Kernel of the L2-closed free difference quotient

Determine whether the kernel of the L2-closure of the operator-valued free difference quotient ∂_{X:B}: L2(B X, τ) → L2(B X ⊗ B X, τ ⊗ τ) coincides with L2(B, τ); equivalently, establish whether ker(∂_{X:B}) = L2(B, τ) in general for the B-valued free difference quotient associated to a formal variable X over B.

Background

In operator-valued free probability, the free difference quotient ∂{X:B} provides a non-commutative analogue of a derivative acting on B-valued non-commutative polynomials in a formal variable X over B. Its L2-closure ∂{X:B} is defined on the Hilbert space completions L2(B X, τ) and L2(B X ⊗ B X, τ ⊗ τ) with respect to a faithful tracial state τ.

The equality ker(∂_{X:B}) = L2(B, τ) is a central structural question raised in the AIM 2006 “Free Analysis” workshop, seeking to identify exactly the constants (elements of B) as the kernel of the closed free difference quotient in the operator-valued setting. The present paper proves this equality in specific settings related to B-valued semicircular systems and presents counterexamples to a naive form of Voiculescu’s operator-valued free Poincaré conjecture, but the general kernel characterization remains unresolved.