Metric geometry in homogeneous spaces of the unitary group of a \(C^*\)-algebra. I: Minimal curves

DOI10.1016/S0001-8708(03)00148-8zbMath1060.53076MaRDI QIDQ1826889

Luis E. Mata-Lorenzo, Lázaro Recht, Carlos Eduardo Durán

Publication date: 6 August 2004

Published in: Advances in Mathematics (Search for Journal in Brave)

zbMATH Keywords

von Neumann algebraFinsler metricHomogeneous spacesUnitary groupMinimal curves

Mathematics Subject Classification ID

Selfadjoint operator algebras ((C^*)-algebras, von Neumann ((W^*)-) algebras, etc.) (46L99) Infinite-dimensional Lie groups and their Lie algebras: general properties (22E65) Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds (58B20) Global differential geometry of Finsler spaces and generalizations (areal metrics) (53C60) Linear operators in (C^*)- or von Neumann algebras (47C15)

Related Items (27)

Constructions of minimal Hermitian matrices related to a C*-subalgebra of 𝑀_{𝑛}(ℂ) ⋮ Grassmann geometry of zero sets in reproducing kernel Hilbert spaces ⋮ The set of partial isometries as a quotient Finsler space ⋮ Minimal elements related to a conditional expectation in a \(\mathrm{C}^\ast\)-algebra ⋮ Hopf fibration in a C^*‐module^† ⋮ Moment of a subspace and joint numerical range ⋮ Minimal self-adjoint compact operators, moment of a subspace and joint numerical range ⋮ A characterization of minimal Hermitian matrices ⋮ Best approximation by diagonal compact operators ⋮ Unitary subgroups and orbits of compact self-adjoint operators ⋮ Metrics in the sphere of a \(C^*\)-module ⋮ Hopf–Rinow theorem in Grassmann manifolds of C*-algebras ⋮ Minimal Hermitian compact operators related to a \(\mathrm{C}^\ast\)-subalgebra of \(K(H)\) ⋮ Some characterizations of minimal compact normal operators ⋮ Concrete minimal \(3 \times 3\) Hermitian matrices and some general cases ⋮ Metric geometry in infinite dimensional Stiefel manifolds ⋮ Best approximation by diagonal operators in Schatten ideals ⋮ Projections with fixed difference: a Hopf-Rinow theorem ⋮ Metric geometry of infinite-dimensional Lie groups and their homogeneous spaces ⋮ Minimal curves in \(\mathcal{U}(n)\) and \(\mathcal{G}l(n)^+\) with respect to the spectral and the trace norms ⋮ Minimal length curves in unitary orbits of a Hermitian compact operator ⋮ Geodesic neighborhoods in unitary orbits of self-adjoint operators of \(\mathcal{K}+\mathbb{C}\) ⋮ Minimal matrices and the corresponding minimal curves on flag manifolds in low dimension ⋮ Supports for minimal Hermitian matrices ⋮ Canonical sphere bundles of the Grassmann manifold ⋮ GEOMETRY OF UNITARIES IN A FINITE ALGEBRA: VARIATION FORMULAS AND CONVEXITY ⋮ Metric geodesics of isometries in a Hilbert space and the extension problem

Cites Work

• Differential geometry of systems of projections in Banach algebras
• The geometry of spaces of projections in \(C^*\)-algebras
• Convexity of the geodesic distance on spaces of positive operators
• The space of spectral measures is a homogeneous reductive space
• Conditional expectations and operator decompositions
• Isometric submersions of Finsler manifolds
• Minimality of Geodesics in Grassmann Manifolds
• A Geometric Interpretation of Segal's Inequality || e X + Y || ≤ || e X/2 e Y e X/2 ||
• Selected new aspects of the calculus of variations in the large
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