Generalizing earlier results about the set of idempotents in a Banachalgebra, or of self-adjoint idempotents in a $C^*$-algebra, we announceconstructions of nice connecting paths in the connected components of the setof elements in a Banach algebra, or of self-adjoint elements in a$C^*$-algebra, that satisfy a given polynomial equation, without multipleroots. In particular, we will prove that in the Banach algebra case every suchnon-central element lies on a complex line, all of whose points satisfy thegiven equation. We also formulate open questions.
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机译：概括关于Banachalgebra中的幂等集或$ C ^ * $-代数中的自伴幂等式的较早结果，我们宣布Banach代数或self中的setof元素的连通分量中的漂亮连接路径的构造。 $ C ^ * $-代数中的-adjoint元素，它们满足给定的多项式方程，没有多重根。特别是，我们将证明在Banach代数情况下，每个这样的非中心元素都位于一条复线上，所有点均满足给定方程。我们还提出未解决的问题。
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