Moyal planes are spectral triples
Axioms for nonunital spectral triples, extending those introduced in the unital case by \textit{A. Connes} [Commun. Math. Phys. 182, 155--176 (1996; Zbl 0881.58009)], are given. A \textit{compact noncommutative spin manifold} consists of a \textit{spectral triple} \((\mathcal A, \mathcal H, D)\), where \(\mathcal A\) is a unital algebra, represented on a Hilbert space \(\mathcal H\), together with a distinguished self-adjoint operator, the abstract Dirac operator \(D\), whose resolvent is completely continuous, such that each operator \([D, a]\) for \(a \in \mathcal A\) is bounded. The authors show how to build noncompact noncommutative spin geometries: The indispensable commutative example of noncompact manifolds is considered. Then using a suitably chosen Dirac operator, the geometry associated to the Moyal product is laid out. As a consequence, the classical background of present-day noncommutative field theory is recast in the framework of the rigorous Connes formalism for geometrical noncommutative spaces. The spaces \(\mathbb R^{2N}\) endowed with Moyal products are investigated, and some physical applications, such as the construction of noncommutative Wick monomials and the computation of the Connes-Lott functional action, are given for these noncommutative hyperplanes. The main results have been summarized in [\textit{V. Gayral}, Lett. Math. Phys. 65, 147--157 (2003; Zbl 1043.46051)].
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