Spaces of multiplicative maps between highly structured ring spectra

We uncover a somewhat surprising connection between spaces of multiplicative maps between $A_{\mathrm{\infty }}$-ring spectra and topological Hochschild cohomology. As a consequence we show that such spaces become infinite loop spaces after looping only once. We also prove that any multiplicative cohomology operation in complex cobordisms theory $MU$ canonically lifts to an $A_{\mathrm{\infty }}$-map $MU\to MU$\mathrm{.}Thisimplies,inparticular,thattheBrown-Petersonspectrum$BP$ splits off $MU$ as an $A_{\mathrm{\infty }}$-ring spectrum.