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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
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.
[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. MR 2006311 | Zbl 1084.46519
[2] Boulmaarouf, Z., Miranda, M. Fernandez, Labrousse, J.-Ph.: An algorithmic approach to orthogonal projections and Moore-Penrose inverses. Numer. Funct. Anal. Optimization 18 (1997), 55-63. DOI 10.1080/01630569708816746 | MR 1442018
[3] Esterle, J.: Polynomial connections between projections in Banach algebras. Bull. Lond. Math. Soc. 15 (1983), 253-254. DOI 10.1112/blms/15.3.253 | MR 0697127 | Zbl 0517.46034
[4] Kovarik, Z. V.: Similarity and interpolation between projectors. Acta Sci. Math. 39 (1977), 341-351. MR 0482324 | Zbl 0392.47008
[5] Maeda, S.: On arcs in the space of projections of a $C^*$-algebra. Math. Jap. 21 (1976), 371-374. MR 0454651 | Zbl 0353.46051
[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.
[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 {makai}.
[8] E. Makai, Jr., J. Zemánek: On polynomial connections between projections. Linear Algebra Appl. 126 (1989), 91-94. MR 1040774 | Zbl 0714.47011
[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. MR 1660382 | Zbl 0853.46044
[10] Tremon, M.: Polynômes de degré minimum connectant deux projections dans une algèbre de Banach. Linear Algebra Appl. French 64 (1985), 115-132. MR 0776520 | Zbl 0617.46054
[11] Zemánek, J.: Idempotents in Banach algebras. Bull. Lond. Math. Soc. 11 (1979), 177-183. DOI 10.1112/blms/11.2.177 | MR 0541972 | Zbl 0429.46029
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