We will define L_1 Banach lattices and recall some of its model theoretic properties.  We will then consider group algebras associated to locally compact groups, where the multiplication is convolution and we will consider them as L_1 Banach lattices.  We will show that such expansions carry deep information about the underlying group. For example, when the group is discrete, the group will be definable inside the expansion. In particular, we show, for discrete groups, that if two group algebras are elementary equivalent, then the corresponding groups are elementary equivalent.

This is joint work with K. Gannon and S. Song.