Minimality of Geodesics in Grassmann Manifolds

Porta, Horácio; Recht, Lázaro

doi:10.2307/2046430

Cited by 37 publications

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“…Proof. This may be deduced from [8]. K Observe that by its very definition, given \ 0 , \ 1 # Gr(A) and \ 1 # 0 \ 0 , there exists a unique nonwandering geodesic joining \ 0 and \ 1 .…”

Section: Minimality Of Geodesicsmentioning

confidence: 96%

“…K Using Theorem 4.8, we can give a new proof (in the case of a C*-algebra with trace) of the minimality theorem for the L functional. This has been proven in general in [8]. First, we need a lemma about the p-norms, which is well known in the C*-algebra of n_n complex matrices: The p-norm of a matrix tends to its operator norm as p tends to infinity.…”

Section: Minimality Of Geodesicsmentioning

confidence: 99%

“…By the Gelfand Naimark Theorem (see [6]) we may consider A as a closed subalgebra of the algebra L(H ) of bounded linear operators of an infinite dimensional Hilbert space H, and the Grassmannian of A can be identified with a submanifold of the set of subspaces of H. In this case, the paper may be considered as a continuation of the study of the differential geometry of operator spaces from the point of view of homogeneous reductive spaces (see for example [8,9,11,12]). In particular, the main theorem in [8] can be deduced from this work (in the case of C*-algebras with a faithful trace). Let us remark that if A is infinite-dimensional, the topology of its Grassmannian may be unusual.…”

Section: Introductionmentioning

confidence: 99%

“…As an application of Theorem 4.8 we give an alternative proof, in the context of C*-algebras with trace (see Theorem 4.10 in Section 4) of the minimality Theorem (presented in [8]) for the operator norm, namely: If \ 0 and \ 1 are sufficiently close, then the (nondifferentiable) functional L given by…”

Section: Introductionmentioning

confidence: 99%

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Natural Variational Problems in the Grassmann Manifold of a C*-Algebra with Trace

Durán

Mata-Lorenzo

Recht

2000

Advances in Mathematics

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dedicated to luis antonio santalo on his 87th birthday We consider a family of natural variational problems in the Grassmannian of a C*-algebra with trace which can be considered as slightly degenerate Finsler metrics. We show that all these problems have as solutions the standard geodesics and that the short standard geodesics are absolute minima of the functionals restricted to a special class of curves. Academic PressThe finite dimensional case. If A is finite-dimensional, then each connected component of the Grassmannian of A is an ordinary (real or complex) Grassmann manifold G k, n of k-planes in n-space or a product of

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“…Proof. This may be deduced from [8]. K Observe that by its very definition, given \ 0 , \ 1 # Gr(A) and \ 1 # 0 \ 0 , there exists a unique nonwandering geodesic joining \ 0 and \ 1 .…”

Section: Minimality Of Geodesicsmentioning

confidence: 96%

Section: Minimality Of Geodesicsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Natural Variational Problems in the Grassmann Manifold of a C*-Algebra with Trace

Durán

Mata-Lorenzo

Recht

2000

Advances in Mathematics

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“…Para comenzar, vamos a probar el siguiente lema, el cual es una pequeña modificación de un hermoso truco usado por Porta y Recht en [54] (ver también [5]):…”

Section: Demostración Del Teorema 531unclassified

Fórmulas ergódicas para medias geométricas y curvas minimales

Ghiglioni¹

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La tesis se divide como sigue: En el capítulo 2 damos las herramientas que necesitamos para el capítulo 3. De finimos los espacios de Hadamard, damos ejemplos de los mismos y algunos resultados preliminares. Luego de finimos el baricentro. A continuación pasamos a dar algunas nociones básicas de Teoría ergódica. Particularmente nos interesa el Teorema de Birkhoff y su relación con la ley de los grandes números. Por último pasamos a sistemas de Kronecker. Estos resultados son necesarios para el desarrollo del capítulo 3. No obstante, si el lector conoce estos temas, puede obviar este capítulo y pasar al siguiente. En el capítulo 3 damos la demostración del Teorema ergódico para espacios de Hadamard. Comenzamos el capítulo con una motivación del mismo. La relevancia de esta motivación yace en el hecho que utilizamos algunos lemas relacionados con la misma. Los resultados originales se encuentran en las secciones 3.3 y 3.4, donde demostramos el Teorema ergódico para el caso continuo y L1 respectivamente. A su vez, utilizamos una nueva noción de molli fiers en espacios de Hadamard, la cual es interesante por sí misma y había sido probada por Karcher para variedades Riemannianas. En el capítulo 4 estudiamos la estructura de las curvas minimales en el caso de la norma traza. Comenzamos el capítulo motivando este problema, a partir de un estudio análogo realizado por Y. Lim, en el espacio de matrices positivas. Los resultados originales se encuentran en la sección 4.2 y los resultados más importantes obtenidos son: la caracterización de las curvas minimales para matrices Hermitianas y positivas. En el capítulo 5 estudiamos la estructura de las curvas minimales para matrices unitarias medidas con la norma espectral. Los resultados originales se encuentran en las secciones 5.3, 5.4 y 5.5. En la primera, estudiamos dicha estructura. En la segunda, la geometría de los puntos medios y en la tercera, la Grassmaniana. En el capítulo 6 incluímos resultados que utilizamos del análisis matricial. Entre estos se encuentran resultados de mayorización, normas unitariamente invariantes u otros. A su vez se pueden encontrar resultados que utilizamos sobre la geometría de las matrices positivas. También agregamos algunos resultados que son conocidos, pero tal vez no en toda su generalidad (para espacios métricos), como el Teorema de Lusin.

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Projective spaces of aC *-algebra

Andruchow¹,

Corach

Stojanoff

2000

Integr equ oper theory

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Based on the projective matrix spaces studied by B. Schwarz and A. Zaks, we study the notion of projective space associated to a C * -algebra A with a fixed projection p. The resulting space P(p) admits a rich geometrical structure as a holomorphic manifold and a homogeneous reductive space of the invertible group of A. Moreover, several metrics (chordal, spherical, pseudo-chordal, non-Euclidean -in Schwarz-Zaks terminology) are considered, allowing a comparison among P(p), the Grassmann manifold of A and the space of positive elements which are unitary with respect to the bilinear form induced by the reflection ε = 2p − 1. Among several metrical results, we prove that geodesics are unique and of minimal length when measured with the spherical and non-Euclidean metrics. * Partially supported by UBACYT TW49 and TX92, PIP 4463 (CONICET) and ANPCYT PICT 97-2259 (Argentina)

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Minimality of Geodesics in Grassmann Manifolds

Cited by 37 publications

References 0 publications

Natural Variational Problems in the Grassmann Manifold of a C*-Algebra with Trace

Natural Variational Problems in the Grassmann Manifold of a C*-Algebra with Trace

Fórmulas ergódicas para medias geométricas y curvas minimales

Projective spaces of aC *-algebra

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