A remark about supramenability and the Macaev norm

We show that a finitely generated group G which satisfies a certain condition with respect to the Macaev norm is supramenable. The condition is equivalent to the existence of quasicentral approximate unit with respect to the Macaev norm relative to the left regular representation of the group and has been studied by the author in connection with perturbation questions for Hilbert space operators. The condition can be also viewed as an analogue with respect to the Macaev norm of Yamasaki parabolicity. We also show that existence of quasicentral approximate units relative to the Macaev norm for n-tuples of operators is not preserved when taking tensor products.