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By Anthony Bak (auth.), R. Keith Dennis (eds.)

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Isomorphic to a direct summand of U(A,G) and, if A is B-free, as a B[G]-module to U(B,G). 2 implies spaces agree with those constructed Proposition argument f#:~(A,G) ÷ ~(B,G). that the resulting maps of K-theory in the previous paragraph. 2 may be applied in situations where no such precise categorical is available. Now let A be a commutative homomorphisms ~ m × H n + ~ ring and write ~ n = GL(n,A). 4. and terminal since 9 0 is the trivial group, C ( G , ~ 0) has object space object * = G/G, hence B(G,~ 0) is G-contractible.

KGB(G,H ) contains no lim I term. That is, if B(G,H) is the union of an expanding sequence of compact G-spaces Bn, then KGB(G,N) = lim KGB n. This is not hard to prove directly when G and ~ are finite. When G and ~ are compact Lie groups, Haeberle and Hauschild have proven the result as a byproduct of a suitable generalization of the Atiyah-Segal completion theorem zation gives that KGB(G,~) [4]. The generali- is naturally isomorphic to the completion of R(G x ~) with respect to the topology generated by the kernels of the restriction homomorphisms R(G x H) ÷ R(H) for H C G.

While the third author has conbtructed an equivariant plus construction for G-connected Gspaces X, namely those X for which all X H are path connected, we shall not use it. Instead, we shall prove that, in the cases of interest to us, equivalent to a topological G-monoid B ~(G). J~n B(G,~n ) is G- This means that B ~ (G) is a G-space and a topological monoid such that its unit is a fixed point and its product is a G- 28 map. 2. to G-spaces, to G-spaces and this will allow us to construct together with a natural G-map ~:B ~(G) Definition space functor takes G-monoids flBB~ (G).

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