Banach manifolds of algebraic elements in the algebra \[\mathcal {L}\]
(H) of bounded linear operatorsof bounded linear operators

as a disjoint union of closed connected subsets M of Z each of which is an orbit under the action of G, the group of all C*-algebra automorphisms of Z. Those orbits M consisting of hermitian algebraic elements with a fixed finite rank r, (0< r<∞) are real-analytic direct submanifolds of Z. Using the C*-algebra structure of Z, a Banach-manifold structure and a G-invariant torsionfree affine connection ∇ are defined on M, and the geodesics are computed. If M is the orbit of a finite rank projection, then a G-invariant Riemann structure is defined with respect to which ∇ is the Levi-Civita connection.

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José Isidro. "Banach manifolds of algebraic elements in the algebra \[\mathcal {L}\] (H) of bounded linear operatorsof bounded linear operators." Open Mathematics 3.2 (2005): 188-202. <http://eudml.org/doc/268863>.

@article{JoséIsidro2005,
abstract = {Given a complex Hilbert space H, we study the manifold \[\mathcal \{A\}\] of algebraic elements in \[Z = \mathcal \{L\}\left( H \right)\] . We represent \[\mathcal \{A\}\] as a disjoint union of closed connected subsets M of Z each of which is an orbit under the action of G, the group of all C*-algebra automorphisms of Z. Those orbits M consisting of hermitian algebraic elements with a fixed finite rank r, (0< r<∞) are real-analytic direct submanifolds of Z. Using the C*-algebra structure of Z, a Banach-manifold structure and a G-invariant torsionfree affine connection ∇ are defined on M, and the geodesics are computed. If M is the orbit of a finite rank projection, then a G-invariant Riemann structure is defined with respect to which ∇ is the Levi-Civita connection.},
author = {José Isidro},
journal = {Open Mathematics},
keywords = {17C27; 17C36; 17B65},
language = {eng},
number = {2},
pages = {188-202},
title = {Banach manifolds of algebraic elements in the algebra \[\mathcal \{L\}\] (H) of bounded linear operatorsof bounded linear operators},
url = {http://eudml.org/doc/268863},
volume = {3},
year = {2005},
}

TY - JOUR
AU - José Isidro
TI - Banach manifolds of algebraic elements in the algebra \[\mathcal {L}\] (H) of bounded linear operatorsof bounded linear operators
JO - Open Mathematics
PY - 2005
VL - 3
IS - 2
SP - 188
EP - 202
AB - Given a complex Hilbert space H, we study the manifold \[\mathcal {A}\] of algebraic elements in \[Z = \mathcal {L}\left( H \right)\] . We represent \[\mathcal {A}\] as a disjoint union of closed connected subsets M of Z each of which is an orbit under the action of G, the group of all C*-algebra automorphisms of Z. Those orbits M consisting of hermitian algebraic elements with a fixed finite rank r, (0< r<∞) are real-analytic direct submanifolds of Z. Using the C*-algebra structure of Z, a Banach-manifold structure and a G-invariant torsionfree affine connection ∇ are defined on M, and the geodesics are computed. If M is the orbit of a finite rank projection, then a G-invariant Riemann structure is defined with respect to which ∇ is the Levi-Civita connection.
LA - eng
KW - 17C27; 17C36; 17B65
UR - http://eudml.org/doc/268863
ER -

References

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