On order units in the augmentation ideal

Abstract

It was shown by Ozawa that the Laplacian is an order unit in the augmentation ideal of the group ring, and this fact was crucial in his characterization of property (T) in terms of positivity of certain group ring elements. Here we show that for groups with finite abelianization there is a deeper reason for this fact: the augmentation ideal is an image of a positive map from a matrix algebra that maps the identity matrix to the Laplacian.