Cartan subalgebras of operator ideals

Abstract

Abstract. Denote by U I (H) the group of all unitary operators in 1+I where H is a separable infinite-dimensional complex Hilbert space and I is any twosided ideal of B(H). A Cartan subalgebra C of I is defined in this paper as a maximal abelian self-adjoint subalgebra of I and its conjugacy class is defined herein as the set of Cartan subalgebras {V CV * | V ∈ U I (H)}. For nonzero proper ideals I we construct an uncountable family of Cartan subalgebras of I with distinct conjugacy classes. This is in contrast to the by now classical observation of P. de La Harpe who noted that when I is any of the Schatten ideals, there is precisely one conjugacy class under the action of the full group of unitary operators on H. Our perspective is that the action of the full unitary group on Cartan subalgebras of I is transitive, while by shrinking to U I (H) we obtain an action with uncountably many orbits if {0} = I = B(H).In the case when I is a symmetrically normed ideal and is the dual of some Banach space, we show how the conjugacy classes of the Cartan subalgebras of I 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 I (H) 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 I (H) 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.

On restricted diagonalization

“…On the other hand, we study the structure of the self-adjoint part of D J and the way it sits inside the self-adjoint part of the ideal J . These results allow us to answer open problems stated in [10].…”

“…Each operator ideal has associated a subgroup of the unitary operators defined by U J (H) := U(H) ∩ (I + J ), where U(H) denotes the unitary group of H. Throughout, we fix an orthonormal basis e = {e n } n≥1 of H and we let D denote the corresponding algebra of diagonal operators. We call an operator A ∈ B(H) diagonalizable if there exists a unitary U ∈ U(H) such that UAU * ∈ D. In their study of Lie theoretic properties of operator ideals, Beltit ¸ȃ, Patnaik and Weiss introduced in [10] the notion of U J (H)-diagonalizable operators with respect to e, or simply, restricted diagonalizable operators, which refers to those diagonalizable operators that can be diagonalized by a unitary U ∈ U J (H). Furthermore, given two operator ideals I, J , they offered the following versions of restricted diagonalization defined by the sets:…”

Restricted diagonalization of finite spectrum normal operators and a theorem of Arveson

“…For this we use the term restricted diagonalization. This concept has been studied by others in the aforementioned paper of Brown-Douglas-Fillmore [BDF73], as well as by Beltit ¸a-Patnaik-Weiss [BPW16], and Hinkkanen [Hin85]. To our knowledge, the term restricted diagonalization was introduced by Beltit ¸a-Patnaik-Weiss.…”

Subideals of Operators – A Survey and Introduction to Subideal-Traces

“…Therefore, the definition of trace on a B(H)-ideal, that is, a unitarily invariant linear functional, need not make sense on a subideal. Motivated by our work in [2] on unitary operators of the form U = 1 + A for A ∈ K(H) we observe that subideals I are invariant under these unitaries (i.e., U IU * ⊂ I). This led the authors of this paper to introduce the notion of a subideal-trace as defined below in Definition 5.3 (see also Remark 5.9).…”