Crossed-products by finite index endomorphisms and KMS states

• Published in: Journal of functional analysis 2003-04, Vol.199 (1), p.153-188
• Main Author: Exel, Ruy
• Format: Article
• Language: English
• ISSN: 0022-1236; 1096-0783
• DOI: 10.1016/S0022-1236(02)00023-X

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 −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, b∈ A (e.g. when A is commutative) then the KMS β states are precisely those of the form ψ= φ∘ G, where φ is a trace on A satisfying the identity φ(a)=φ( L(h −β ind(E)a)), where ind( E) is the Jones-Kosaki-Watatani index of E.