The following question/problem has been bugging me on and off for some time now: so I thought it might be worth broaching here on MO, as a case of "ask the experts".

Let $G$ be a discrete group. Kaplansky observed that since the group von Neumann algebra $VN(G)$ is a finite von Neumann algebra, each left-invertible element in $VN(G)$ is invertible. A proof is outlined in

M.S. Montgomery, Left and right inverses in group algebras, Bull. AMS 75 (1969)

(Well, she actually states a weaker result, but inspection shows that her argument extends to give what we claim. See also my remarks on this previous MO answer.) The basic idea is to exploit the faithful trace $T\mapsto \langle T\delta_e,\delta_e\rangle$ and how it behaves on idempotents: for if $ab=I$, then $ba$ is an idempotent.

In particular, each left-invertible element of the reduced group $C^*$-algebra is invertible.

Question. What can we say for the full group $C^*$-algebra? Is every left-invertible element in $C^*(G)$ automatically invertible?

Some basic observations:

• The case where $G$ is the free group on two generators follows from a result of M-D Choi [no relation] who showed that $C^*({\mathbb F}_2)$ embeds into a direct product of matrix algebras.

• More generally, if $C^*(G)$ has a faithful trace then one can use the same argument as for the reduced $C^*$-algebra to get a positive answer.

• If $C^*(G)$ has no non-trivial projections then $ab=I$ implies $ba=I$. (I think this was known to be true for $G={\mathbb F}_2$ but I've forgotten the reference at present.)

• There are examples of $G$ where $C^*(G)$ has no faithful trace; these can be found in work of Bekka and Louvet, and come from exploiting Property (T).

Bekka, M. B.(F-METZ-MM); Louvet, N.(CH-NCH) Some properties of $C^*$-algebras associated to discrete linear groups. $C^*-algebras (Münster, 1999), 1–22, Springer, Berlin, 2000.

The following question/problem has been bugging me on and off for some time now: so I thought it might be worth broaching here on MO, as a case of "ask the experts".

Let $G$ be a discrete group. Kaplansky observed that since the group von Neumann algebra $VN(G)$ is a finite von Neumann algebra, each left-invertible element in $VN(G)$ is invertible. A proof is outlined in

M.S. Montgomery, Left and right inverses in group algebras, Bull. AMS 75 (1969)

(Well, she actually states a weaker result, but inspection shows that her argument extends to give what we claim. See also my remarks on this previous MO answer.) The basic idea is to exploit the faithful trace $T\mapsto \langle T\delta_e,\delta_e\rangle$ and how it behaves on idempotents: for if $ab=I$, then $ba$ is an idempotent.

In particular, each left-invertible element of the reduced group $C^*$-algebra is invertible.

Question. What can we say for the full group $C^*$-algebra? Is every left-invertible element in $C^*(G)$ automatically invertible?

Some basic observations:

• The case where $G$ is the free group on two generators follows from a result of M-D Choi [no relation] who showed that $C^*({\mathbb F}_2)$ embeds into a direct product of matrix algebras.

• More generally, if $C^*(G)$ has a faithful trace then one can use the same argument as for the reduced $C^*$-algebra to get a positive answer.

• If $C^*(G)$ has no non-trivial projections then $ab=I$ implies $ba=I$. (I think this was known to be true for $G={\mathbb F}_2$ but I've forgotten the reference at present.)

• There are examples of $G$ where $C^*(G)$ has no faithful trace; these can be found in work of Bekka and Louvet, and come from exploiting Property (T).

Bekka, M. B.(F-METZ-MM); Louvet, N.(CH-NCH) Some properties of $C^*$-algebras associated to discrete linear groups. $C^*-algebras (Münster, 1999), 1–22, Springer, Berlin, 2000.