Lázaro Recht

An operator inequality

Abstract For Hilbert space operators, with S invertible hermitian, it is proved that ⇁STS -1 +S -1 TS⇁⩾2⇁T⇁ .

Conditional expectations and operator decompositions

For aC*-algebraA with a conditional expectation Φ:A → A onto a subalgebraB we have the linear decompositionA=B⊕H whereH=ker(Φ). Since Φ preserves adjoints, it is also clear that a similar decomposition holds for the selfadjoint parts:As=Bs⊕Hs (we useVs={aεV;a*=a} for any subspaceV of A). Apply now the exponential function to each of the three termsAs,Bs, andHs. The results are: the setG+ of positive invertible elements ofA, the setB+ of positive invertible elements ofB, and the setC={eh;h*=h, Φ(h)=0}, respectively.We consider here the question of lifting the decompositionAs=Bs⊕Hs to the exponential sets. Concretely, is every element ofG+ the product of elements ofB+ andC, respectively, just as any selfadjoint element ofA is the sum of selfadjoint elements ofB andH? The answer is yes in the following sense: Eacha ε G+ is the positive part of a productbe of elementsb ε B+ and c εC, and bothb andc are uniquely determined and depend analytically ona. This can be rephrased as follows: The map (6, c) →(bc)+ is an analytic diffeomorphism fromB+x C ontoG+, where for any invertiblex ε A we denote with x+ the positive square root ofxx*. This result can be expressed equivalently as: The map (b, c) →bcb is a diffeomorphism between the same spaces.Notice that combining the polar decomposition with these results we can write every invertibleg ε A asg=bcu, whereb ε B+,c ε C, andu is unitary. This decomposition is unique and the factorsb, c, u depend analytically ofg. In the case of matrix algebras with Φ=trace/dimension, the factorization corresponds tog=| det(g)|cu withc > 0,det(c)=1, andu unitary.This paper extends some results proved by G. Corach and the authors in [2]. Also, Theorem 2 states that the reductive homogeneous space resulting from a conditional expectation satisfies the regularity hypothesis introduced by L. Mata-Lorenzo and L. Recht in [5], Definition 11.1. The situation considered here is the ”general context” for regularity indicated in the introduction of the last mentioned paper.

Metric geometry in homogeneous spaces of the unitary group of a C * -algebra: Part I¿minimal curves

Abstract We study the metric geometry of homogeneous spaces P of the unitary group of a C ∗ -algebra A modulo the unitary group of a C ∗ -subalgebra B , where the invariant Finsler metric is induced by the quotient norm of A / B . The main results include the following. In the von Neumann algebra context, for each tangent vector X at a point ρ∈ P , there is a geodesic γ(t), γ (0)=X , which is obtained by the action on ρ of a 1-parameter group in the unitary group of A . This geodesic is minimizing up to length π/2.

Finsler geometry and actions of the p-Schatten unitary groups

Let p be an even positive integer and U p (H) the Banach-Lie group of unitary operators u which verify that u — 1 belongs to the p-Schatten ideal B p (H). Let O be a smooth manifold on which U p (H) acts transitively and smoothly. Then one can endow O with a natural Finsler metric in terms of the p-Schatten norm and the action of U p (H). Our main result establishes that for any pair of given initial conditions x ∈ O and X ∈ (TO) x there exists a curve 6(t) = e t z · x in O, with z a skew-hermitian element in the p-Schatten class, such that δ(0) = x and δ(0) = X, which remains minimal as long as t∥z∥ p < π/4. Moreover, δ is unique with these properties. We also show that the metric space (O, d) (where d is the rectifiable distance) is complete. In the process we establish minimality results in the groups U p (H) and a convexity property for the rectifiable distance. As an example of these spaces, we treat the case of the unitary orbit O = {uAu*: u ∈ U p (H)} of a self-adjoint operator A ∈ B(H).

Convexity properties of Tr[(a∗a)n]

Abstract In this paper, for any even positive integer p=2n , we consider the function E p (a)= Tr [(a * a) n ] defined on a C * -algebra A with a faithful trace Tr : A → C . We show that the quadratic form Q(E p ) a associated to the Hessian of E p at a∈ A is non-negative and for p>2 we give a simple characterization of the null-space. Finally, we show that g(t)=E p (a+t b) is at least as convex as t p uniformly with respect to b .

Exponential sets and their geometric motions

AbstractLet us call an “exponential set” in a C*-algebraA any set consisting of the exponentialseX of all the self-adjoint elementsX of a subspaceH ofA. For example, ifH = A the resulting exponential setG+ consists of all the positive invertible elements ofA, and all other exponential sets are contained in G+. An exponential setC ⊂ G+ inherits the geometric structure of the space G+ when the defining subspaceH has suitable properties. Here we investigate reasonable conditions onH that permit, for example, reduction of the canonical connection of G+ toC. As a consequence, in these cases the setC has a rich family of motions that are “rigid” for the geometry of G+. In particular we find thatC itself operates on C by the actionLga = (g−1)*ag− of the groupG of all invertible elements ofA in G+, and that the subgroup generated byC is transitive. Similarly, in several cases the productscu withc ε C andu unitary form a closed Lie subgroup ofG that acts onC, withC contained in it. This is the case forH, the space of elements of trace zero, when there is a trace.The conditions onH are all additions to the following basic situation:H is the kernel of a (bounded linear) projection Φ:A → A. For example, ifH is closed under triple brackets [X, [Y, Z]] then parallel transport in G+ along geodesics inC through 1 ∈C preserves vectors tangent toC. Similarly, if the symmetric part of [eXYe−X,Z] is inH for allX, Y, Z ∈Hs thenC is “geodesically convex” in the sense that geodesics tangent toC stay inC. The most interesting cases correspond to a conditional expectation. Two additional conditions produce the groups described in the first paragraph: the case of a Z2-graded C*-algebra with Φ the projection on the elements of degree 0 (which is automatically a conditional expectation) and the case of a conditional expectation such that the anti-symmetric part ofeXYe−Y is in the range of Φ wheneverX, Y are self-adjoint and Φ(X)= Φ(Y) = 0. This is verified for example in the case of central traces.

The space of spectral measures is a homogeneous reductive space

We prove that the space of spectral measures on a W*-algebra is a smooth Banach manifold in a natural way and that the action of the group of invertible elements of the algebra by inner automorphisms makes it into a reductive homogeneous space. This gives a geometric structure for the set of normal operators with the same spectrum.

GEOMETRY OF UNITARIES IN A FINITE ALGEBRA: VARIATION FORMULAS AND CONVEXITY

Given a C*-algebra

GRASSMANNIANS OF A FINITE ALGEBRA IN THE STRONG OPERATOR TOPOLOGY

\mathcal{A}

Classification of Linear Connections

with trace τ, we compute the first and second variation formulas for the p-energy functional Fp of the unitary group