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The APS theorem works for any Dirac-type operator; see e.g. the excellent monograph by Booss-Wojchiecowski on this topic.

More than four decades ago, Boutet de Monvel has described a general set-up for dealing with boundary value problems thatmimicks the K-theoretic approach to the index theorem on closed manifolds. For a modern presentation of this point of view I recommend this paper by Melo-Shrohe-Schick arXiv: 1203.5649 and the references therein. It also involves noncommutative geometry because the symbols in the Boutet-de-Monvel calculus of boundary value problems belong to a more complicated algebra than the symbols of operators on compact manifolds.