Product of real spectral triples (Q2891200)

It is known, that the spectral triple \(({\mathcal A}, {\mathcal H}, D)\) extends the Riemannian spin structure of an \(n\)-dimensional manifold \(X\) to the case when \({\mathcal A}\) is no longer the commutative algebra \(C(X)\). It is interesting to find a spectral triple analog for the Cartesian product of such manifolds; this is the main topic of the present paper. The authors introduce a real structure on the triple \(({\mathcal A}, {\mathcal H}, D)\) as an anti-linear isometry \(J: {\mathcal H}\to {\mathcal H}\) satisfying certain properties. It is shown, that the (algebraic) tensor product \({\mathcal A}_1\otimes {\mathcal A}_2\) admits two real structures on the corresponding spectral triple (for even dimensions). The text contains extensive tables of possible tensor products, which complete the known results of \textit{F.~J.~Vanhecke} [Lett. Math. Phys. 50, No. 2, 157--162 (1999; Zbl 1079.58507)].