Nice connecting paths in connected components of sets of algebraic elements in a Banach algebra

E., Jr. Makai; Jaroslav Zemánek

Czechoslovak Mathematical Journal (2016)

  • Volume: 66, Issue: 3, page 821-828
  • ISSN: 0011-4642

Abstract

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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.

How to cite

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Makai, E., Jr., and Zemánek, Jaroslav. "Nice connecting paths in connected components of sets of algebraic elements in a Banach algebra." Czechoslovak Mathematical Journal 66.3 (2016): 821-828. <http://eudml.org/doc/286787>.

@article{Makai2016,
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.},
author = {Makai, E., Jr., Zemánek, Jaroslav},
journal = {Czechoslovak Mathematical Journal},
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},
language = {eng},
number = {3},
pages = {821-828},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Nice connecting paths in connected components of sets of algebraic elements in a Banach algebra},
url = {http://eudml.org/doc/286787},
volume = {66},
year = {2016},
}

TY - JOUR
AU - Makai, E., Jr.
AU - Zemánek, Jaroslav
TI - Nice connecting paths in connected components of sets of algebraic elements in a Banach algebra
JO - Czechoslovak Mathematical Journal
PY - 2016
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 66
IS - 3
SP - 821
EP - 828
AB - 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.
LA - eng
KW - 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
UR - http://eudml.org/doc/286787
ER -

References

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  7. E. Makai, Jr., J. Zemánek, On the structure of the set of elements in a Banach algebra which satisfy a given polynomial equation, and their liftings. Available at www.renyi.mta.hu/ {makai}, . 
  8. E. Makai, Jr., J. Zemánek, On polynomial connections between projections, Linear Algebra Appl. 126 (1989), 91-94. (1989) Zbl0714.47011MR1040774
  9. Trémon, M., On the degree of polynomials connecting two idempotents of a Banach algebra, Proc. R. Ir. Acad. Sect. A 95 (1995), 233-235. (1995) Zbl0853.46044MR1660382
  10. Tremon, M., Polynômes de degré minimum connectant deux projections dans une algèbre de Banach, Linear Algebra Appl. French 64 (1985), 115-132. (1985) Zbl0617.46054MR0776520
  11. Zemánek, J., 10.1112/blms/11.2.177, Bull. Lond. Math. Soc. 11 (1979), 177-183. (1979) Zbl0429.46029MR0541972DOI10.1112/blms/11.2.177

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