Smooth symbols of a circular element: calculi, quantizations, and obstructions

Definitions and mathematical statement

Let $c=x+iy$ be a normalized circular element in a finite tracial von Neumann algebra, where $x,y$ are free centered semicircular elements of variance $1/2$. Then $\sigma(c)=\overline{\mathbb D}$, and put $M=W^*(c)$. The source asks whether every $f\in C^\infty({\mathbb{C}})$ can be assigned an operator $f(c)$ affiliated with $M$, allowing unbounded operators. Affiliation means that a closed densely defined operator commutes with the unitaries in $M'$ in the usual domain-preserving sense. The desired algebraic, continuity, adjoint, and spectral-locality requirements are not specified.

The aid first records an obstruction: a unital multiplicative map from the commutative algebra ${\mathbb{C}}[z,\bar z]$ sending $z$ to $c$ and $\bar z$ to $c^*$ would imply $cc^*=c^*c$, contrary to circular nonnormality. Thus no such full commutative star-calculus exists.

It nevertheless constructs ordered quantizations. Let $\mu_x,\mu_y$ be the variance-$1/2$ semicircle laws. For separated symbols $f(s+it)=\sum_kg_k(s)h_k(t)$ define \[ Q_{xy}(f)=\sum_kg_k(x)h_k(y). \] The aid states the exact identity \[ \|Q_{xy}(f)\|_{L^2(M,\tau)} =\|f\|_{L^2(\mu_x\otimes\mu_y)}, \] which extends to every symbol in that $L^2$ space, as a measurable affiliated operator. For smooth symbols it reports boundedness of the ordered double-operator-integral realization. Reverse ordering replaces $g_k(x)h_k(y)$ by $h_k(y)g_k(x)$ and generally changes the result. For example the symbol $st$ gives $xy$ versus $yx$. These quantizations are not asserted to be multiplicative, nor to agree with holomorphic functional calculus on every holomorphic polynomial.

The product semicircle law occupies a square extending outside the unit disk. Hence the aid reports failure of spectral locality: a smooth symbol vanishing near $\sigma(c)$ can still have nonzero ordered quantization. For Weyl ordering it supplies a different exact trace distribution, for the appropriate smooth test class, \[ \tau(Q_{\mathrm W}(f))=\frac1{2\pi}\int_{|z|\le\sqrt2}f(z)\,{\,\mathrm{d}}^2z. \] This radius-$\sqrt2$ disk differs from the circular element's Brown measure, uniform on the unit disk. The full Weyl construction and its symbol-class hypotheses are not included in the record.

In a separate holomorphic direction, the aid states that every nonzero polynomial $q(c)$ is injective with dense range. Its inverse is therefore an affiliated operator, and \[ (p/q)(c)=p(c)q(c)^{-1} \] defines a canonical multiplicative rational calculus, with products understood as affiliated-operator products and their closures. Poles in the spectral disk are not asserted to produce bounded inverses. This rational extension, the ordered smooth quantizations, and the obstruction address different compatibility requirements; they do not specify one canonical smooth calculus satisfying all possible interpretations of the source question.

Short English statement

A circular element cannot have a commutative multiplicative star-calculus for all smooth symbols. The aid instead gives ordering-dependent quantizations and a separate multiplicative rational calculus, with different locality and boundedness properties. The original question needs those desired properties specified.

Sources

• Local UnsolvedMath registry, permanent ID 20002686, source code catalogue entry; created and updated 14 August 2026. Canonical aim-probability-notes.json, zero-based notes[127], tag question, item 26.
• AIM "Free Analysis," section 0.9, "Focus Group on Invariant Subspaces (day 4)," according to the supplied verification: https://aimath.org/WWN/freeanalysis/freeanalysis.pdf. Wording is marked exact; original remarks and literature field are empty.
• Research attempt 1: partial_result, problem status at run partially_solved, rigor label proved, novelty confidence low. No complete authoritative references or full double-operator/Weyl construction hypotheses are supplied.
• Ulam AI / UnsolvedMath Contributors, CC BY 4.0. Machine-generated research aid, not peer reviewed; independently verify before citation.