131 research outputs found
Geometry of oblique projections
Let A be a unital C*-algebra. Denote by P the space of selfadjoint
projections of A. We study the relationship between P and the spaces of
projections P_a determined by the different involutions #_a induced by positive
invertible elements a in A. The maps f_p: P \to P_a sending p to the unique q
in P_a with the same range as p and \Omega_a: P_a \to P sending q to the
unitary part of the polar decomposition of the symmetry 2q-1 are shown to be
diffeomorphisms. We characterize the pairs of idempotents q, r in A with
|q-r|<1 such that there exists a positive element a in A verifying that q, r
are in P_a. In this case q and r can be joined by an unique short geodesic
along the space of idempotents Q of A.Comment: 25 pages, Latex, to appear in Studia Mathematic
Projective spaces of a C*-algebra
Based on the projective matrix spaces studied by B. Schwarz and A. Zaks, we
study the notion of projective space associated to a C*-algebra A with a fixed
projection p. The resulting space P(p) admits a rich geometrical structure as a
holomorphic manifold and a homogeneous reductive space of the invertible group
of A. Moreover, several metrics (chordal, spherical, pseudo-chordal,
non-Euclidean - in Schwarz-Zaks terminology) are considered, allowing a
comparison among P(p), the Grassmann manifold of A and the space of positive
elements which are unitary with respect to the bilinear form induced by the
reflection e = 2p-1. Among several metrical results, we prove that geodesics
are unique and of minimal length when measured with the spherical and
non-Euclidean metrics.Comment: 26 pages, Late
Redundant decompositions, angles between subspaces and oblique projections
Let Η be a complex Hilbert space. We study the relationships between the angles between closed subspaces of H, the oblique projections associated to non direct decompositions of H and a notion of compatibility between a positive (semidefinite) operator A acting on H and a closed subspace S of H. It turns out that the compatibility is ruled by the values of the Dixmier angle between the orthogonal complement S _l_ of S and the closure of AS. We show that every redundant decomposition H = S+M_l_ (where redundant means that S ∩M_l_ is not trivial) occurs in the presence of a certain compatibility. We also show applications of these results to some signal processing problems (consistent reconstruction) and to abstract splines problems which come from approximation theory
Oblique projections and abstract splines
Given a closed subspace L of a Hilbert space ℋ and a bounded linear operator A ∈ L(ℋ) which is positive, consider the set of all A-self-adjoint projections onto Y: ℘(A,Y) = {Q ∈ L(ℋ): Q2 = Q, Q(ℋ) = Y, AQ = Q*A}. In addition, if ℋ1 is another Hilbert space, T : ℋ → ℋ1 is a bounded linear operator such that T*T = A and ξ ∈ ℋ, consider the set of (T, Y) spline interpolants to ξ: sp(T, Y, ξ) = { η ε ξ + Y : ∥Tη∥ = min ∥T(ξ + σ)∥}. A strong relationship exists between ℘(A, Y) and s p(T, Y, ξ). In fact, ∥(A, Y) is not empty if and only if s p(T, Y, ξ) is not empty for every ξ ∈ ℋ. In this case, for any ξ ∈ ℋ\Y it holds s p(T, Y, ξ) = {(1 - Q)ξ:Q ∈ ℘(A, Y)} and for any ξ ∈ ℋ, the unique vector of s p(T, Y, ξ) with minimal norm is (1 - PA,Y)ξ, where PA,L is a distinguished element of ℘(A, Y). These results offer a generalization to arbitrary operators of several theorems by de Boor, Atteia, Sard and others, which hold for closed range operators.Facultad de Ciencias Exacta
Weak Riemannian manifolds from finite index subfactors
Let be a finite Jones' index inclusion of II factors, and
denote by their unitary groups. In this paper we study the
homogeneous space , which is a (infinite dimensional) differentiable
manifold, diffeomorphic to the orbit
of the Jones projection of the inclusion. We endow with a
Riemannian metric, by means of the trace on each tangent space. These are
pre-Hilbert spaces (the tangent spaces are not complete), therefore is a weak Riemannian manifold. We show that enjoys certain
properties similar to classic Hilbert-Riemann manifolds. Among them, metric
completeness of the geodesic distance, uniqueness of geodesics of the
Levi-Civita connection as minimal curves, and partial results on the existence
of minimal geodesics. For instance, around each point of ,
there is a ball (of uniform radius ) of
the usual norm of , such that any point in the ball is joined to
by a unique geodesic, which is shorter than any other piecewise smooth curve
lying inside this ball. We also give an intrinsic (algebraic) characterization
of the directions of degeneracy of the submanifold inclusion , where the last set denotes the Grassmann manifold
of the von Neumann algebra generated by and .Comment: 19 page
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