Alejandro Varela

Metric geodesics of isometries in a Hilbert space and the extension problem

We consider the problem of finding short smooth curves of isometries in a Hilbert space H. The length of a smooth curve γ(t), t ∈ [0,1], is measured by means of ∫ 1 0 ∥γ(t)∥ dt, where ∥ ∥ denotes the usual norm of operators. The initial value problem is solved: for any isometry V 0 and each tangent vector at V 0 (which is an operator of the form iXV 0 with X* = X) with norm less than or equal to π, there exist curves of the form e itZ V 0 , with initial velocity iZVo = iXV 0 , which are short along their path. These curves, which we call metric geodesics, need not be unique, and correspond to the so called extension problem considered by M.G. Krein and others: in our context, given a symmetric operator X 0 |R(V 0 ): R(V 0 ) → H, find all possible Z* = Z extending X 0 | R(V0) to all H, with ∥Z∥ = ∥X0∥. We also consider the problem of finding metric geodesics joining two given isometries Vo and V 1 . It is well known that if there exists a continuous path joining V 0 and V 1 , then both ranges have the same codimension. We show that if this number is finite, then there exist metric geodesics joining V 0 and V 1 .

Geometry and the Jones projection of a state

LetA be a von Neumann algebra and φ a faithful normal state. ThenOϕ = {ϕ ºAd(g1) :g ∈GA}andUϕ = {ϕ ºAd(u*) :u ∈UA are homogeneous reductive spaces. IfA is aC* algebra,eφ the Jones projection of the faithful state φ viewed as a conditional expectation, then we prove that the similarity orbit ofeφ by invertible elements ofA can be imbedded inA⊗A in such a way thateφ is carried to 1 ⊗ 1 and the orbit ofeφ to a homogeneous reductive space and an analytic submanifold ofA⊗A.

Metrics in the sphere of a C*-module

AbstractGiven a unital C*-algebra

Weight centralizer expectations with finite index

\mathcal{A}

Best approximation by diagonal compact operators

and a right C*-module

Minimal hermitian matrices with fixed entries outside the diagonal

\mathcal{X}