Unitary element

Remarks

$U(A)$ is a group under multiplication, with $u^{-1} = u^*$. Example: in $C(\mathbb{T})$, the function $u(z) = z$ satisfies $\overline{u(z)} u(z) = \abs{z}^2 = 1$ for all $z$ on the unit circle, so it is unitary. Non-example: the unilateral shift $S$ on $\ell^2(\N)$ satisfies $S^* S = 1$ but $S S^* \ne 1$, since $S S^*$ kills the first basis vector; so $S$ is an isometry and not a unitary. In a C*-algebra, the C* identity gives $\norm{u}^2 = \norm{u^* u} = 1$, and the spectrum of a unitary lies in the unit circle. If $h$ is self-adjoint, then $e^{ih}$ is unitary.