Nice connecting paths in connected components of sets of algebraic elements in a Banach algebra
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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.
KeywordsBanach 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
MSC 201046H20 46L05
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