= Möbius transform of a unitary element
{c}
{title2=$v=(u+\lambda1)(1+\overline\lambda u)^{-1},\quad|\lambda|<1$}
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 $\|\overline\lambda u\|<1$. 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 $f(u)^*f(u)=f(u)f(u)^*=1$. The complex conjugate in the denominator is necessary for the unit-modulus calculation.
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