Convergence of Peter--Weyl Truncations of Compact Quantum Groups

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 ${\mathrm{C}}^{\ast }$-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.