Möbius transform of a unitary element ($v=(u+\lambda 1)(1+\overline{\lambda }u{)}^{-1},|\lambda |<1$)

For a Unitary element of a C-star algebra $u$, the displayed element is also unitary. Its denominator has a multiplicative inverse by the Neumann series, since $\Vert \overline{\lambda }u\Vert <1$. The spectrum of $u$ lies on the unit circle, where $|t+\lambda |=|1+\overline{\lambda }t|$. The resulting continuous function $f(t)=(t+\lambda )/(1+\overline{\lambda }t)$ therefore has $\overline{f}f=f\overline{f}=1$ on the spectrum. The continuous functional calculus preserves products and the involution, giving $f(u{)}^{\ast }f(u)=f(u)f(u{)}^{\ast }=1$. The complex conjugate in the denominator is necessary for the unit-modulus calculation.