Temperature states on gauge groups

We introduce the notion of a temperature state on the Kac-Moody extension of the infinite dimensional Lie group Map (R,U(N)) and on its subgroups. We show that for Map (R,U(N)), by utilising earlier work of one of us (A.L.C.) with S.N.M. Ruijsenaars, these temperature states are associated with type III1 factor representations of the group. In particular this may be interpreted as yielding type III1 factor representations of Kac-Moody algebras. In the general setting of KMS states on twisted group C*-algebras we address the question of uniqueness of these temperature states and obtain both a general formula for such states and a criterion for uniqueness. We find that uniqueness holds for states on Map (R, U (N)) but fails for certain subgroups leading to other possibilities which have a natural physical interpretation in terms of Bose-Einstein condensation