The Theory of Topological Semigroups, Volume 2 by James Harvey Carruth, J. A. Hildebrant

By James Harvey Carruth, J. A. Hildebrant

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Recall the notation [Aξ ] for the projection onto the closed subspace generated by vectors aξ , a ∈ A, ξ ∈ H . If A = C we use the notation [ξ ] instead of [Cξ ] for the 1-dimensional projection on the subspace generated by the vector ξ . 7 Let φ be an automorphism of B(H ). Then there exists a unitary operator U such that φ = AdU . Proof Since φ maps minimal projections onto minimal projections, for each ξ ∈ H there is η ∈ H such that φ([ξ ]) = [η]. Composing φ by an inner automorphism AdU , we may assume φ([ξ ]) = [ξ ] for a unit vector ξ .

4 Let A and B be C ∗ -algebras and φ : A → B be an extreme point of the convex set of positive unital maps of A into B. 1 General Properties of Extremal Maps 29 A and assume φ(a) belongs to the center of B. Then a belongs to the multiplicative domain for φ. Proof We have a= 1 1 a + a∗ + i a − a∗ . 2 2i Since a ∗ satisfies the same assumptions as a, we may assume a is self-adjoint and a < 1. Then φ(a) < 1, so 1 − a and 1 − φ(a) are positive and invertible. Define ψ : A → B by ψ(b) = φ (1 − a)b 1 − φ(a) −1 .

To show this, note that the span of the ηi ’s must be r-dimensional by minimality of r, so we can write the ηi ’s as linear combinations of r orthonormal vectors ω1 , . . , ωr in H . Using this we can give more specific information on V and ω in the last proposition. 4. Then CAdV = λ[ω] for some λ ≥ 0. ω has Schmidt rank r if and only if rank V = r. Proof Suppose rank V = r. Choose an orthonormal basis η1 , . . , ηm for H such that V ∗ V ηk = λk ηk with λ1 , . . , λr > 0 and λk = 0 for k > r.

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