Crystallization of $$\hbox {C}^*$$-Algebras

Abstract Given a $$\hbox {C}^*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mtext>C</mml:mtext> <mml:mo>∗</mml:mo> </mml:msup> </mml:math> -algebra A with an almost periodic time evolution $$\sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>σ</mml:mi> </mml:math> , we define a new $$\hbox {C}^*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mtext>C</mml:mtext> <mml:mo>∗</mml:mo> </mml:msup> </mml:math> -algebra $$A_c$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>c</mml:mi> </mml:msub> </mml:math> , which we call the crystal of $$(A,\sigma )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:mi>σ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , that represents the zero temperature limit of $$(A, \sigma )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:mi>σ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . We prove that there is a one-to-one correspondence between the ground states of $$(A,\sigma )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:mi>σ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> and the states on $$A_c$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>c</mml:mi> </mml:msub> </mml:math> , justifying the name. In order to investigate further the relation between low temperature equilibrium states on A and traces on $$A_c$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>c</mml:mi> </mml:msub> </mml:math> , we define a Fock module $$\mathcal {F}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>F</mml:mi> </mml:math> over the crystal and construct a vacuum representation of A on $$\mathcal {F}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>F</mml:mi> </mml:math> . This allows us to show, under relatively mild assumptions, that for sufficiently large inverse temperatures $$\beta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>β</mml:mi> </mml:math> the $$\sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>σ</mml:mi> </mml:math> - $$\hbox {KMS}_\beta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mtext>KMS</mml:mtext> <mml:mi>β</mml:mi> </mml:msub> </mml:math> -states on A are induced from traces on $$A_c$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>c</mml:mi> </mml:msub> </mml:math> by means of the Fock module. In the second part, we compare the K-theoretic structures of A and $$A_c$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>c</mml:mi> </mml:msub> </mml:math> . 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 $$\hbox {C}^*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mtext>C</mml:mtext> <mml:mo>∗</mml:mo> </mml:msup> </mml:math> -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 $$I_c$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>I</mml:mi> <mml:mi>c</mml:mi> </mml:msub> </mml:math> , which we call the crystal of ( I , N ).