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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.

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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