Positivity 8: 31–48, 2004.
© 2004 Kluwer Academic Publishers. Printed in the Netherlands.
Orbits of Positive Operators from a Differentiable
, A. MAESTRIPIERI
and D. STOJANOFF
Depto. de Matemática, Facultad de Ingeniería-UBA, Buenos Aires, Argentina.
Instituto de Ciencias, UNGS, Los Polvorines, Argentina.
Depto. de Matemática, FCE-UNLP, La Plata, Argentina.
Instituto Argentino de Matematica, CONICET, Argentina.
(Received 18 July 2002; accepted 23 March 2003)
Abstract. Let be a unital C
-algebra and G the group of units of . A geometrical study of the
action of G over the set
of all positive elements of is presented. The orbits of elements with
closed range by this action are provided with a structure of differentiable homogeneous space with
a natural connection. The orbits are partitioned in “components” which also have a rich geometrical
Mathematics Subject Classiﬁcation 1991: 46L99, 46B70, 58B20
Key words: closed range positive operators, differential geometry, Thompson components
The differential geometry of the space of n×n positive deﬁnite complex matrices
turns to be a relevant subject in problems coming from many different areas. To
mention only a few, we refer the reader to the work of Ohara et al. [29, 30], in linear
systems, Amari [1, 2], Kass , Campbell , Murray and Rice  in
statistics, Bougerol  in Kalman-Bucy ﬁlters, Liverani and Wojtkowski  in
Lagrangian geometry, Hiai and Petz [19, 20], Petz [31, 32] in quantum systems.
Moreover, Uhlmann [34, 35] proposed the extension to mixed states of the Berry
phase using techniques from differential geometry on the set of density operators,
i.e., trace class positive operators, on a Hilbert space. Ulhmann’s work has been
extended by Dabrowski and Jadczyk , Dabrowski and Grosse , Dittmann
[13, 14], Dittmann and Rudolph [15, 16] and Uhlmann himself . Uhlmann’s
approach is an invitation to study the space L
of all positive bounded linear
operators on the Hilbert space from a differentiable viewpoint. A description
of the differential geometry of GL
(i.e., the set of invertible positive linear
operators) has been done in [6, 9] and extended to the set of closed range posit-
ive operators in . In , the set L
is partitioned in certain “components”,