Gabriel Larotonda
Thompson-type formulae.
Tue, 07/05/2011 - 15:01 — SomNambulAuthors: Gabriel Larotonda, Jorge Antezana, Alejandro Varela
Subjects: Functional Analysis
AbstractLet X and Y be two nxn Hermitian matrices. In the article "Proof of a
conjectured exponential formula" (Linear and Multilinear Algebra (19) 1986,
187-197) R. C. Thompson proved that there exist two nxn unitary matrices U and
V such that $$ e^{i X}e^{i Y}=e^{i (UXU^*+VBV^*)}. $$ In this note we consider
extensions of this result to compact operators as well as to operators in an
embeddable II$_1$ factor.Smooth paths of conditional expectations.
Thu, 10/07/2010 - 15:01 — SomNambulAuthors: Gabriel Larotonda, Esteban Andruchow
Subjects: Operator Algebras
AbstractLet A be a von Neumann algebra with a finite trace $\tau$, represented in
$H=L^2(A,\tau)$, and let $B_t\subset A$ be sub-algebras, for $t$ in an interval
$I$. Let $E_t:A\to B_t$ be the unique $\tau$-preserving conditional
expectation. We say that the path $t\mapsto E_t$ is smooth if for every $a\in
A$ and $v \in H$, the map $$ I\ni t\mapsto E_t(a)v\in H $$ is continuously
differentiable. This condition implies the existence of the derivative operator
$$ dE_t(a):H\to H, \ dE_t(a)v=\frac{d}{dt}E_t(a)v.Short paths for symmetric norms in the unitary group.
Tue, 01/05/2010 - 16:01 — SomNambulAuthors: Gabriel Larotonda, Jorge Antezana, Alejandro Varela
Subjects: Metric Geometry
AbstractFor a given symmetrically normed ideal I on an infinite dimensional Hilbert
space H, we study the rectifiable distance in the classical Banach-Lie unitary
group $$ U_I={u is a unitary operator in H, u-1\in I}. $$ We prove that
one-parameter subgroups of U_I are short paths, provided the spectrum of the
exponent is bounded by $\pi$, and that any two elements of U_I can be joined
with a short path, thus obtaining a Hopf-Rinow theorem in this infinite
dimensional setting, for a wide and relevant class of (non necessarily smooth)
metrics.Spaces of nonpositive curvature arising from a finite algebra.
Thu, 08/27/2009 - 13:23 — SomNambulAuthors: Cristian Conde, Gabriel Larotonda
Subjects: Metric Geometry
AbstractIn this paper we introduce a family of examples that can be regarded as
spaces of nonpositive curvature, but with the distinct quality that they are
not complete as metric spaces. This amounts to the fact that they are modelled
on a finite von Neumann algebra, and the metrics introduced arise from the
trace of the algebra. In spite of the noncompleteness of these manifolds, their
geometry can be studied from the view-point of metric geometry, and several
techniques derived from the functional analysis are applied to gain insight on
their geodesic structure.
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