Abstract
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.
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Dedicated to the memory of Professor Miroslav Fiedler, from two grateful participants in mathematical olympiads
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Makai, E., Zemánek, J. Nice connecting paths in connected components of sets of algebraic elements in a Banach algebra. Czech Math J 66, 821–828 (2016). https://doi.org/10.1007/s10587-016-0294-6
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DOI: https://doi.org/10.1007/s10587-016-0294-6
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