In a class talking about $C^*$ algebra and (higher) index theory, I heard a marvelous theorem (which is proved by Kaplansky according to the speaker), that is

Suppose $\Gamma$ is a group (admitting Haar measure if necessary) while $\mathbb{C} \Gamma := \widehat{C_c(\Gamma)}$ is the group $C^*$ algebra of $\Gamma$ ($\widehat{(-)}$ means completion with largest $C^*$-norm $\|f\|:= \sup_{\pi \, \text{a Repn of} \, L^1(\Gamma)} \|\pi(f)\|_{H_{\pi}}$).

Then if $a,b \in \mathbb{C} \Gamma, ab = 1$ one derives $ba = 1$.

This result is far more deep than ordinary ring theory. And the speaker said the most astonishing thing is that the only way to prove it is operator algebra theory.

I want to know if what he said is true. If true, how to prove?

The speaker also remarked that the $C^*$ algebra condition is necessary. There are counterexamples for usual group algebra $F\Gamma$ where $F$ a field and $\Gamma$ as above. How to find one?

I also have few knowledge about Kaplansky's work on noncommutative ring/module theory. And His Unit Conjecture goes

For $\mathbb{F}$ a field and $\Gamma$ a group, $a \in \mathbb{F} \Gamma^{\times}$ iff $a \in \Gamma$.

In 2022 (Maybe) A paper gave an counterexample, which was a big news that time. Can that counterexample solve this problem?

Any comments will be helpful.

In a class talking about $C^*$ algebra and (higher) index theory, I heard a theorem (related to Kaplansky, proved?), that is

Suppose $\Gamma$ is a group (admitting Haar measure if necessary) while $\mathbb{C} \Gamma := \widehat{C_c(\Gamma)}$ is the group $C^*$ algebra of $\Gamma$ ($\widehat{(-)}$ means completion with largest $C^*$-norm $\|f\|:= \sup_{\pi \, \text{a Repn of} \, L^1(\Gamma)} \|\pi(f)\|_{H_{\pi}}$).

Then if $a,b \in \mathbb{C} \Gamma, ab = 1$ one derives $ba = 1$.

I want to know if what he said is true. If true, how to prove?

The speaker also remarked that the $C^*$ algebra condition is necessary. There are counterexamples for usual group algebra $F\Gamma$ where $F$ a field and $\Gamma$ as above. How to find one?

I also have few knowledge about Kaplansky's work on noncommutative ring/module theory. And His Unit Conjecture goes

For $\mathbb{F}$ a field and $\Gamma$ a group, $a \in \mathbb{F} \Gamma^{\times}$ iff $a \in \Gamma$.

Any comments will be helpful.