I was looking at *-algebra and C*-algebra and as far as I understood, the latter guarantees each element to have a singular value decomposition, or at least, that every element $a$ is such that $u a v$ is self-adjoint for some unitary elements $u, v$.
Is there anything between these two structures (weaker than C* and stronger than *) that guarantees such claim?
It is not true than in an arbitrary C$^*$-algebra given $a$ there exist unitaries $u,v$ such that $uav$ is selfadjoint.
For instance consider $B(\ell^2(\mathbb N))$ and let $a$ be the unilateral shift. Given any unitaries $u,v$ we have $$ (uav)^*(uav)=v^*a^*u^*uav=v^*a^*av=v^*v=1, $$ while $$ (uav)(uav)^*=uavv^*au^*=uaa^*u^*, $$ which is a projection of codimension $1$. So we cannot have $(uav)^*=uav$.