebra Research4 references15 citations

TL;DR

This paper presents a diagrammatic orthogonalization technique for the graded algebra of a subfactor planar algebra, enabling a concise proof that the GNS construction of the algebra yields a II₁ factor whose standard invariant recovers the original planar algebra. By introducing an orthogonal basis via a change of basis using epi and non-nested epi Temperley-Lieb diagrams, the authors simplify the inner product and multiplication structures, proving isomorphism between the orthogonal and original constructions without relying on Fock space or C*-algebra machinery.

ABSTRACT

By changing to an orthogonal basis, we give a short proof that the subfactor of the graded algebra of a planar algebra reproduces the planar algebra.

Motivation & Objective

• To simplify the GNS construction of the subfactor associated to a planar algebra by replacing the standard non-orthogonal basis with an orthogonal one.
• To eliminate reliance on full Fock space or graph C*-algebras in proving the II₁ factor tower result.
• To provide a direct, diagrammatic proof that the subfactor of the graded algebra reproduces the original planar algebra.
• To establish an isomorphism between the orthogonal construction and the original GJS construction via an explicit change of basis.
• To demonstrate that the orthogonal basis preserves algebraic structure while simplifying inner product computations.

Proposed method

• Introduce an orthogonal basis for the graded algebra $Gr_k(rak{P})$ using a change of basis defined by sums of epi and non-nested epi Temperley-Lieb diagrams.
• Define the map $X$ as the sum of all epi TL diagrams, and $Y$ as the sum of non-nested epi TL diagrams with sign $(-1)^{i-j}$, forming an inverse pair.
• Prove that $X$ and $Y$ are mutual inverses via inclusion-exclusion on innermost/outermost turn-backs in TL diagrams.
• Show that the multiplication $\star$ in the orthogonal basis satisfies $X(a \bullet b) = X(a) \star X(b)$, preserving the algebraic structure.
• Establish that the inner product $\langle\langle a,b\rangle\rangle$ in the original basis equals $\langle X(a), X(b) \rangle$ in the orthogonal basis.
• Use boundedness of left multiplication maps to verify that the GNS construction applies directly to the orthogonal algebra.

Experimental results

Research questions

• RQ1 Can the subfactor construction of a planar algebra be re-proven using an orthogonal basis to simplify the inner product structure?
• RQ2 Does the orthogonal basis preserve the algebraic multiplication and *-structure of the original graded algebra?
• RQ3 Is the GNS construction on the orthogonal algebra isomorphic to the original GJS construction without invoking Fock space?
• RQ4 Can the isomorphism between the orthogonal and original algebras be explicitly constructed via diagrammatic means?
• RQ5 Does the orthogonal basis allow for a more direct proof of the II₁ factor tower result with minimal functional analytic machinery?

Key findings

• The map $X$, defined as the sum of all epi TL diagrams, provides a change of basis from the original algebra to an orthogonal one.
• The inverse map $Y$, summing non-nested epi diagrams with alternating signs, satisfies $XY = YX = 1$, proving the basis change is invertible.
• The multiplication $\star$ in the orthogonal basis satisfies $X(a \bullet b) = X(a) \star X(b)$, ensuring algebra isomorphism.
• The inner product $\langle\langle a,b\rangle\rangle$ in the original basis equals $\langle X(a), X(b) \rangle$ in the orthogonal basis, preserving the trace structure.
• Left multiplication by elements of $Gr_k(\frak{P})$ is bounded in the orthogonal basis, enabling the GNS construction to yield a II₁ factor.
• The resulting II₁ factor has standard invariant isomorphic to the original planar algebra, confirming the tower construction via orthogonal methods.