Reciprocal Cuntz–Krieger algebras

The reciprocality in this paper means a duality in Kirchberg algebras between strong extension groups and K-theory groups. We describe a construction of the reciprocal dual algebra  for a Kirchberg algebra A with finitely generated K-groups via K-theoretic duality for extensions. In particular, we may concretely construct and realize the reciprocal algebra ÔA for a simple Cuntz–Krieger algebra OA in several different ways. As a result, the algebra ÔA is realized as a unital simple purely infinite universal C*-algebra generated by a family of partial isometries subject to certain operator relations. We will finally study the gauge action on the reciprocal algebra ÔA and prove that there exists an isomorphism between the fundamental groups π1(Aut(OA)) and π1(Aut( ÔA)) preserving their gauge actions.