Sphere and projective space of a C * -algebra with a faithful state

[摘要] Let ? be a unital C * -algebra with a faithful state ϕ . We study the geometry of the unit sphere ? ϕ = { x ∈ ? : ϕ ( x * x ) = 1} and the projective space ℙ ϕ = ? ϕ /?. These spaces are shown to be smooth manifolds and homogeneous spaces of the group ? ϕ (?) of isomorphisms acting in ? 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.

[学科分类] 外科医学

[关键词] homogeneous space;minimal curves;C*-algebra;projective space