Noncommutative motives, numerical equivalence, and semi-simplicity

Making use of Hochschild homology, we introduce the correct category ${\rm NNum}(k)_F$ of noncommutative {\it numerical} motives (over a base ring $k$ and with coefficients in a field $F$). We prove that ${\rm NNum}(k)_F$ is abelian semi-simple and that Grothendieck's category ${\rm Num}(k)_{\Bbb{Q}}$ of numerical motives embeds into ${\rm NNum}(k)_{\Bbb{Q}}$ after being factored out by the action of the Tate object. As an application we obtain an alternative proof of Jannsen's celebrate semi-simplicity result, which uses the noncommutative world instead of a classical Weil cohomology.