A characterization of pure states of 𝐶*-algebras
Erling Størmer · Proceedings of the American Mathematical Society · 1968
ERLING ST0RMERIf p is a state of a C*-algebra 21 then its definite set 2DP is the set of selfadjoint operators A in 21 such that p(A2)=p(A)2.Kadison and Singer [3, Theorem 4] showed that a state of all bounded operators (B( §) on a Hilbert space § is pure if and only if its definite set is maximal (in the set of definite sets due to states ordered by inclusion), and they left the problem open for states on general C*-algebras.Of course, if there exists a representation of 21 onto the complex numbers then $lsA-the selfadjoint operators in 21-is itself the definite set of some state, so unless 21 is in this case abelian, there will exist pure states whose definite sets are not maximal.We shall in the present note solve the problem to the affirmative if 21 has no one-dimensional representations.Theorem.Let 21 be a C*-algebra with identity and with no one-dimensional representations.Then a state o/ 21 is pure i/ and only if its definite set is maximal.Kadison and Singer showed [3, proof of Theorem 4] that if 3P