Schreier’s Formula for Some Free Probability Invariants

Let Gα (M,τ ) be a trace-preserving action of a finite group G on a tracial von Neumann algebra. Suppose that A ⊂ M is a finitely generated α-invariant unital *-subalgebra. We give a formula relating the von Neumann dimension of the space of derivations on A valued on its coarse bimodule to the von Neumann dimension of the space of derivations on A ᵃˡᵍ_α G valued on its coarse bimodule, which is reminiscent of Schreier’s formula for finite index subgroups of free groups. This formula induces a formula for the free Stein dimension (defined by Charlesworth and Nelson), Derc,1⊗ 1(A,τ ), and Δ (defined by Connes and Shlyakhtenko). The latter is done by establishing that Δ is equal to the von Neumann dimension of a certain subspace of the derivation space of A, similar to that of the free Stein dimension. Using the formula for Δ, we strengthen the recent results of Shlyakhtenko on the microstates free entropy dimension.