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Maliyah Robles 2022-07-07 Answered
Let A be a C -algebra.
(i) Let φ be a state on a u A-algebra A. Suppose that | φ ( u ) | = 1 for all unitary elements u∈A. Show that φ is a pure state. [Hint: span U ( A ) = A ]
(ii) Let φ be a multiplicative functional on a C -algebra A. Show that φ is a pure state on A.
(iii) Show that the pure and multiplicative states coincide for commutative A.
I managed to work out the first two problems but I have no idea about the last one. How to see from being an extreme element in the state space of a commutative A that the extreme element is multiplicative?
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Answers (1)

Johnathan Morse
Answered 2022-07-08 Author has 18 answers
Suppose that φ is an extreme point in the state space.
Let a 0 with 0 < φ ( a ) < 1 (this exists unless φ = 0). Define
ψ a ( x ) = φ ( a x ) φ ( a ) .Because A is abelian, for x 0 we have ψ a ( x ) = φ ( a 1 / 2 x a 1 / 2 ) 0.. So ψ a is positive and ψ ( 1 ) = 1, thus a state. We have, with t = φ ( a ),
φ ( x ) = t ψ a ( x ) + ( 1 t ) , ψ 1 a ( x ) , x A .
As φ is extreme, we have ψ a = φ. This equality is
φ ( a x ) = φ ( a ) φ ( x ) .
This works for all x A and all a A + with φ ( a ) < 1. As the positive elements span A, we get
φ ( a x ) = φ ( a ) φ ( x ) , a , x A .
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