Positive projective space of a C*-algebra with a tracial conditional expectation

E. ANDRUCHOW, L. RECHT

POSITIVITY

Birkhauser

Lugar: Basel; Año: 2007 vol. 11 p. 285 - 298

1385-1292

Let A be a C∗-algebra, B a subalgebra of its center and ¥Õ : A ¡æ BA be a C∗-algebra, B a subalgebra of its center and ¥Õ : A ¡æ B a tracial faithful conditional expectation. We define the positive projective space as the quotient IP++ B = IP+ = G+/ ¡­B= IP+ = G+/ ¡­B where G+ is the space of positive invertible elements of A, and a ¡­B a if there exists g invertible in B such that a = |g|2a. When A is abelian, this space is a set of representatives for probability densities equivalent to a given one. The aim of this paper is to endow IP+ with differentiable structure, a linear connection and a Finsler metric. This is done in a way that given any pair of elements in IP+, there is a unique geodesic of this connection, which is the shortest curve joining such endpoints for the given metric. The metric space IP+ with the given geodesic distance is non positively curvedG+ is the space of positive invertible elements of A, and a ¡­B a if there exists g invertible in B such that a = |g|2a. When A is abelian, this space is a set of representatives for probability densities equivalent to a given one. The aim of this paper is to endow IP+ with differentiable structure, a linear connection and a Finsler metric. This is done in a way that given any pair of elements in IP+, there is a unique geodesic of this connection, which is the shortest curve joining such endpoints for the given metric. The metric space IP+ with the given geodesic distance is non positively curvedg invertible in B such that a = |g|2a. When A is abelian, this space is a set of representatives for probability densities equivalent to a given one. The aim of this paper is to endow IP+ with differentiable structure, a linear connection and a Finsler metric. This is done in a way that given any pair of elements in IP+, there is a unique geodesic of this connection, which is the shortest curve joining such endpoints for the given metric. The metric space IP+ with the given geodesic distance is non positively curved+ with differentiable structure, a linear connection and a Finsler metric. This is done in a way that given any pair of elements in IP+, there is a unique geodesic of this connection, which is the shortest curve joining such endpoints for the given metric. The metric space IP+ with the given geodesic distance is non positively curved+, there is a unique geodesic of this connection, which is the shortest curve joining such endpoints for the given metric. The metric space IP+ with the given geodesic distance is non positively curved+ with the given geodesic distance is non positively curved