A Note on Tortrat Groups

A locally compact group G is called a Tortrat group if for any probability measure on G which is not idempotent, the closure of {gg ^–1 | gG} does not contain any idempotent measure. We show that a connected Lie group G is a Tortrat group if and only if for all gG all eigenvalues of Ad g are of absolute value 1. Together with well-known results this also implies that a connected locally compact group is a Tortrat group if and only if it is of polynomial growth.