Conjugacy classes in algebraic monoids

Let $M$ be a connected linear algebraic monoid with zero and a reductive group of units $G$. The following theorem is established.

Theorem. There exist affine subsets ${M_1}, \ldots ,{M_k}$ of $M$, reductive groups ${G_1}, \ldots ,{G_k}$ with antiautomorphisms $^{\ast}$, surjective morphisms ${\theta _i}:{M_i} \to {G_i}$, such that: (1) Every element of $M$ is conjugate to an element of some ${M_i}$, and (2) Two elements $a$, $b$ in ${M_i}$ are conjugate in $M$ if and only if there exists $x \in {G_i}$ such that $x{\theta _i}(a){x^{\ast}} = {\theta _i}(b)$. As a consequence, it is shown that $M$ is a union of its inverse submonoids.