Units of equivariant ring spectra

It is well known that very special $Γ$-spaces and grouplike $\E_\infty$ spaces both model connective spectra. Both these models have equivariant analogues. Shimakawa defined the category of equivariant $Γ$-spaces and showed that special equivariant $Γ$-spaces determine positive equivariant spectra. Costenoble and Waner showed that grouplike equivariant $\E_\infty$-spaces determine connective equivariant spectra. We show that with suitable model category structures the category of equivariant $Γ$-spaces is Quillen equivalent to the category of equivariant $\E_\infty$ spaces. We define the units of equivariant ring spectra in terms of equivariant $Γ$-spaces and show that the units of an equivariant ring spectrum determines a connective equivariant spectrum.

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