Cartan subalgebras of operator ideals
Daniel BeltitaSASMITA PATNAIKG. Weiss
Primary 22E65Secondary 47B1047L2020G20operator idealCartan subalgebrainfinite-dimensional linear algebraic group
Denote by $U_{Ic}(Hc)$ the group of all unitary operators in $1+Ic$
where $Hc$ is a separable infinite-dimensional complex Hilbert space and
$Ic$ is any two-sided ideal of $Bc(Hc)$.
A Cartan subalgebra $Cc$ of $Ic$
is defined in this paper as a maximal abelian self-adjoint subalgebra of~$Ic$,
and its conjugacy class is defined herein as the set of Cartan subalgebras ${VCc V^*mid Vin U_{Ic}(Hc)}$.
For nonzero proper ideals $Ic$, we construct an uncountable family
of Cartan subalgebras of $Ic$ with distinct conjugacy classes.
This is in contrast to the (by now classical) observation of P. de La Harpe,
who noted that when $Ic$ is any of the Schatten ideals,
there is precisely one conjugacy class under the action of the full group of unitary operators on~$Hc$.
Our perspective is that the action of the full unitary group on Cartan subalgebras of $Ic$ is transitive,
while by shrinking to $U_{Ic}(Hc)$, we obtain an action with uncountably many orbits if ${0}
eIc
eBc(Hc)$.
In the case when $Ic$ is a symmetrically normed ideal and
is the dual of some Banach space, we show how the conjugacy classes of the Cartan subalgebras of $Ic$ become smooth manifolds modeled on suitable Banach spaces.
These manifolds are endowed with groups of smooth transformations given by the action of the group $U_{Ic}(Hc)$ on the orbits, and are equivariantly diffeomorphic to each other. We then find that there exists a unique diffeomorphism class of full flag manifolds of $U_{Ic}(Hc)$, and we give its construction. This resembles the case of compact Lie groups when one has a unique full flag manifold, since all the Cartan subalgebras are conjugated to each other.
Indiana University Mathematics Journal
2016
text
pdf
10.1512/iumj.2016.65.5784
10.1512/iumj.2016.65.5784
en
Indiana Univ. Math. J. 65 (2016) 1 - 37
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