Affiliation:
1. Instituto de Ciencias , Universidad Nacional de General Sarmiento
Abstract
Abstract
Let 𝒜 be a unital C
*-algebra with a faithful state ϕ. We study the geometry of the unit sphere 𝕊
ϕ
= {x ∈ 𝒜 : ϕ(x
*
x) = 1} and the projective space ℙ
ϕ
= 𝕊
ϕ
/𝕋. These spaces are shown to be smooth manifolds and homogeneous spaces of the group 𝒰
ϕ
(𝒜) of isomorphisms acting in 𝒜 which preserve the inner product induced by
ϕ
, which is a smooth Banach-Lie group. An important role is played by the theory of operators in Banach spaces with two norms, as developed by M.G. Krein and P. Lax. We define a metric in ℙ
ϕ
, and prove the existence of minimal geodesics, both with given initial data, and given endpoints.
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