### Noncommutative motives, numerical equivalence, and semi-simplicity, by Matilde Marcolli and Goncalo Tabuada

In this article we further the study of the relationship between pure
motives and noncommutative motives. Making use of Hochschild homology, we
introduce the category NNum(k)_F of noncommutative numerical motives (over
a base ring k and with coefficients in a field F). We prove that NNum(k)_F
is abelian semi-simple and that Grothendieck's category Num(k)_Q of
numerical motives embeds in NNum(k)_Q after being factored out by the
action of the Tate object. As an application we obtain an alternative
proof of Jannsen's semi-simplicity result, which uses the noncommutative
world instead of a Weil cohomology.

Matilde Marcolli <matilde@caltech.edu>

Goncalo Tabuada <tabuada@fct.unl.pt>