## Directly finite algebras of pseudofunctions on locally compact groups

#### Authors

Yemon Choi

DOI: 10.1017/S0017089514000573

MSC 2010: 22D15, 43A15 (primary); 22D25, 46L05 (secondary.

#### Status

Appeared as Glasgow Math. Journal 57 (2015) no. 3, 693–707, © Cambridge University Press.

Preprint version available at arXiv 1205.4354

#### Local copy

CUP have informed me that as the author, I retain

the right to post the definitive version of the contribution as published at Cambridge Journals Online (in PDF or HTML form) on their personal or departmental web page, no sooner than upon its appearance at Cambridge Journals Online, subject to file availability and provided the posting includes a prominent statement of the full bibliographical details, a copyright notice in the name of the copyright holder (Cambridge University Press or the sponsoring Society, as appropriate), and a link to the online edition of the journal at Cambridge Journals Online.

So on that basis, here is a copy of the definitive online-advance version.

#### Abstract

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 ${\rm C}^\ast$-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.

This paper uses material from, and hence supersedes, the unpublished preprint arXiv 1003.1650

For anyone comparing the arXiv date with the date of publication: the original version of this article, which was somewhat discursive and expository, was submitted for publication to another journal in 2012, and rejected in mid-2013. For some reason I was only notified of this in 2014...

In any case, this prompted a rethink of style, a rewrite, and a resubmission. So really the published version should be regarded as written in 2014, although most of the mathematical "meat" was worked out by 2012.

Note added September 2017. It is implicitly asked, in the remarks before Proposition 2.7, if a Banach algebra $A$ with a dense subalgebra that is directly finite must itself be finite. The answer turns out to be negative: in

N. J. Laustsen, J. T.White. An infinite C*-algebra with a dense, stably finite *-subalgebra, Proc. Amer. Math. Soc. 146 (2018), 2523–2528

the authors exhibit a $*$-semigroup $S$ for which $\ell^1(S)$ is directly finite, together with a $*$-homomorphic embedding of $\ell^1(S)$ into a $C^*$-algebra that is not directly finite.

Yemon Choi