Crystallization of \(\hbox {C}^*\)-Algebras

Given a \(\hbox {C}^*\)-algebra A with an almost periodic time evolution \(\sigma \), we define a new \(\hbox {C}^*\)-algebra \(A_c\), which we call the crystal of \((A,\sigma )\), that represents the zero temperature limit of \((A, \sigma )\). We prove that there is a one-to-one correspondence between the ground states of \((A,\sigma )\) and the states on \(A_c\), justifying the name. In order to investigate further the relation between low temperature equilibrium states on A and traces on \(A_c\), we define a Fock module \(\mathcal {F}\) over the crystal and construct a vacuum representation of A on \(\mathcal {F}\). This allows us to show, under relatively mild assumptions, that for sufficiently large inverse temperatures \(\beta \) the \(\sigma \)-\(\hbox {KMS}_\beta \)-states on A are induced from traces on \(A_c\) by means of the Fock module. In the second part, we compare the K-theoretic structures of A and \(A_c\). Previous work by various authors suggests that they have (rationally) isomorphic K-groups. We analyze this phenomenon in detail, confirming it under favorable conditions, but showing that, in general, there is apparently no easy way to relate these groups. As examples, we discuss in particular Exel’s results on semi-saturated circle actions, and recent results of Miller on the K-theory of inverse semigroup \(\hbox {C}^*\)-algebras. In relation to the latter, we introduce the notion of a scale N on an inverse semigroup I and define a new inverse semigroup \(I_c\), which we call the crystal of (I, N).