The \(C^\ast\)-algebras of connected real two-step nilpotent Lie groups (Q262634)

Let \(G\) be a locally compact group, \(L^1(G)\) the group algebra of \(G\), and \(\widehat G\) the set of all (equivalence classes of) continuous topologically irreducible unitary representations of \(G\). Each \(\pi \in \widehat G\), induces an \(*\)-representation of \(L^1(G)\) which we may still denote by \(\pi\) [\textit{J. Dixmier}, \(C^\ast\)-algebras. Translated by Francis Jellett. Amsterdam: North-Holland Publishing Company (1977; Zbl 0372.46058)]. One can then define a new norm on \(L^1(G)\) by putting \(\|f\|=\sup_{\pi \in \widehat G}\|\pi (f)\|\), where \(f\in L^1(G)\). The completion of \(L^1(G)\) under this norm is called the \(C^\ast\)-algebra of \(G\) and is denoted by \(C^{\ast}(G)\). The importance of \(C^{\ast}(G)\) is partly due to the fact that it provides a link between group theory, harmonic analysis, and operator algebras. Since \(C^{\ast}(G)\) is obtained as a result of completion of \(L^1(G)\), a concrete description of \(C^{\ast}(G)\) will help to better understand these algebras. In recent years, the second named author has led the efforts in providing characterizations of \(C^{\ast}(G)\) for various Lie groups. For example, the structure of \(C^{\ast}(G)\) is already known for the Heisenberg group and the thread-like Lie groups [the second author and \textit{L. Turowska}, Math. Z. 268, No. 3--4, 897--930 (2011; Zbl 1230.22004)], the \(ax+b\)-like groups [\textit{Y.-F. Lin} and the second author, J. Funct. Anal. 259, No. 1, 104--130 (2010; Zbl 1208.22005)], \(5\)-dimensional nilpotent Lie groups [the second author and \textit{H. Regeiba}, J. Lie Theory 25, No. 3, 613--655 (2015; Zbl 1331.22009)], and \(6\)-dimensional nilpotent Lie groups [\textit{H. Regeiba}, Les \(C^\ast\)-algèbres des groupes de Lie nilpotents de dimension [inférieure ou égale à] 6. Nancy: Université de Lorraine (PhD thesis) (2014)]. In the paper under review, the authors give a complete characterization of \(C^{\ast}(G)\) when \(G\) is a connected real two-step nilpotent Lie group. To this end, they introduce two finite families of subsets \((S_i)_{i\in \{0,\ldots , r\}}\) and \((\Gamma_i)_{i\in \{0,\ldots , r\}}\) of \(\widehat G\), and a family of Hilbert spaces \((\mathcal H _i)_{i\in \{0,\ldots , r\}}\), and prove the following result (Theorem 4): Theorem. The \(C^{\ast}\)-algebra \(C^{\ast}(G)\) of a connected real two-step nilpotent Lie group \(G\) is isomorphic (under the Fourier transform) to the set of all operator fields \(\varphi\) defined over \(\widehat G\) such that {\parindent=6mm \begin{itemize}\item[(1)] \(\varphi (\gamma )\in \mathcal K(\mathcal H_i)\) (compact operators on \(\mathcal H_i\)) for every \(i\in \{1,\ldots , r\}\) and every \(\gamma \in \Gamma_i\). \item[(2)] \(\varphi \in \ell^\infty (\widehat G)\). \item[(3)] The mappings \(\gamma \mapsto \varphi (\gamma )\) are norm continuous on the different sets \(\Gamma_i\). \item[(4)] For any sequence \((\gamma_k)_{k\in \mathbb{N}}\subset \widehat G\) going to infinity, \(\lim_{k\to \infty}\|\varphi (\gamma_k)\|_{\text{op}}=0\). \item[(5)] For \(i\in \{1,\ldots , r\}\) and any properly converging sequence \(\overline \gamma =(\gamma_k )_k\subset \Gamma_i\) whose limit set \(L(\overline \gamma)\) is contained in \(S_{i-1}\) (taking a subsequence if necessary) and for the mappings \(\tilde\nu_k=\tilde\nu_{\overline \gamma , k}: CB(S_{i-1})\to \mathcal B(\mathcal H_i)\), one has \[ \lim_{k\to \infty}\|\varphi (\gamma_k)-\tilde\nu_k({\varphi}|_{{S_{i-1}}}) \|_{\text{op}}=0. \] \end{itemize}} The main work of this paper consists of proving (5), and in particular in the construction of the mappings \((\tilde\nu)_k\). Section 1 contains a very informative introduction to the subject. Section 2 covers the preliminary material. Section 3 deals with the proofs of properties (1)--(4). The long and technical proof of (5) is carried out in Section 4. Some examples are briefly discussed in the final section of the paper.