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76 H. Emerson and R. MeyerExample 4. More generally, any proper, G-compact G-space X carries a unique coarsestructure for which G acts isometrically; its entourages are defined as in Example 3.With this coarse structure, the orbit map G ! X, g 7! g x, is a coarse equivalencefor any choice of x 2 X. If the G-compactness assumption is omitted, the result is a-coarse space. We always equip a proper G-space with this additional structure.2.3 The stable Higson corona. We next recall the definition of the stable Higsoncorona of a coarse space X from [5], [7]. Let D beaC -algebra.Let M.D ˝ K/ be the multiplier algebra of D ˝ K, and let xB red .X; D/ be theC -algebra of norm-continuous, bounded functions f W X ! M.D ˝ K/ for whichf.x/ f.y/2 D ˝ K for all x;y 2 X. We also letB red .X; D/ WD xB red .X; D/=C 0 .X; D ˝ K/:Definition 5. A function f 2 xB red .X; D/ has vanishing variation if the functionE 3 .x; y/ 7! kf.x/ f.y/k vanishes at 1 for any closed entourage E X X.The reduced stable Higson compactification of X with coefficients D is the subalgebraNc red .X; D/ xB red .X; D/ of vanishing variation functions. The quotientc red .X; D/ WD Nc red .X; D/=C 0 .X; D ˝ K/ B red .X; D/is called reduced stable Higson corona of X. This defines a functor on the coarsecategory of coarse spaces: a coarse map f W X ! X 0 induces a map c red .X 0 ;D/ !c red .X; D/, and two maps X ! X 0 induce the same map c red .X 0 ;D/! c red .X; D/if they are close. Hence a coarse equivalence X ! X 0 induces an isomorphismc red .X 0 ;D/Š c red .X; D/.For some technical purposes, we must allow unions X D S X n of coarse spacessuch that the embeddings X n ! X nC1 are coarse equivalences; such spaces are called-coarse spaces. The main example is the Rips complex P .X/ of a coarse space X,which is used to define its coarse K-theory. More generally, if X is a proper but notG-compact G-space, then X may be endowed with the structure of a -coarse space.For coarse spaces of the form jGj for a locally compact group G with a G-compactuniversal proper G-space EG, we may use EG instead of P .X/ because EG is coarselyequivalent to G and uniformly contractible. Therefore, we do not need -coarse spacesmuch; they only occur in Lemma 7.It is straightforward to extend the definitions of Nc red .X; D/ and c red .X; D/ to-coarse spaces (see [5], [7]). Since we do not use this generalisation much, we omitdetails on this.Let H be a locally compact group that acts coarsely and properly on X. It is crucialfor us to allow non-compact groups here, whereas [7] mainly needs equivariance forcompact groups. Let D be an H -C -algebra, and let K H WD K.`2N ˝L 2 H/. Then Hacts on xB red .X; D ˝ K H / by.h f /.x/ WD h f.xh/ ;