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Keywords:
Banach algebra; $C^*$-algebra; (self-adjoint) idempotent; connected component of (self-adjoint) algebraic elements; (local) pathwise connectedness; similarity; analytic path; polynomial path; polygonal path; centre of a Banach algebra; distance of connected components
Banach algebra; $C^*$-algebra; (self-adjoint) idempotent; connected component of (self-adjoint) algebraic elements; (local) pathwise connectedness; similarity; analytic path; polynomial path; polygonal path; centre of a Banach algebra; distance of connected components
Summary:
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 algebra, or of self-adjoint elements in a $C^*$-algebra, that satisfy a given polynomial equation, without multiple roots. In particular, we prove that in the Banach algebra case every such non-central element lies on a complex line, all of whose points satisfy the given equation. We also formulate open questions.
References:
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