A Note on Tortrat Groups

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A locally compact group G is called a Tortrat group if for any probability measure λ on G which is not idempotent, the closure of {gλg−1 | g∈G} does not contain any idempotent measure. We show that a connected Lie group G is a Tortrat group if and only if for all g∈G 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.

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