The Weyl group and the normalizer of a conditional expectation

We define a discrete group W(E) associated to a faithful normal conditional expectation E : M → N for N ⊆ M von Neuman algebras. This group shows the relation between the unitary group UN and the normalizer NE of E, which can be also considered as the isotropy of the action of the unitary group UM of M on E. It is shown that W(E) is finite if dim Z(N) < ∞ and bounded by the index in the factor case. Also sharp bounds of the order of W(E) are founded. W(E) appears as the fibre of a covering space defined on the orbit of E by the natural action of the unitary group of M. W(E) is computed in some basic examples.Facultad de Ciencias Exacta

Young's inequality in trace-class operators

If a and b are trace-class operators, and if u is a partial isometry, then , where ∥⋅∥1 denotes the norm in the trace class. The present paper characterises the cases of equality in this Young inequality, and the characterisation is examined in the context of both the operator and the Hilbert–Schmidt forms of Young's inequality.Facultad de Ciencias Exacta

Towards the Carpenter's theorem

Let M be a II1 factor with trace τ, A ⊆ Ma masa and EA the unique conditional expectation onto A. Under some technical assumptions on the inclusion A⊆M, which hold true for any semiregular masa of a separable factor, we show that for elements a in certain dense families of the positive part of the unit ball of A, it is possible to find a projection p ∈ M such that EA(p) = a. This shows a new family of instances of a conjecture by Kadison, the so-called "carpenter's theorem".Facultad de Ciencias Exacta

Schur-Horn theorems in II∞-factors

We describe majorization between selfadjoint operators in a ρ-finite II∞ factor (M, τ) in terms of simple spectral relations. For a diffuse abelian von Neumann subalgebra A ⊂ M that admits a (necessarily unique) tracepreserving conditional expectation, denoted by EA, we characterize the closure in the measure topology of the image through EA of the unitary orbit of a selfadjoint operator in M in terms of majorization (i.e., a Schur-Horn theorem). We also obtain similar results for the contractive orbit of positive operators in M and for the unitary and contractive orbits of τ-integrable operators in M.Facultad de Ciencias Exacta

Towards the Carpenter's theorem

Let M be a II1 factor with trace τ, A ⊆ Ma masa and EA the unique conditional expectation onto A. Under some technical assumptions on the inclusion A⊆M, which hold true for any semiregular masa of a separable factor, we show that for elements a in certain dense families of the positive part of the unit ball of A, it is possible to find a projection p ∈ M such that EA(p) = a. This shows a new family of instances of a conjecture by Kadison, the so-called "carpenter's theorem".Facultad de Ciencias Exacta

The local form of doubly stochastic maps and joint majorization in II1 factors

We find a description of the restriction of doubly stochastic maps to separable abelian C ∗ -subalgebras of a II1 factor M. We use this local form of doubly stochastic maps to develop a notion of joint majorization between ntuples of mutually commuting self-adjoint operators that extends those of Kamei (for single self-adjoint operators) and Hiai (for single normal operators) in the II1 factor case. Several characterizations of this joint majorization are obtained. As a byproduct we prove that any separable abelian C ∗ -subalgebra of M can be embedded into a separable abelian C ∗ -subalgebra of M with diffuse spectral measure.Facultad de Ciencias Exacta

El grupo de Weyl y la estructura diferenciable de la órbita de similaridad de una esperanza condicional

A mediados de la década de 1980, se inició una fructífera colaboración entre H. Porta y L. Recht, a quienes pronto se les sumó G. Corach, en la que atacaron una aproximación a la teoría de álgebras de operadores desde el punto de vista de la geometría diferencial. Más tarde se fueron uniendo a este proyecto varios investigadores, como D. Stojanoff, E. Andruchow, A. Maestripieri, A. Varela. Se obtuvieron resultados de caracterización, en función de la estructura geométrica, de propiedades algebraicas de álgebras de von Neumann y C* (ver, por ejemplo, [ACS]), y aún se continúan hallando diversas generalizaciones y aplicaciones. De esa línea participa este trabajo.Facultad de Ciencias Exacta

Schur-Horn theorems in II∞-factors

We describe majorization between selfadjoint operators in a ρ-finite II∞ factor (M, τ) in terms of simple spectral relations. For a diffuse abelian von Neumann subalgebra A ⊂ M that admits a (necessarily unique) tracepreserving conditional expectation, denoted by EA, we characterize the closure in the measure topology of the image through EA of the unitary orbit of a selfadjoint operator in M in terms of majorization (i.e., a Schur-Horn theorem). We also obtain similar results for the contractive orbit of positive operators in M and for the unitary and contractive orbits of τ-integrable operators in M.Facultad de Ciencias Exacta