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On the Product of Real Spectral Triples

Authors:	Vanhecke F J

Institution:	(1) Instituto de Física, UFRJ, Ilha do Fundão, Rio de Janeiro, Brasil

Abstract:	The product of two real spectral triples $\left\{ {\mathcal{A}_1 ,\mathcal{H}_1 ,\mathcal{D}_1 ,\mathcal{J}_1 ,{\gamma }_{\text{1}} } \right\}$ and $\left\{ {\mathcal{A}_2 ,\mathcal{H}_2 ,\mathcal{D}_2 ,\mathcal{J}_2 ,\left( {{\gamma }_{2} } \right)} \right\}$ , the first of which is necessarily even, was defined by A.Connes as $\left\{ {\mathcal{A},\mathcal{H},\mathcal{D},\mathcal{J}\left( {,{\gamma }} \right)} \right\}$ given by $\mathcal{A} = \mathcal{A}_1 \otimes \mathcal{A}_2 ,\mathcal{H} = \mathcal{H}_1 \otimes \mathcal{H}_2 ,\mathcal{D} = \mathcal{D}_1 \otimes {\text{Id}}_{\text{2}} + {\gamma }_{1} \otimes \mathcal{D}_2 ,\mathcal{J} = \mathcal{J}_1 \otimes \mathcal{J}_2$ and, in the even-even case, by ${\gamma }_{1} \otimes {\gamma }_{2}$ . Generically it is assumed that the real structure $\mathcal{J}$ obeys the relations $\mathcal{J}^2 = \varepsilon {\text{Id}}$ , $\mathcal{J}\mathcal{D} = \varepsilon '\mathcal{D}\mathcal{J}$ , $\mathcal{J}{\gamma = }\varepsilon '{\gamma }\mathcal{J}$ , where the $\varepsilon$ -sign table depends on the dimension n modulo 8 of the spectral triple. If both spectral triples obey Connes' $\varepsilon$ >-sign table, it is seen that their product, defined in the straightforward way above, does not necessarily obey this $\varepsilon$ -sign table. In this Letter, we propose an alternative definition of the product real structure such that the $\varepsilon$ -sign table is also satisfied by the product.

Keywords:	noncommutative geometry real spectral triples
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