Geometry of the projective unitary group of a C ∗ -algebra
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Esteban Andruchow [1]
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[1] Universidad Nacional de General Sarmiento
Universidad Nacional de General Sarmiento
Argentina
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- Localización: Revista de la Unión Matemática Argentina, ISSN 0041-6932, ISSN-e 1669-9637, Vol. 58, Nº. 2, 2017, págs. 319-329
- Idioma: inglés
- Enlaces
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Resumen
Let A be a C∗-algebra with a faithful state φ. It is proved that the projective unitary group PUA of A,PUA=UA/T.1,(UA denotes the unitary group of A) is a C∞-submanifold of the Banach space Bs(A) of bounded operators acting in A, which are symmetric for the φ-inner product, and are usually called symmetrizable linear operators in A. A quotient Finsler metric is introduced in PUA, following the theory of homogeneous spaces of the unitary group of a C∗-algebra. Curves of minimal length with any given initial conditions are exhibited. Also it is proved that if A is a von Neumann algebra (or more generally, an algebra where the unitary group is exponential) two elements in PUA can be joined by a minimal curve.
In the case when A is a von Neumann algebra with a finite trace, these minimality results hold for the quotient of the metric induced by the p-norm of the trace (p≥2), which metrizes the strong operator topology of PUA.
