Polynomial and Polygonal Connections Between Idempotents in Finite-Dimensional Real Algebras

Abstract

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.

Similarity implies homotopy of idempotents in Banach algebras of stable rank one

“…For finite-dimensional real algebras, J. Esterle and the author have proved in [3] that any two homotopic idempotents can be connected by a piecewise affine homotopy within at most 3 affine steps (and the number 3 is optimal). Thus Theorem 1 provides a generalization of the latter together with a different proof.…”

“…In [3] it is also shown that any two homotopic idempotents in a finitedimensional real algebra can be connected by a polynomial homotopy of degree at most 3 (optimal again). Thanks to a basic lemma of [4], a direct consequence of Therorem 1 is that the latter global estimate remains true for Banach algebras of stable rank one with 5 instead of 3.…”

Segments of bounded linear idempotents on a Hilbert space

“…If A has finite dimension, then Esterle and the author have proved in [3] that the estimate s(p, q) 3 holds for every pair of homotopic idempotents in A. So our first purpose is to find an algebra where the numbers s(p, q) may no longer be uniformly bounded by 3; this is done in Section 2.…”

Path connectivity of idempotents on a Hilbert space

“…In 1983, J. Esterle established that there exists a polynomial connection between two homotopic idempotents of P in a Banach algebra [8]. Particularly, in 2004, J. Esterle had obtained that, for two homotopic idempotents P and Q in a finite dimensional real algebra,s(P, Q) ≤ 3 [9]. In 2005, J. Giol had proved that in an infinite dimensional Hilbert space, for two homotopic idempotents P and Q,s(P, Q) ≤ 4 (see [11]), wheres(P, Q) denotes the minimal number of segments required to connect not only from P to Q, but also from Q to P in P. Moreover, J. Giol had proved in [11] the following result.…”

“…Recently, a number of researchers have considered questions concerning the path connectivity between idempotents (see [7,8,9,11]). In 1979, J. Zemánek found that the components of P are arcwise connected ( [12]).…”