On the product of real spectral triples

The product of two real spectral triples {A(1), H-1, D-1, J(1), gamma(1)} and {A(2), H-2, D-2, J(2)(,gamma(2))}, the first of which is necessarily even, was defined by A.Connes as {A, H, D, J(,gamma)} given by A = A(1) x A(2), H = H-1 x H-2, D = D-1 x Id(2) + gamma(1) x D-2, J = J(1) x J(2) and, in the even-even case, by gamma = gamma(1) x gamma(2). Generically it is assumed that the real structure J obeys the relations J(2) = epsilon Id, JD = epsilon'DJ, J gamma = epsilon "gamma J, where the epsilon-sign table depends on the dimension n modulo 8 of the spectral triple. If both spectral triples obey Connes' epsilon-sign table, it is seen that their product, defined in the straightforward way above, does not necessarily obey this epsilon-sign table. In this Letter, we propose an alternative definition of the product real structure such that the epsilon-sign table is also satisfied by the product.