Geodesic
Nice connecting paths in connected components of sets of algebraic elements in a Banach algebra
Czechoslovak Mathematical Journal, Tome 66 (2016) no. 3, pp. 821-828

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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.
DOI : 10.1007/s10587-016-0294-6
Classification : 46H20, 46L05
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 
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     title = {Nice connecting paths in connected components of sets of algebraic elements in a {Banach} algebra},
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Makai, Endre Jr.; Zemánek, Jaroslav. Nice connecting paths in connected components of sets of algebraic elements in a Banach algebra. Czechoslovak Mathematical Journal, Tome 66 (2016) no. 3, pp. 821-828. doi: 10.1007/s10587-016-0294-6

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