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

Endre 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, Endre 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, Endre 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, Endre 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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  1. B. Aupetit, E. Makai, Jr., M. Mbekhta, J. Zemánek, The connected components of the idempotents in the Calkin algebra, and their liftings, Operator Theory and Banach Algebras, Proc. Int. Conf. in Analysis, Rabat, 1999 M. Chidami et al. Theta, Bucharest (2003), 23-30. (2003) Zbl1084.46519MR2006311
  2. Boulmaarouf, Z., Miranda, M. Fernandez, Labrousse, J.-Ph., 10.1080/01630569708816746, Numer. Funct. Anal. Optimization 18 (1997), 55-63. (1997) MR1442018DOI10.1080/01630569708816746
  3. Esterle, J., 10.1112/blms/15.3.253, Bull. Lond. Math. Soc. 15 (1983), 253-254. (1983) Zbl0517.46034MR0697127DOI10.1112/blms/15.3.253
  4. Kovarik, Z. V., Similarity and interpolation between projectors, Acta Sci. Math. 39 (1977), 341-351. (1977) Zbl0392.47008MR0482324
  5. Maeda, S., On arcs in the space of projections of a C * -algebra, Math. Jap. 21 (1976), 371-374. (1976) Zbl0353.46051MR0454651
  6. E. Makai, Jr., Algebraic elements in Banach algebras (joint work with J. Zemánek), 6th Linear Algebra Workshop, Book of Abstracts Kranjska Gora (2011), p. 26. (2011) 
  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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