Topologisch linksengelsche Elemente. (Topologically left Engel elements) (Q1071110)

Let G be a topological group. For \(g\in G\) and \(\phi\in Aut(G)\), define \([g,_ 0\phi]=g\), \([g,\phi]=g \phi (g)^{-1}\), and \([g,_ n\phi]=[[g,_{n-1}\phi],\phi]\), and \(\phi\) is called a topologically nilpotent automorphism (t-nil) of G if for all \(g\in G\), \(\lim_{n\to \infty}[g,_ n\phi]=1\). An element \(h\in G\) is called a topologically left Engel (t.l.E.) element if the inner automorphism \(i_ h: g\mapsto hgh^{-1}\) is t-nil. This concept was introduced by \textit{V. P. Platanov} [cf. Isv. Akad. Nauk. SSSR, Ser. Mat. 30, 1257-1274 (1966); translated in Am. Math. Soc., Transl., II. Ser. 66, 111-129 (1968; Zbl 0222.22002)] who asked various questions about these elements. The main part of this paper deals with Lie groups where the t.l.E. elements can be completely described. Let G be a Lie group with Lie algebra \({\mathfrak g}\) and \(\phi\) an automorphism of G with \(d\phi\in Aut({\mathfrak g})\). Let \({\mathfrak r}\) be the radical of \({\mathfrak g}\) and \({\mathfrak k}\) be the ideal of \({\mathfrak g}\) characterized by \({\mathfrak r}\leq {\mathfrak k}\) and \({\mathfrak k/r}\) is the maximal compact ideal of \({\mathfrak g/r}\). If the automorphism \(\phi\) induces a nil automorphism of \(G/G_ 0\) (in the sense of discrete topology), then \(\phi\) is a topological nil automorphism and \(d\phi\) induces the identity on \({\mathfrak g/k}\), \(\rho(I- d\phi)\leq 1\) and \(\rho (I-d\phi |_{{\mathfrak r}})<1\), where \(G_ 0\) is the connected component of G containing the identity and \(\rho (f)=\max \{| a|\); \(\det (f-aI)=0\}\) for any endomorphism f of a vector space. With the aid of these results, answers to Platonov's questions are given for many classes of locally compact groups.