Operator System Perspectives at Truncated Noncommutative Geometry

This thesis collects some contributions to the approach to noncommutative geometry in terms of spectral truncations. Connes—van Suijlekom proposed to study the operator systems arising as compressions of the C*-algebra by spectral projections of the Dirac operator in a spectral triple, and whether these compressions converge in a suitable sense to the original spectral triple when the spectral projections converge strongly to the identity. This question can be posed more precisely in Rieffel’s setting of compact quantum metric spaces, and with Kerr—Li’s quantum version of Gromov—Hausdorff distance for compact quantum metric spaces modeled on operator systems. We show that spectral truncations of (the canonical spectral triples associated to) tori converge in this quantum Gromov—Hausdorff sense. Moreover, we obtain a similar convergence result for coamenable compact quantum groups; in this case we consider compressions by projections arising from the Peter—Weyl decomposition, and general Lip-norms satisfying appropriate invariance properties. Last, we study the duality of operator systems of Toeplitz matrices which arise in truncations of discrete groups; in particular, we make the connections to sums of squares and extension problems for partially defined positive semi-definite functions precise.