Crossed-products by finite index endomorphisms and KMS states

Given a unital C∗-algebra A, an injective endomorphism α : A→A preserving the unit, and a conditional expectation E from A to the range of α we consider the crossed-product of A by α relative to the transfer operator L=α−1E. 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,b∈A (e.g. when A is commutative) then the KMSβ states are precisely those of the form ψ=φ∘G, where φ is a trace on A satisfying the identityφ(a)=φ(L(h−β ind(E)a)),where ind(E) is the Jones-Kosaki-Watatani index of E.