497 research outputs found
Weak Riemannian manifolds from finite index subfactors
Let be a finite Jones' index inclusion of II factors, and
denote by their unitary groups. In this paper we study the
homogeneous space , which is a (infinite dimensional) differentiable
manifold, diffeomorphic to the orbit
of the Jones projection of the inclusion. We endow with a
Riemannian metric, by means of the trace on each tangent space. These are
pre-Hilbert spaces (the tangent spaces are not complete), therefore is a weak Riemannian manifold. We show that enjoys certain
properties similar to classic Hilbert-Riemann manifolds. Among them, metric
completeness of the geodesic distance, uniqueness of geodesics of the
Levi-Civita connection as minimal curves, and partial results on the existence
of minimal geodesics. For instance, around each point of ,
there is a ball (of uniform radius ) of
the usual norm of , such that any point in the ball is joined to
by a unique geodesic, which is shorter than any other piecewise smooth curve
lying inside this ball. We also give an intrinsic (algebraic) characterization
of the directions of degeneracy of the submanifold inclusion , where the last set denotes the Grassmann manifold
of the von Neumann algebra generated by and .Comment: 19 page
Non positively curved metric in the space of positive definite infinite matrices
We introduce a Riemannian metric with non positive curvature in the (infinite dimensional) manifold Σ∞ of positive invertible operators of a Hilbert space H, which are scalar perturbations of Hilbert-Schmidt operators. The (minimal) geodesics and the geodesic distance are computed. It is shown that this metric, which is complete, generalizes the well known non positive metric for positive definite complex matrices. Moreover, these spaces of finite matrices are naturally imbedded in Σ∞.Fil: Andruchow, Esteban. Consejo Nacional de Investigaciones CientÃficas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina. Universidad Nacional de General Sarmiento. Instituto de Ciencias; ArgentinaFil: Varela, Alejandro. Universidad Nacional de General Sarmiento. Instituto de Ciencias; Argentina. Consejo Nacional de Investigaciones CientÃficas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentin
Quotient p-Schatten metrics on spheres
Let S(H) be the unit sphere of a Hilbert space H and Up(H) thegroup of unitary operators in H such that u−1 belongs to the p-Schatten idealBp(H). This group acts smoothly and transitively in S(H) and endows it witha natural Finsler metric induced by the p-norm kzkp = tr(zz∗)p/21/p. Thismetric is given bykvkx,p = min{kz − ykp : y ∈ gx},where z ∈ Bp(H)ah satisfies that (dÏ€x)1(z) = z · x = v and gx denotes theLie algebra of the subgroup of unitaries which fix x. We call z a lifting of v.A lifting z0 is called a minimal lifting if additionally kvkx,p = kz0kp. Inthis paper we show properties of minimal liftings and we treat the problemof finding short curves α such that α(0) = x and ˙α(0) = v with x ∈ S(H)and v ∈ TxS(H) given. Also we consider the problem of finding short curveswhich join two given endpoints x, y ∈ S(H).Fil: Andruchow, Esteban. Consejo Nacional de Investigaciones CientÃficas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina. Universidad Nacional de General Sarmiento. Instituto de Ciencias; ArgentinaFil: Antunez, Andrea. Universidad Nacional de General Sarmiento. Instituto de Ciencias; Argentin
Metric geodesics of isometries in a Hilbert space and the extension problem
We consider the problem of finding short smooth curves of isometries in a Hilbert space H. The length of a smooth curve γ(t), t ∈ [0, 1], is measured by means of ∫^1-0 γ^. (t)ǀǀ dt, where ǀǀ ǀǀ denotes the usual norm of operators. The initial value problem is solved: for any isometry Vo and each tangent vector at V0 (which is an operator of the form iXV0 with X* = X) with norm less than or equal to Ï€, there exist curves of the form e^itZ V0, with initial velocity iZV0 = iXV0, which are short along their path. These curves, which we call metric geodesics, need not be unique, and correspond to the so called extension problem considered by M.G. Krein and others: in our context, given asymmetric operator X0|R(V0) : R(V0)→H, find all possible Z* = Z extending X0|R(V0) to all H, with ǀǀZǀǀ= ǀǀX0ǀǀ. We also consider the problem of finding metric geodesics joining two given isometries V0 and V1. It is well known that if there exists a continuous path joining V0 and V1, then both ranges have the same codimension. We show that if this number is finite, then there exist metric geodesics joining V0 and V1.Fil: Andruchow, Esteban. Universidad Nacional de General Sarmiento. Instituto de Ciencias; Argentina. Consejo Nacional de Investigaciones CientÃficas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; ArgentinaFil: Recht, Lázaro. Universidad Simón BolÃvar; Venezuela. Consejo Nacional de Investigaciones CientÃficas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; ArgentinaFil: Varela, Alejandro. Consejo Nacional de Investigaciones CientÃficas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina. Universidad Nacional de General Sarmiento. Instituto de Ciencias; Argentin
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