Monotone Complete C*-algebras and Generic Dynamics by Kazuyuki Saitô & J. D. Maitland Wright

Author:Kazuyuki Saitô & J. D. Maitland Wright
Language: eng
Format: epub
Publisher: Springer London, London

Proof

Suppose that ϕ is a (complete) isometry. Take any a ∈ A with a ≥ 0. We shall show ϕ(a) ≥ 0. To do this, we may assume . Then, and so .

Take any , the state space of B. Let with .

Let for each positive integer m. Then we have , which implies that

Hence it follows that for all m and so β = 0 follows. So, for all , that is, ϕ(a) ∈ B sa . The fact that now tells us that . So ϕ(a) ≥ 0 follows.

Let a ∈ A such that ϕ(a) ≥ 0. To show that a ≥ 0, we may assume that . Then where a 1 and a 2 are self-adjoint. So . Since ϕ(a),  ϕ(a 1) and ϕ(a 2) are self-adjoint, it follows that ϕ(a 2) = 0. So, , that is, a 2 = 0, which implies a = a 1 ∈ A sa . Since 0 ≤ ϕ(a) ≤ 1, . Hence it follows that and a ≥ 0 follows. So ϕ is a completely bipositive map.

Conversely suppose that ϕ is completely bipositive. We shall show that ϕ is a complete isometry. To do this, take any a ∈ A and take any positive real number . Put . Then , which implies that

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