E∞-ring structures for Tate spectra

Let G be a compact Lie group and kG a G spectrum (as defined in [3, Section I.2]). Greenlees and May ([2]) have defined an associated G-spectrum t(kG) called the Tate spectrum. They observe that if kG is a ring G-spectrum then there is an induced ring G-spectrum structure on t(kG), and that if kG is homotopy-commutative then t(kG) will also be homotopycommutative (see [2, Proposition 3.5]). It is therefore natural to ask whether an equivariant E∞-ring structure on kG induces an equivariant E∞-ring structure on t(kG) (we will recall the definition in a moment). We offer both positive and negative answers to this question. On the positive side, we show that t(kG) inherits a structure which is somewhat weaker than an equivariant E∞-ring structure, but which should be adequate for most practical purposes. To explain this, let us recall from [3, Example VII.1.4] that to each G-universe U is associated an equivariant operad L(U). Let us fix a complete G universe U and let V denote the trivial G-universe U. An equivariant E∞-ring structure is defined to be an action of an equivariant operad equivalent to L(U) (see [3, Definitions VII.2.1 and VII.1.2 and Remark VII.1.3]). Let us define an E′ ∞-ring structure to be an action of an equivariant operad equivalent to L(V ); since G acts trivially on L(V ) we can rephrase this by saying that an E′ ∞-ring structure is an action of a nonequivariant E∞ operad through G-maps. Since there is a map of operads L(V ) → L(U), an equivariant E∞ structure specializes to an E′ ∞ structure. On the other hand, Remark VII.2.5 of [3] shows that if kG is an E ′ ∞-ring spectrum then the fixed point spectra (kG) H have (nonequivariant) E∞-ring structures which are consistent as H varies; this is likely to be the point most relevant for applications. Our positive result is: