far as we know, there is no treatment in the literature of the analogous orientation theory for ${A_{\infty}}$ ring spectra.

In §\[sec:ainfty-thom-spectrum\] and in §\[sec:units-via-infty\], we give two independent approaches to the theory of Thom spectra and orientations for ${A_{\infty}}$ ring spectra. Our first approach begins by adapting the ideas of the ${E_{\infty}}$ construction in §\[sec:constr-may-quinn\]. In the associative case the analogue of the adjunction is $$\xymatrix{ {{(\mbox{group-like $A_{\infty}$ spaces})}} \ar@<.5ex>[r] &
{{{(\mbox{${A_{\infty}}$ spaces})}}} \ar@<.5ex>[r]^-{{{\Sigma^{\infty}}_{+}}} \ar@<.5ex>[l]^-{{GL_{1}}} &
{{(\mbox{${A_{\infty}}$ ring spectra})}: {GL_{1}},} \ar@<.5ex>[l]^-{{\Omega^{\infty}}} }$$ where the right-hand adjunction is a special case of [@LMS:esht p. 366]. If $R$ is an ${A_{\infty}}$ spectrum, then we have the related adjunction $$\xymatrix{ {{{\Sigma^{\infty}}_{+}}: {{(\mbox{right ${\Omega^{\infty}}R$-modules})}}} \ar@<.5ex>[r] & {{{(\mbox{right $R$-modules})}}: {\Omega^{\infty}}.}
\ar@<.5ex>[l] }$$ The main difficulty in using this classical operadic approach is that ${{GL_{1}}R}$ is a not a topological group but rather only a group-like ${A_{\infty}}$ space, and so it is not immediately apparent how to form the (quasi)fibration $${{GL_{1}}R} \to E{{GL_{1}}R} \to B{{GL_{1}}R},$$ and then make sense of the constructions suggested in §\[sec:units\].

In §\[sec:ainfty-thom-spectrum\] we present technology to realize the picture sketched in §\[sec:units\]. The essential strategy is to adapt the operadic smash product of [@MR1361938; @EKMM] to the category ${{\mathscr{T}}}$ of 