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Directly finite algebras of pseudofunctions on locally compact groups. (English) Zbl 1405.22005
Summary: An algebra $$A$$ is said to be directly finite if each left-invertible element in the (conditional) unitization of $$A$$ is right invertible. We show that the reduced group $$\mathrm{C}^\ast$$-algebra of a unimodular group is directly finite, extending known results for the discrete case. We also investigate the corresponding problem for algebras of $$p$$-pseudofunctions, showing that these algebras are directly finite if $$G$$ is amenable and unimodular, or unimodular with the Kunze-Stein property. An exposition is also given of how existing results from the literature imply that $$L^{1}(G)$$ is not directly finite when $$G$$ is the affine group of either the real or complex line.

##### MSC:
 22D15 Group algebras of locally compact groups 43A15 $$L^p$$-spaces and other function spaces on groups, semigroups, etc. 22D25 $$C^*$$-algebras and $$W^*$$-algebras in relation to group representations 46L05 General theory of $$C^*$$-algebras
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