Recent zbMATH articles in MSC 43A20https://www.zbmath.org/atom/cc/43A202021-04-16T16:22:00+00:00Werkzeug\(L^1\)-determined primitive ideals in the \(C^\ast\)-algebra of an exponential Lie group with closed non-\(\ast\)-regular orbits.https://www.zbmath.org/1456.430012021-04-16T16:22:00+00:00"Inoue, Junko"https://www.zbmath.org/authors/?q=ai:inoue.junko"Ludwig, Jean"https://www.zbmath.org/authors/?q=ai:ludwig.jeanSummary: Let \(G = \exp (\mathfrak{g})\) be an exponential solvable Lie group and Ad\((G) \subset \mathbb{D}\) an exponential solvable Lie group of automorphisms of \(G\). Assume that for every non-\(\ast\)-regular orbit \(\mathbb{D} \cdot q$, $q \in \mathfrak{g}^\ast\), of \(\mathbb{D} = \exp(\mathfrak{d})\) in \(g^\ast\), there exists a nilpotent ideal \(\mathfrak{n}\) of \(\mathfrak{g}\) containing \(\mathfrak{d} \cdot \mathfrak{g}\) such that \(\mathbb{D} \cdot q_{|\mathfrak{n}}\) is closed in \(\mathfrak{n}^\ast\). We then show that for every \(\mathbb{D}\)-orbit \(\Omega\) in \(g^\ast\) the kernel \(\ker_{C^\ast} (\Omega)\) of \(\Omega\) in the \(C^\ast\)-algebra of \(G\) is \(L^1\)-determined, which means that \(\ker_{C^\ast} (\Omega)\) is the closure of the kernel \(\ker L^1(\Omega )\) of \(\Omega\) in the group algebra \(L^1(G)\). This establishes also a new proof of a result of Ungermann, who obtained the same result for the trivial group \(\mathbb{D} = \text{Ad}(G)\). We finally give an example of a non-closed non-\(\ast\)-regular orbit of an exponential solvable group \(G\) and of a coadjoint orbit \(O \subset \mathfrak{g}^\ast\), for which the corresponding kernel \(\text{ker}_{C^\ast}(\pi_O)\) in \(C^\ast (G)\) is not \(L^1\)-determined.Hyperreflexivity constants of the bounded \(n\)-cocycle spaces of group algebras and \(C^*\)-algebras.https://www.zbmath.org/1456.460452021-04-16T16:22:00+00:00"Samei, Ebrahim"https://www.zbmath.org/authors/?q=ai:samei.ebrahim"Farsani, Jafar Soltani"https://www.zbmath.org/authors/?q=ai:farsani.jafar-soltaniSummary: We introduce the concept of strong property \((\mathbb{B})\) with a constant for Banach algebras and, by applying a certain analysis on the Fourier algebra of the unit circle, we show that all \(C^*\)-algebras and group algebras have the strong property \((\mathbb{B})\) with a constant given by \(288 \pi (1 + \sqrt{2})\). We then use this result to find a concrete upper bound for the hyperreflexivity constant of \(\mathcal{Z}^n (A, X)\), the space of bounded \(n\)-cocycles from \(A\) into \(X\), where \(A\) is a \(C^*\)-algebra or the group algebra of a group with an open subgroup of polynomial growth and \(X\) is a Banach \(A\)-bimodule for which \(\mathcal{H}^{n+1} (A, X)\) is a Banach space. As another application, we show that for a locally compact amenable group \(G\) and \(1 < p < \infty\), the space \(CV_P(G)\) of convolution operators on \(L^p(G)\) is hyperreflexive with a constant given by \(384 \pi^2 (1 + \sqrt{2})\). This is the generalization of a well-known result of \textit{E. Christensen} [Math. Scand. 50, 111--122 (1982; Zbl 0503.47032)] for \(p = 2\).