Crossed-products by finite index endomorphisms and KMS states

摘要

Given a unital C*-algebra A, an injective endomorphism alpha: A --> A preserving the unit, and a conditional expectation E from A to the range of alpha we consider the crossed-product of A by alpha relative to the transfer operator L = alpha(-1)E. When E is of index-finite type we show that there exists a conditional expectation G from the crossed-product to A which is unique under certain hypothesis. We define a "gauge action" on the crossed-product algebra in terms of a central positive element h and study its KMS states. The main result is: if h > 1 and E(ab) = E(ba) for all a,bis an element ofA (e.g. when A is commutative) then the KMSbeta states are precisely those of the form psi = phicircleG, where phi is a trace on A satisfying the identity phi(a) = phi(L(h(-beta) ind(E)a)), where ind(E) is the Jones-Kosaki-Watatani index of E. (C) 2002 Elsevier Science (USA). All rights reserved. [References: 13]