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Pairs of Projections: Geodesics, Fredholm and Compact Pairs
(Birkhauser Verlag Ag, 2014-08)
A pair (P,Q) of orthogonal projections in a Hilbert space H is called a Fredholm pair if QP:R(P)→R(Q)is a Fredholm operator. Let F be the set of all Fredholm pairs. A pair is called compact if P−Q is compact. Let C be the ...
Geodesics of projections in von neumann algebras
(American Mathematical Society, 2021-07)
Let A be a von Neumann algebra and PA the manifold of projections in A. There is a natural linear connection in PA, which in the finite dimensional case coincides with the the Levi-Civita connection of the Grassmann manifold ...
A note on geodesics of projections in the Calkin algebra
(Birkhauser Verlag Ag, 2020-11)
Let C(H) = B(H) / K(H) be the Calkin algebra (B(H) the algebra of bounded operators on the Hilbert space H, K(H) the ideal of compact operators, and π: B(H) → C(H) the quotient map), and PC(H) the differentiable manifold ...
Weak Riemannian manifolds from finite index subfactors
(Springer, 2008-10)
Let N ⊂ M be a finite Jones' index inclusion of II1 factors and denote by UN ⊂ UM their unitary groups. In this article, we study the homogeneous space UM/UN, which is a (infinite dimensional) differentiable manifold, ...
Nonpositively curved metric in the positive cone of a finite von Neumann algebra
(Oxford University Press, 2006-08)
In this paper we study the metric geometry of the space Σ of positive invertible elements of a von Neumann algebra. A with a finite, normal and faithful tracial state T. The trace induces an incomplete Riemannian metric ...
The Grassmann manifold of a Hilbert space
(Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca. Universidad Nacional del Sur. Departamento de Matemática. Instituto de Matemática Bahía Blanca, 2014)
The present paper surveys the geometric properties of the Grassmann manifold Gr(H ) of an infinite dimensional complex Hilbert space H . Gr(H ) is viewed as a set of operators, identifying each closed subspace S ⊂ H with ...
Nonlinear projection using geodesic distances and the neural gas network
(SPRINGER-VERLAG BERLIN, 2006)
A nonlinear projection method that uses geodesic distances and the neural gas network is proposed. First, the neural gas algorithm is used to obtain codebook vectors, and a connectivity graph is concurrently created by ...