We demonstrate that the perforative phenomena shown to occur among simple amenable C Ã -algebras by Villadsen and Toms can be realized within a dynamical framework. More specifically, we construct a minimal homeomorphism for which the K 0 group of the crossed product fails to be weakly unperforated, and a minimal homeomorphism for which the crossed product has the same Elliott invariant as an AT-algebra but has Cuntz semigroup which fails to be almost unperforated.

Let P(A) be the set of idempotents in a finite-dimensional real algebra A. Let p and q be idempotents that lie in the same component of P(A). Then, among the continuous paths connecting p and q in P(A), there exist a polynomial path of degree at most 3 and a polygonal path consisting of at most three segments.

Abstract. We investigate the topological and metric structure of the set of idempotent operators and projections which have prescribed diagonal entries with respect to a ſxed 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.

Let H be a separable Hilbert space. We prove that any two homotopic idempotents in the algebra L(H ) 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.

We establish the property stated in the title by means of an elementary construction: in a Banach algebra of stable rank one, any two similar idempotents can be connected by a piecewise affine path of idempotents consisting of at most 3 affine steps. Mathematics Subject Classification (2000). 19A99, 46H20.