A spectral triple for noncommutative compact surfaces

A Dirac operator is presented that will yield a $1^+$-summable regular even spectral triple for all noncommutative compact surfaces defined as subalgebras of the Toeplitz algebra. Connes’ conditions for noncommutative spin geometries are analyzed and it is argued that the failure of some requirements is mainly due to a wrong choice of a noncommutative spin bundle.