Quantum isometries and noncommutative geometry

Abstract

The space $\mathbb C^N$ has no free analogue, but we can talk instead about the free sphere $S^{N-1}_{\mathbb C,+}$, as the manifold defined by the equations $\sum_ix_ix_i^*=\sum_ix_i^*x_i=1$. We discuss here the structure and hierarchy of the submanifolds $X\subset S^{N-1}_{\mathbb C,+}$, with particular attention to the manifolds having an integration functional $tr:C(X)\to\mathbb C$.Comment: 400 pages. arXiv admin note: text overlap with arXiv:1909.0815