Left Invertibility Implies Right Invertibility in Certain $C^*$-algebras

Question

This is a follow up to this question. In it, it was noted that in certain $C*$-algebras, left invertibility implies right invertibility. It was stated that AF-algebras have this property. Are there any other classes of $C*$-algebras that have this property? I don't know much about $C^*$-algebras at this point in my career, but I'm wondering if the group $C^*$-algebra or the $C^*$-algebra of generated by a unitary representation of a group have this property.

Answer 1

There is indeed a good characterization of when C*-algebras have this property. A projection $p \in A$ is called finite if whenever $0 \leq q \leq p$ and $q$ is Murray-von Neumann equivalent to $p$ (i.e., there exists a partial isometry such that $v^*v = q$ and $vv^* = p$), then $q = p$. A unital C*-algebra is called finite if the unit is.

See Lemma 5.1.2. in Rørdam's "An Introduction to K-Theory for C*-Algebras." The following are equivalent for a unital C*-algebra $A$.

1. $A$ is finite.
2. All isometries in $A$ are unitary.
3. All projections in $A$ are finite.
4. Every left-invertible element in $A$ is invertible.
5. Every right-invertible element in $A$ is invertible.

Note that an element having both a left and right inverse means that it is invertible; so "left invertibility implies right invertibility" is the same as "left (or right) invertibility implies invertibility".