INVESTIGADORES
ANDRUCHOW Esteban
artículos
Título:
Metrics in the sphere of a Hilbert C*-module
Autor/es:
E. ANDRUCHOW, A. VARELA
Revista:
Central European Journal of Mathematics - (Online)
Editorial:
Versita
Referencias:
Lugar: Varsovia; Año: 2007 vol. 5 p. 639 - 653
ISSN:
1644-3616
Resumen:
Given a unital C∗-algebra A and a right C∗-module X over A, we consider the problem
of finding short smooth curves in the sphere SX = {x ¡Ê X : x, x = 1}. Curves in SX are measured
considering the Finsler metric which consists of the norm of X at each tangent space of SX . The initial
value problem is solved, for the case when A is a von Neumann algebra and X is selfdual: for any elementC∗-algebra A and a right C∗-module X over A, we consider the problem
of finding short smooth curves in the sphere SX = {x ¡Ê X : x, x = 1}. Curves in SX are measured
considering the Finsler metric which consists of the norm of X at each tangent space of SX . The initial
value problem is solved, for the case when A is a von Neumann algebra and X is selfdual: for any elementshort smooth curves in the sphere SX = {x ¡Ê X : x, x = 1}. Curves in SX are measured
considering the Finsler metric which consists of the norm of X at each tangent space of SX . The initial
value problem is solved, for the case when A is a von Neumann algebra and X is selfdual: for any elementX at each tangent space of SX . The initial
value problem is solved, for the case when A is a von Neumann algebra and X is selfdual: for any elementA is a von Neumann algebra and X is selfdual: for any element
x0 ¡Ê SX and any tangent vector v at x0, there exists a curve ¦Ã(t) = etZ(x0), Z ¡Ê LA(X), Z∗ = −Z0 ¡Ê SX and any tangent vector v at x0, there exists a curve ¦Ã(t) = etZ(x0), Z ¡Ê LA(X), Z∗ = −Z
and Z ¡Ü ¦Ð, such that ¦Ã(0) = x0 and ¦Ã¨B (0) = v, which is minimizing along its path for t ¡Ê [0, 1]. The
existence of such Z is linked to the extension problem of selfadjoint operators. Such minimal curves need
not be unique. Also we consider the boundary value problem: given x0, x1 ¡Ê SX , find a curve of minimal
length which joins them. We give several partial answers to this question. For instance, let us denote
by f0 the selfadjoint projection I − x0 ⊗ x0, if the algebra f0LA(X)f0 is finite dimensional, then there
exists a curve ¦Ã joining x0 and x1, which is minimizing along its path.Z ¡Ü ¦Ð, such that ¦Ã(0) = x0 and ¦Ã¨B (0) = v, which is minimizing along its path for t ¡Ê [0, 1]. The
existence of such Z is linked to the extension problem of selfadjoint operators. Such minimal curves need
not be unique. Also we consider the boundary value problem: given x0, x1 ¡Ê SX , find a curve of minimal
length which joins them. We give several partial answers to this question. For instance, let us denote
by f0 the selfadjoint projection I − x0 ⊗ x0, if the algebra f0LA(X)f0 is finite dimensional, then there
exists a curve ¦Ã joining x0 and x1, which is minimizing along its path.Z is linked to the extension problem of selfadjoint operators. Such minimal curves need
not be unique. Also we consider the boundary value problem: given x0, x1 ¡Ê SX , find a curve of minimal
length which joins them. We give several partial answers to this question. For instance, let us denote
by f0 the selfadjoint projection I − x0 ⊗ x0, if the algebra f0LA(X)f0 is finite dimensional, then there
exists a curve ¦Ã joining x0 and x1, which is minimizing along its path.x0, x1 ¡Ê SX , find a curve of minimal
length which joins them. We give several partial answers to this question. For instance, let us denote
by f0 the selfadjoint projection I − x0 ⊗ x0, if the algebra f0LA(X)f0 is finite dimensional, then there
exists a curve ¦Ã joining x0 and x1, which is minimizing along its path.f0 the selfadjoint projection I − x0 ⊗ x0, if the algebra f0LA(X)f0 is finite dimensional, then there
exists a curve ¦Ã joining x0 and x1, which is minimizing along its path.¦Ã joining x0 and x1, which is minimizing along its path.