An involution semigroup or $\star$-semigroup is a unary semigroup $\langle S,{\cdot}\,,{}^\star\rangle$ that satisfies the equations $$ (x^\star)^\star = x \quad \text{and} \quad (xy)^\star = y^\star x^\star. $$ Common examples include $\langle M_n(\mathbb{R}),{\cdot}\,,{}^T\rangle$ and $\langle M_n(\mathbb{C}),{\cdot}\,,{}^*\rangle$. An inverse $\star$-semigroup is a $\star$-semigroup that satisfies the equations $$xx^\star x = x \quad \text{and} \quad xx^\star yy^\star = yy^\star xx^\star. {\tag1\label{1}}$$

Using standard methods from linear algebra, it is quite easy to show:

• Every periodic sub-$\star$-semigroup of $\langle M_n(\mathbb{C}),{\cdot}\,,{}^*\rangle$ is an inverse $\star$-semigroup.

In other words, every periodic sub-$\star$-semigroup of $\langle M_n(\mathbb{C}),{\cdot}\,,{}^*\rangle$ satisfies the equations \eqref{1}. Recall that a $\star$-semigroup $\langle S,{}\cdot\,,{}^\star\rangle$ is periodic if $(\forall a \in S)(\exists m \geq 1) \ a^{2m} = a^m$.

A colleague suggested that this may be a special case of some known general result, probably in operator theory. Therefore, I would like to ask if anyone is aware of a reference for the above result.