X. ?f-x/y-be, Remark 6.5.6 states that we have a homeomorphism E S (X) ? ?(X)

, We obtain the following result for the spaces

, This corresponds to potentials that have limit 0 on X/Y . The spectrum of this algebra (after we adjoin a unit) identifies with the closure of the image of X in Y ?S (X/Y ) + , where Z + denotes the one point compactification of a locally compact space, Z. Since A S ? E S (X)

, Generally, the topology on A S is rather complicated and singular

, we take X := R 3N and consider the subspaces P j and P ij of Equation (1.5) We let F be the semilattice generated by the subspaces P i and P ij , i, j ? {1, 2, . . . , N }. Let S be the finite semilattice of p-submanifolds of X

, S] will be endowed with natural smooth actions of X := R 3N by translation, of S N , the symmetric group on N variables, by permutation, and O(3) acting diagonally on the components of X := R 3N

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