Convergence of Peter--Weyl Truncations of Compact Quantum Groups
Malte Leimbach
TL;DR
The paper develops a rigorous framework for approximating coamenable compact quantum groups by Peter–Weyl truncations in the setting of compact quantum metric spaces. It endows compressed algebras $A^{(\Lambda)}=P_{\Lambda}A P_{\Lambda}$ with bi-invariant Lip-norms and coaction structure, and proves convergence to the full function algebra $A=\mathrm{C}(\mathbb{G})$ in Kerr’s complete Gromov–Hausdorff distance by constructing Lip-norm morphisms via compression and symbol maps and exploiting a density of liftable states. A key technical tool is a suitable approximate inverse to compression provided by slice maps, together with Li–type invariance results for Lip-norms under coactions; these yield a convergence theorem for PW truncations that applies to both general CQGs and the special case of compact groups. The work generalizes prior results (e.g., GEvS23) to the operator-system CQMS setting, providing a robust finite-dimensional approximation scheme for noncommutative geometry of quantum groups with potential implications for numerical and analytical studies in quantum group theory and noncommutative metric geometry.
Abstract
We consider a coamenable compact quantum group $\mathbb{G}$ as a compact quantum metric space if its function algebra $\mathrm{C}(\mathbb{G})$ is equipped with a Lip-norm. By using a projection $P$ onto direct summands of the Peter--Weyl decomposition, the $\mathrm{C}^*$-algebra $\mathrm{C}(\mathbb{G})$ can be compressed to an operator system $P\mathrm{C}(\mathbb{G})P$, and there are induced left and right coactions on this operator system. Assuming that the Lip-norm on $\mathrm{C}(\mathbb{G})$ is bi-invariant in the sense of Li, there is an induced bi-invariant Lip-norm on the operator system $P\mathrm{C}(\mathbb{G})P$ turning it into a compact quantum metric space. Given an appropriate net of such projections which converges strongly to the identity map on the Hilbert space $\mathrm{L}^2(\mathbb{G})$, we obtain a net of compact quantum metric spaces. We prove convergence of such nets in terms of Kerr's complete Gromov--Hausdorff distance. An important tool is the choice of an appropriate state whose induced slice map gives an approximate inverse of the compression map $\mathrm{C}(\mathbb{G}) \ni a \mapsto PaP$ in Lip-norm.