Orbits of conditional expectations

Abstract

Let N ⊆ M be von Neumann algebras and E : M → N a faithful normal conditional expectation. In this work it is shown that the similarity orbit S(E) of E by the natural action of the invertible group of G M of M has a natural complex analytic structure and the map given by this action: G M → S(E) is a smooth principal bundle. It is also shown that if N is finite then S(E) admits a Reductive Structure. These results were known previously under the conditions of finite index and N ′ ∩ M ⊆ N , which are removed in this work. Conversely, if the orbit S(E) has an Homogeneous Reductive Structure for every expectation defined on M , then M is finite. For every algebra M and every expectation E, a covering space of the unitary orbit U (E) is constructed in terms of the connected component of 1 in the normalizer of E. Moreover, this covering space is the universal covering in any of the following cases: 1) M is a finite factor and Ind(E) < ∞; 2) M is properly infinite and E is any expectation; 3) E is the conditional expectation onto the centralizer of a state. Therefore, in those cases, the fundamental group of U (E) can be characterized as the Weyl group of E.

Local Minimal Curves in Homogeneous Reductive Spaces of the Unitary Group of a Finite von Neumann Algebra

“…Our next example is about the unitary orbit of E. For a treatment of geometric properties of this example in a more general setting than finite algebras, we refer the reader to [4] and [5]. Consider the unitary orbit of E with this action…”

“…In [5] was proved that for any faithful normal conditional expectation, its unitary orbit is a homogeneous reductive space of U M . In the finite algebra case we shall prove the orthogonality condition restricting to the unique trace invariant conditional expectation E. The arguments involved are adapted from Proposition 4.5 in [5], to this easier case.…”

“…In the finite algebra case we shall prove the orthogonality condition restricting to the unique trace invariant conditional expectation E. The arguments involved are adapted from Proposition 4.5 in [5], to this easier case. The Lie algebra of N E is the kernel of the differential of the natural fibration…”

Holomorphic geometric models for representations of C∗-algebras

“…(Such a faithful state exists if the Hilbert space H is separable.) The set A E is a von Neumann subalgebra of A and, using the modular group of A induced by the gauge state ψ := ϕ • E, it can be proven that there exists a faithful, normal, conditional expectation F : [4,Proposition 4.5]). Set Δ = E + F − EF .…”

“…By considering the connected 1-component G(E) 0 = G A E · G B of the isotropy group G(E) (see [4,Proposition 3.3]), the existence of Δ implies that G(E) is in fact a Banach-Lie subgroup of G A , the orbits S(E) and U(E) are homogeneous Banach manifolds, and the quotient map G A → S(E) G A /G(E) is an analytic submersion, see [4,Corollary 4.7 and Theorem 4.8]. Also, the following assertions hold: Proposition 7.9.…”