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We generalize earlier results about connected components of idempotents in Banach algebras, due to B. SzH{o}kefalvi Nagy, Y. Kato, S. Maeda, Z. V. Kovarik, J. Zemanek, J. Esterle. Let $A$ be a unital complex Banach algebra, and $p(lambda) = prodlimits_{i = 1}^n (lambda - lambda_i)$ a polynomial over $Bbb C$, with all roots distinct. Let $E_p(A) := {a in A mid p(a) = 0}$. Then all connected components of $E_p(A)$ are pathwise connected (locally pathwise connected) via each of the following three types of paths: 1)~similarity via a finite product of exponential functions (via an exponential function); 2)~a polynomial path (a cubic polynomial path); 3)~a polygonal path (a polygonal path consisting of $n$ segments). If $A$ is a $C^*$-algebra, $lambda_i in Bbb R$, let $S_p(A):= {ain A mid a = a^*$, $p(a) = 0}$. Then all connected components of $S_p(A)$ are pathwise connected (locally pathwise connected), via a path of the form $e^{-ic_mt}dots e^{-ic_1t} ae^{ic_1t}dots e^{ic_mt}$, where $c_i = c_i^*$, and $t in [0, 1]$ (of the form $e^{-ict} ae^{ict}$, where $c = c^*$, and $t in [0,1]$). For (self-adjoint) idempotents we have by these old papers that the distance of different connected components of them is at least~$1$. For $E_p(A)$, $S_p(A)$ we pose the problem if the distance of different connected components is at least $min bigl{|lambda_i - lambda_j| mid 1 leq i,j leq n, i eq jbigr}$. For the case of $S_p(A)$, we give a positive lower bound for these distances, that depends on $lambda_1, dots, lambda_n$. We show that several local and global lifting theorems for analytic families of idempotents, along analytic families of surjective Banach algebra homomorphisms, from our recent paper with B. Aupetit and M. Mbekhta, have analogues for elements of $E_p(A)$ and $S_p(A)$.
Generalizing earlier results about the set of idempotents in a Banach algebra, or of self-adjoint idempotents in a $C^*$-algebra, we announce constructions of nice connecting paths in the connected components of the set of elements in a Banach algebr
Let $G$ be a complex simple Lie group and let $g = hbox{rm Lie},G$. Let $S(g)$ be the $G$-module of polynomial functions on $g$ and let $hbox{rm Sing},g$ be the closed algebraic cone of singular elements in $g$. Let ${cal L}s S(g)$ be the (graded) id
A subset $A$ of a Banach space is called Banach-Saks when every sequence in $A$ has a Ces{`a}ro convergent subsequence. Our interest here focusses on the following problem: is the convex hull of a Banach-Saks set again Banach-Saks? By means of a comb
Let $A$ be a real commutative Banach algebra with unity. Let $a_0in Asetminus{0}$. Let $mathbb Z a_0:={na_0}_{nin mathbb Z}$. Then, $mathbb Z a_0$ is a discrete subgroup of $A$. For any $nin mathbb Z$, the Frechet derivative of the mapping $$x , in ,
The use of the properties of actions on an algebra to enrich the study of the algebra is well-trodden and still fashionable. Here, the notion and study of endomorphic elements of (Banach) algebras are introduced. This study is initiated, in the hope