Letters in Mathematical Physics 10 (1985) 29-31. 0377-9017/85.15. 9 1985 t~v D. Retdel Publishing Company. 29 On the Existence of KMS States S L. WORONOWICZ Department of Mathematical Methods m Physics, ~#uversitv of Warsaw, Ho2a 74, PL 00-682 Warsaw ~Received: 4 February 1985) Abstract. A necessary and sufficient condition for the existence of KMS states is formulated in terms of a certam left ideal m the algebra A~174 We consider a C*-dynamical system (A, o9: A is a C*-algebra and a = (o-,)t~ R is a one-parameter group of automorphisms ofA. We use the usual continuity assumption: For any a ~ A the map R ~ t ~ ~,(. ) s A is norm continuous. An element a e A is called analytic if there exists an entire function C ~ t -~ a, ~ A such that a, = 6~(a) for real t. The set of all analytic elements is a dense ,-subalgebra ofA. A state o)ofA is called a KMS state if co(ab) = co(ba~)for any b e A and any analytic aeA. We believe that KMS states describe the thermal equilibrium of considered systems. In many papers (e.g., [2, 3]) it is shown that the KMS property implies (and is implied by) characteristic properties of equilibrium states. Therefore, the problem of the existence of KMS states for a given C*-dynamical system is very interesting. In this Letter we show that a KMS state exists if and only if a certain left ideal in the algebra .~ = A ~ | A is not trivial. Here | denotes the maximal tensor product of C*-algebras (i.e., the completion of the algebraic tensor product with respect to the maximal C*-norm) and A ~ is the opposite algebra of A. The latter means that an antilinear multiplicative .-invariant isometry of A onto A ~ is given. The image of an element a ~ A will be denoted by ~ ~ A ~ Let L be the smallest closed left ideal in .4 = A ~ | A containing all elements of the form a| - I@ (a*),,, 2 where a is an analytic element of A. Then we have THEOREM 1. Let co be a state A such that goIL = 0 and co be the restriction of 63 to A, i.e., co(a) = ~7j(7 | a) for any a ~ A. Then (1) co is a KMS state and (2) 63 is the purification of co. THEOREM 2. Let 6o be a KMS state of A and go be the purification of co. Then go!L = O. THEOREM 3. (The dynamical system (A, ~r) admits a KMS state)*>(L r A ~ |