On universal enveloping algebras in a topological setting (Q2787140)

The authors study smooth functions, compactly supported distributions and left invariant differential operators on an arbitrary topological group \(G\). Let \(L(G)\) be the set of all continuous one-parameter groups in \(G\) (i.e., continuous homomorphisms \(\gamma: {\mathbb R}\to G\)), endowed with the compact-open topology. One-parameter groups can be used to differentiate functions on \(G\): If \(f: U\to F\) is a function from an open subset \(U\subseteq G\) to a locally convex topological vector space \(F\), define NEWLINE\[NEWLINE (D_\gamma f)(x):=\frac{d}{dt}\Big|_{t=0}f(x\gamma(t)) NEWLINE\]NEWLINE for \(x\in U\) and \(\gamma\in L(G)\), whenever the derivative exists. Following [\textit{H. Boseck} et al., Analysis on topological groups -- general Lie theory. Leipzig: BSB B (1981; Zbl 0558.22012)], \(f\) is called smooth if \(d^{(0)}f:=f\) is continuous, the iterated derivatives NEWLINE\[NEWLINE d^{(k)}f(x,\gamma_1,\ldots,\gamma_k):= (D_{\gamma_k}\cdots D_{\gamma_1}f)(x) NEWLINE\]NEWLINE exist for all \(k\in\{1,2,\ldots\}\), \(x\in U\) and \(\gamma_1,\ldots, \gamma_k\in L(G)\), and the maps \(d^{(k)}f: U\times L(G)^k\to F\) so obtained are continuous (compare also [\textit{J. Riss}, Acta Math. 89, 45--105 (1953; Zbl 0050.11203)] for special cases). Give \(C(U\times L(G)^k,F)\) the compact-open topology for \(k\in \{0,1,\ldots\}\). Then the space \(C^\infty(U,F)\) of all smooth maps \(f: U\to F\) is a locally convex space in a natural way, when endowed with the initial topology with respect to the linear mappings NEWLINE\[NEWLINE d^{(k)}: C^\infty(U,F)\to C(U\times L(G)^k,F),\quad f\mapsto d^{(k)}f NEWLINE\]NEWLINE for \(k\in\{0,1,\ldots\}\). Abbreviate \(E(G):=C^\infty(G,{\mathbb C})\) and write \(E'(G)\) for the space of all continuous linear functionals \(u: E(G)\to{\mathbb C}\). The elements \(u\in E'(G)\) are called distributions with compact support. The support \(\mathrm{supp}(u)\) of \(u\in E'(G)\) is defined as the set of all \(x\in G\) such that, for every neighbourhood \(V\subseteq G\) of \(x\), there exists \(f\in E(G)\) with \(\mathrm {supp}(f)\subseteq V\) and \(u(f)\not=0\). Then \(\mathrm{supp}(u)\) is compact (Remark 3.4). Let \(E'_1(G)\subseteq E'(G)\) be the vector subspace of all \(u\in E'(G)\) whose support is contained in the trivial subgroup \(\{1\}\) of \(G\). If \(f\in E(G)\) and \(\check{f}(x):=f(x^{-1})\) for \(x\in G\), then \(\check{f}\in E(G)\); if \(u\in E'(G)\) and \(\check{u}(f):=u(\check{f})\) for \(f\in E(G)\), then \(\check{u}\in E'(G)\) (Proposition 3.8). For \(x\in G\), let \(L_x: G\to G\), \(y\mapsto xy\) be left translation by~\(x\). Then \(f\circ L_x\in E(G)\) for each \(f\in E(G)\), and the map \(E(G)\to E(G)\), \(f\mapsto f\circ L_x\) is an isomorphism of topological vector spaces (Proposition 3.10). Let \(\mathrm{Loc}(G)\) be the algebra of local operators on~\(G\), i.e., continuous linear operators \(D: E(G)\to E(G)\) such that \(\mathrm{supp}(D(f))\subseteq \mathrm{supp}(f)\) for each \(f\in E(G)\). Let \(U(G)\subseteq \mathrm{Loc}(G)\) be the subalgebra of all \(D\in\mathrm{Loc}(G)\) which are left invariant in the sense that \(D(f\circ L_x)=D(f)\circ L_x\) for all \(f\in E(G)\) and \(x\in G\). Set NEWLINE\[NEWLINE D_u(f)(x):=(f*u)(x):=\check{u}(f\circ L_x) NEWLINE\]NEWLINE for \(u\in E'_1(G)\), \(f\;in E(G)\) and \(x\in G\). Then \(D_u(f)\in E(G)\), \(D_u\in U(G)\) and the map NEWLINE\[NEWLINE E'_1(G)\to U(G),\quad u\mapsto D_u NEWLINE\]NEWLINE is an isomorphism of vector spaces with inverse taking \(D\in\mathrm{Loc}(G)\) to the distribution \(f\mapsto D(\check{f})(1)\) (Theorem 5.2). If \(G\) and \(H\) are topological groups, \(V\subseteq G\) and \(W\subseteq H\) are open subsets and \(f: V\times W\to F\) is a smooth map to a locally convex space, then \(f^\vee(x):=f(x,\cdot): W\to F\) is a smooth map for each \(x\in V\). As a tool for the proof of the preceding theorem, the authors show that \(f^\vee\in C^\infty(V,C^\infty(W,F))\) for each \(f\in C^\infty(V\times W,F)\), and that the linear map NEWLINE\[NEWLINE \Phi: C^\infty(V\times W,F)\to C^\infty(V,C^\infty(W,F)),\quad f\mapsto f^\vee NEWLINE\]NEWLINE is a homeomorphism onto its image (see Theorem 4.16 and Remark 4.17). This result generalizes the known special case that \(G\) and \(H\) are locally convex topological vector spaces; even then, \(\Phi\) need not be surjective, but criteria for surjectivity are available (see [\textit{H. Alzaareer}, Lie groups of mappings on non-compact spaces and manifolds. (Doctoral Dissertation) Universität Paderborn (2013), \url{nbn-resolving.de/urn:nbn:de:hbz:466:2-11572}] and [\textit{H. Alzaareer} and \textit{A. Schmeding}, Expo. Math. 33, No. 2, 184--222 (2015; Zbl 1330.46039)]). Stimulated by the work under review, surjectivity of \(\Phi\) for topological groups \(G\) and \(H\) was investigated in [\textit{N. Nikitin}, ``Exponential laws for spaces of differentiable functions on topological groups'', Preprint, \url{arXiv:1608.06095}]; e.g., \(\Phi\) is a homeomorphism whenever both \(G\) and \(H\) are metrizable, or both \(G\) and \(H\) are locally compact. Note that \(L(G)\) merely is a topological space if \(G\) is a general topological group (only under additional hypotheses on \(G\), we might consider \(L(G)\) as a topological vector space). This accounts for some technical complications compared to differential calculus in locally convex spaces.