Invertibility for some homotopy invariant functors related to Roe algebras (Q2037780)

Normally a homotopy between two homomorphisms \(\phi_i :A \rightarrow B\) for \(C^*\)-algebras \(A\) and \(B\) is realized by a homomorphism \(\Phi:A \rightarrow B \otimes C[0,1]\) evaluated at the endpoints, but the author generalizes this situation to the one of being given an endofunctor \(F\) on the category of \(C^*\)-algebras and realizing the so-called \(F\)-homotopy \(\cong_F\) between \(\phi_0\) and \(\phi_1\) by endpoint evaluations of a homomorphim \(\Phi:A \rightarrow F(B \otimes C[0,1])\). He points out that many well-known constructions like \(K\)-theory \(K_0\), extensions semigroups Ext, and \(E\)-theories \(E_0\) and \(E_1\) are realized as \(F\)-homotopy classes \([A,B]_F:= \operatorname{hom} [A,FB]/\cong_F\) for suitably chosen \(F\), with addition defined by taking direct sums in matrices. For \(R_X B \subseteq {\mathcal L}_B(B \otimes \ell^2(X))\), the Roe algebra of operators of finite propagation of a space \(X\) of bounded geometry, the author studies \(F\)-homotopy classes for the Roe functor \(F= R_X\). By rather direct constructions he is able to give criteria on \(X\) such that \([A,B]_{R_X}\) is the zero group, and when it is a group at all. He closes by giving an example of some \(X\) which is not a group, but just a monoid.