Spaces of differentiable functions on compact groups
Let G be a compact group, and let R(G) be the set of its one-parameter subgroups. With respect to an element \(\alpha\) of R(G), one can define right and left derivative operators. Let \({\mathcal E}_ n^{(r)}(G)\) [resp. \({\mathcal E}_ n^{(l)}(G)]\) denote the space of continuous functions for which all right [resp. left] derivatives of all orders up to n are defined and continuous. \({\mathcal E}_{\infty}^{(r)}(G)\) and \({\mathcal E}_{\infty}^{(l)}(G)\) are defined similarly. The paper begins by showing that R(G) is in one-to-one correspondence with a Lie algebra \(\Lambda\) (G), on which there is a natural topology which is locally convex and barrelled. This result depends on work by \textit{K. McKennon} [J. Reine Angew. Math. 307/308, 166-172 (1979; Zbl 0396.43010)], and on the Tannaka duality theorem. Theorem 2.1 states that \({\mathcal E}_ n^{(r)}(G)={\mathcal E}_ n^{(l)}(G)\). After this, the superscript r or l is not needed. Theorem 3.1 characterizes \({\mathcal E}_ n(G)\) in terms of the Schwartz spaces \({\mathcal E}_ n(G/N)\) where N is a closed normal subgroup of G and G/N is a separable, finite-dimensional Lie group. A topology \(\tau^*\) is then defined on \({\mathcal E}_ n(G)\). Theorem 4.1: the complexification of \(\Lambda\) (G) coincides with the Lie algebra of \(\tau^*-continuous\), left-invariant derivations on \(E_{\infty}(G)\). Further theorems extend this result to derivations of higher order.
scientific article; zbMATH DE number 3919036