Noncommutative Gamma_2 and sharp depolarizing curvature frontiers

Definitions and mathematical statement

The source asks for a noncommutative Bakry--Émery criterion and an analogue of $\Gamma_2$ for Dirichlet forms. The aid selects the algebraic operator-order formulation, not an assertion that all quantum curvature notions coincide.

Let $P_t=e^{-tA}$ be a trace-symmetric unital completely positive Markov semigroup on a finite tracial von Neumann algebra, with positive $L^2$-generator $A$. On a common $*$-algebra core where every product and iterated generator term is defined, put \[ \Gamma(x,y)=\tfrac12\left(A(x^*)y+x^*A(y)-A(x^*y)\right), \] \[ \Gamma_2(x,y)=\tfrac12\left(\Gamma(x,Ay)+\Gamma(Ax,y)-A\Gamma(x,y)\right). \] For $K\in{\mathbb{R}}$, $N\in(0,\infty]$, the algebraic condition $\mathrm{BE}(K,N)$ means \[ \Gamma_2(x,x)\succeq K\Gamma(x,x)+N^{-1}(Ax)^*(Ax) \] for all core elements, with $1/\infty=0$. Complete $\mathrm{CBE}(K,N)$ requires the same constants after every amplification $A_r=\operatorname{id}_{M_r}\otimes A$. The operator-domain and matrix-level choices are essential parts of a precise criterion.

The aid's finite-dimensional model takes a nonidentity trace-preserving conditional expectation $E:\mathcal M\to\mathcal B$ and $A=\operatorname{id}-E$. For $z=x-E(x)$, \[ \Gamma(x,x)=\tfrac12(z^*z+E(z^*z)),\qquad \Gamma_2(x,x)=\tfrac14z^*z+\tfrac34E(z^*z). \] Let $C_E$ be the least constant with $z^*z\preceq C_E E(z^*z)$ for all $E(z)=0$, and let $C_E^{\mathrm{cb}}$ be the corresponding least constant over all matrix amplifications. Under the stated finite-dimensional nontrivial centered setup, the aid gives the sharp frontier \[ K_E(N)=\frac12+\frac1{C_E+1}-\frac{2C_E}{N(C_E+1)}, \] and the complete frontier with $C_E$ replaced by $C_E^{\mathrm{cb}}$. A condition holds exactly for $K$ at most its frontier. For $E=\operatorname{id}$, all terms vanish and this nontrivial-centered formula is not the appropriate restriction.

For $\mathcal M=M_n({\mathbb{C}})$, $n\ge2$, and $E(x)=\tau_n(x)I$, the aid's exact centered constants are \[ C_E=n,\qquad C_E^{\mathrm{cb}}=n^2-1. \] Thus \[ K_{\mathrm{op}}(N)=\frac12+\frac1{n+1}-\frac{2n}{N(n+1)},\qquad K_{\mathrm{cb}}(N)=\frac12+\frac1{n^2}-\frac{2(n^2-1)}{Nn^2}. \] In particular, complete curvature $K=1/2$ requires and permits $N\ge2(n^2-1)$.

The complete sharpness witness uses $k=n^2-1$ nonidentity Weyl unitaries $W_1,\ldots,W_k$, orthonormal for $\tau_n$ and of trace zero, and \[ Z=\sum_{j=1}^kE_{1j}\otimes W_j\in M_k\otimes M_n. \] Then $E_k(Z)=0$, $E_k(Z^*Z)=I_k\otimes I_n$, while $\|Z^*Z\|=k$, attaining the complete centered constant. These sharp finite-dimensional bounds do not by themselves classify all noncommutative Dirichlet forms or prove every possible functional-inequality consequence of curvature.

Short English statement

The aid fixes an algebraic noncommutative $\Gamma_2$ and distinguishes operator-order curvature from its complete matrix-level form. For the depolarizing generator on $M_n$, it gives sharp curvature-dimension frontiers with centered constants $n$ and $n^2-1$, respectively.

Sources

• Local registry, permanent UnsolvedMath ID $20002695$, exact public code catalogue entry; canonical location aim-probability-notes.json, notes[136].
• AIM, Free Analysis, Question 35, section 0.11, "Focus Group on Dirichlet Forms, from Classical to Quantum (day 5)"; supplied reference https://aimath.org/WWN/freeanalysis/freeanalysis.pdf. The verification field reports prior visual inspection of physical/PDF page 7. No PDF was accessed here. No proposer, original remarks, or literature entry is supplied.
• Supplied research attempt 1: partial_result, run and registry status partially_solved, rigor label proved, low novelty confidence; classification PARTIAL-PROGRESS, created and updated 14 August 2026. The standard algebraic iterate and sharp centered-index calculations are supplied without primary bibliographic references.
• The research aid explicitly says it is machine generated, not peer reviewed, and requires independent verification before citation.