KU comm ring spectrum

Proof. By Definition 19.7.8, we have \[ \Vect (\pt )^{\simeq } \in \CAlg (\CMon (\An )). \] By Proposition 16.6.4, the composite \[ \CMon (\An ) \xrightarrow {(-)^{\grp }} \CGrp (\An ) \xhookrightarrow {\bB ^{\infty }} \Sp \] is symmetric monoidal. Applying it to the commutative algebra object \(\Vect (\pt )^{\simeq }\) gives a commutative algebra object of \(\Sp \), i.e. a commutative ring spectrum. Under Theorem 5.4.6, the functor \(\bB ^{\infty }\colon \CGrp (\An )\to \Sp _{\geq 0}\) is the inverse of the recognition equivalence, and \((-)^{\grp }\) is the same group-completion left adjoint used in Subsection 9.4.2. Its underlying spectrum is therefore precisely \(\ku \). □