Does anyone know of a point-set level model for complex K-theory as a commutative ring spectrum?

For real $K$-theory I know of "A symmetric ring spectrum representing KO-theory" by Michael Joachim (Topology, 40(2) 2001).

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    $\begingroup$ I think Joachim's proof works verbatim by taking complex vector spaces instead of real vector spaces. $\endgroup$ – Denis Nardin Jun 15 at 20:28
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    $\begingroup$ Schwede's symmetric spectra book project (math.uni-bonn.de/people/schwede/SymSpec-v3.pdf, see Example 1.20) constructs ku and KU as commutative rings in symmetric spectra. $\endgroup$ – Arun Debray Jun 16 at 4:44

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