Metrics in the sphere of a Hilbert C*-module

E. ANDRUCHOW, A. VARELA

Central European Journal of Mathematics - (Online)

Versita

Año: 2007 vol. 5 p. 639 - 653

1644-3616

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.