On maximally unitarily mixed states on W*‐algebras
Peter M. Alberti · Mathematische Nachrichten · 1979
Abstract The matter of discussion is the set of states on W*‐algebras. States are considered in relation to unitary mixing where the full group of unitary elements is used. Among all the states, one could talk about, we are mainly interested in studying those the degree of mixture of which is maximally. We give a general description of the set of maximally unitarily mixed states. For a finite W*‐algebra the set of states in question is identified as the set of unitarily invariant states and for each state there is precisely one maximally mixed state comparable with it in the sense of unitary mixture. The latter fact fails to be true in the general case of an infinite W*‐algebra. In this case the problem is more complicated. For a properly infinite W*‐algebra we introduce an interesting symmetric ideal, the c‐ideal. We prove that a state on a properly infinite W*‐algebra is maximally mixed if and only if its kernel contains the c‐ideal, i. e. maximally mixed states may be identified with the set of states on the quotient algebra generated by the c‐ideal. For a general infinite W*‐algebra the solution of the problem can be represented as a combination of both above mentioned cases. The point of view we adopted in handling the mixture relation has its origin in UHLMANN'S work.