Relative Commutants in Maximal Tensor Products

Definitions and mathematical statement

Let $H\cong\ell^2({\mathbb{N}})$ be a separable infinite-dimensional complex Hilbert space, and let $B(H)$ be its algebra of bounded operators. Let $C\subseteq B(H)$ be a unital maximal abelian self-adjoint subalgebra, so \[ C'\cap B(H)=C, \] isomorphic to $L^\infty((0,1),{\,\mathrm{d}} t)$. Thus the MASA is diffuse, with no nonzero minimal projections; it is not the atomic diagonal MASA.

Let $F_2$ be the free group on two generators and $A=C_r^*(F_2)$ its reduced group $C^*$-algebra, the operator-norm closure of the left regular group algebra on $\ell^2(F_2)$. Form \[ B=B(H)\otimes_{\max}A, \] the maximal $C^*$-completion of the algebraic tensor product. Define the relative commutant by \[ (C\otimes1)^c=\{x\in B:x(c\otimes1)=(c\otimes1)x\text{ for all }c\in C\}. \] The right-hand tensor product in the question means the norm-closed canonical image \[ C\otimes A=\overline{C\odot A}^{\,\|\cdot\|_B}\ \subseteq B. \] Since $C$ is abelian, hence nuclear, its minimal and maximal tensor norms with $A$ agree. The canonical map into $B$ is injective: composing with $B(H)\otimes_{\max}A\to B(H)\otimes_{\min}A$ gives the injective minimal-tensor inclusion. This makes the notation on the right unambiguous.

The question is whether \[ (C\otimes1)^c\stackrel{?}{=}C\otimes C_r^*(F_2). \] The right side is always contained in the left because $C$ is abelian. The substantive assertion is that every element commuting with all $c\otimes1$ in the maximal ambient tensor product is a norm limit of finite sums $\sum_i c_i\otimes a_i$, with $c_i\in C$ and $a_i\in C_r^*(F_2)$. It does not ask for a weak-operator closure, a von Neumann tensor product, or the analogous equality in a minimal ambient tensor product.

The supplied assessment dated 2026-08-21 attributes this exact diffuse-MASA relative-commutant question to Wassermann and reports it as open, with no general resolution located by the supplied curation.

Short English statement

Inside the maximal tensor product of $B(H)$ with the reduced free-group algebra, does commuting with the diffuse MASA force an element to lie in the norm-closed tensor product with that MASA? The reverse inclusion is the entire open assertion.

Sources

• Local UnsolvedMath registry, permanent ID $30001012$, exact source code catalogue entry, proposed year 2008; no individual proposer is supplied, while the assessment attributes the question to Wassermann.
• $C^*$-Algebras (2009), Oberwolfach Reports, DOI 10.4171/owr/2008/37, pp. 2095--2164. https://doi.org/10.4171/owr/2008/37. The supplied title contains escaping artifacts; the mathematical title is rendered here in standard notation.
• Wassermann, Tensor products of maximal abelian subalgebras, https://arxiv.org/abs/0711.0131.
• The supplied corrected_verified note identifies a complete relative-commutant question with restored maximal-tensor notation. The open label and assessment checked 2026-08-21 are supplied dated curation, not independently verified; exact current status requires review.