The elements of bounded trace in the \(C^*\)-algebra of a nilpotent Lie group
Let G be a connected, simply connected nilpotent Lie group with Lie algebra \({\mathfrak g}\). We identify the dual space \(\hat G\) of G with the space \({\mathfrak g}^*/Ad^*(G)\) of the G-orbits in \({\mathfrak g}^*\) under the coadjoint action. For an integer d, denote by \(\hat G_ d\) (resp. by \(\hat G_{<d})\) the subset of \(\hat G,\) consisting of the orbits of dimension d (resp. of the orbits of dimension strictly less than d). In this paper we study the following question. Let A be a closed subset of \(\hat G.\) What positive elements x in the \(C^*\)-algebra \(C^*(G)\) of G are with bounded trace on A, i.e., for what \(x\geq 0\) in \(C^*(G)\), is sup \(\{\) tr \(\pi\) (x) \(|\) \(\pi \in A\}<\infty ?\) It turns out, that the subsets \(C_ d:=(A\cap \hat G_ d)^-\cap \hat G_{<d}\) (where \({}^-\) means closure in \(\hat G)\), \(d=2,4,6,...\), play a crucial role. Let \(C=\cup_{d}C_ d\). In fact, it is easily shown, that a necessary condition for x to be with bounded trace on A, is that \(\pi (x)=0\) for every \(\pi\in C\) (Theorem 3.4.). Let ker C:\(=\{\cap\) ker \(\pi\) \(| \pi \in C\}\). The main result of the paper is Theorem 5.3.: Every \(0\leq x\) in the Pedersen ideal j(C) of ker C is with bounded trace on A. In the proof of this theorem we use the method of variable groups. This allows us to proceed inductively on the dimension of G and to use Kirillov's lemma at every induction step. In the proof we also need the following results. For any closed subset B of \(\hat G,\) the Pedersen ideal j(B) in \(C^*(G)\) is generated by Schwartz functions (Theorem 2.7.). Let \(V=\{v_ 1,...,v_ n\}\) be a Jordan-Hölder basis of \({\mathfrak g}^*\) and let for \(l\in {\mathfrak g}^*\), \(P(z,l)=\sum^{n}_{j=1}P_ j(z,l)\nu_ j\) be the canonical polynomial map, describing the G-orbit of l \((z\in {\mathbb{R}}^ d\), \(d=\dim G\cdot l)\). Let \(1\leq e_ 1<e_ 2<...<e_ d\leq n\) and denote by R the polynomial mapping \(R(z):=(P_{e_ 1}(z,l),...,P_{e_ d}(z,l))\in {\mathbb{R}}^ d\). Then \[ | \det (R'(z))| =| \det [<P(z,l),[b_{e_ r},b_{e_ s}]>_{r,s}]| \cdot (\det [<P(z,l),[b_{j_ r},b_{j_ s}]>_{r,s}])^{-1} \] where \(1<j_ 1<j_ 2<...<j_ d\leq n\) are the jumping indices of the orbit \(G\cdot l\) (Proposition 4.4.). If we identify now the sets A and C with closed subsets of \({\mathfrak g}^*\), we can deduce from the formula above that for any non-negative continuous function \(\phi\) with compact support on \({\mathfrak g}^*\), with supp \(\phi\) \(\cap C=\emptyset\), that sup \(\{\int_{O_{\pi}}\phi (l) d\mu_{\pi}(l) |\) \(\pi \in C\}<\infty\) (Proposition 4.6.). Here \(d\mu_{\pi}\) is the canonical G-invariant measure on \(O_{\pi}\).
scientific article; zbMATH DE number 3941812