Projections and idempotents with fixed diagonal and the homotopy problem for unit tight frames

We investigate the topological and metric structure of the set of idempotent operators and projections which have prescribed diagonal entries with respect to a fixed orthonormal basis of a Hilbert space. As an application, we settle some cases of conjectures of Larson, Dykema, and Strawn on the connectedness of the set of unit-norm tight frames.Comment: New title and introductio

Projections and Idempotents with Fixed Diagonal and the Homotopy Problem for Unit Tight Frames

We investigate the topological and metric structure of the set of idempotent operators and projections which have prescribed diagonal entries with respect to a fixed orthonormal basis of a Hilbert space. As an application, we settle some cases of conjectures of Larson, Dykema, and Strawn on the connectedness of the set of unit-norm tight frames

From a formula of Kovarik to the parametrization of idempotents in Banach algebra.

If p,q are idempotents in a Banach algebra A and if p+q-1 is invertible, then the Kovarik formula provides an idempotent k(p,q) such that pA=k(p,q)A and Aq=Ak(p,q). We study the existence of such an element in a more general situation. We first show that p+q-1 is invertible if and only if k(p,q) and k(q,p) both exist. Then we deduce a local parametrization of the set of idempotents from this equivalence. Finally, we consider a polynomial parametrization first introduced by Holmes and we answer a question raised at the end of his paper

Similarity implies homotopy in Banach algebras of stable rank one

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Segments of bounded idempotents on a Hilbert space.

Let H be a separable Hilbert space. We prove that any two homotopic idempotents in the algebra may be connected by a piecewise affine idempotent-valued path consisting of 4 segments at most. Moreover, we show that this constant is optimal provided H has infinite dimension. We also explain how this result is linked to the problem of finding common complements for two closed subspaces of H

Segments of bounded idempotents on a Hilbert space.

Let H be a separable Hilbert space. We prove that any two homotopic idempotents in the algebra may be connected by a piecewise affine idempotent-valued path consisting of 4 segments at most. Moreover, we show that this constant is optimal provided H has infinite dimension. We also explain how this result is linked to the problem of finding common complements for two closed subspaces of H

Segments of bounded idempotents on a Hilbert space.

Let H be a separable Hilbert space. We prove that any two homotopic idempotents in the algebra may be connected by a piecewise affine idempotent-valued path consisting of 4 segments at most. Moreover, we show that this constant is optimal provided H has infinite dimension. We also explain how this result is linked to the problem of finding common complements for two closed subspaces of H

Arcs d'idempotents dans les algèbres de Banach

Cette thèse étudie les arcs d'idempotents dans les algèbres de Banach réelles. D'après Esterle (1983), si deux idempotents p,q sont homotopes, on peut toujours les relier par un arc polynômial d'idempotents. Notre principale motivation est d'estimer le degré minimal d'un tel polynôme. Le second paramètre étudié est le nombre minimal de segments requis pour relier p et q, bien défini également s'ils sont homotopes. Nous obtenons des estimations uniformes et optimales pour ces nombres dans les algèbres de dimension finie, les AF-algèbres, les algèbres de von Neumann de Type IBORDEAUX1-BU Sciences-Talence (335222101) / SudocSudocFranceF