Some permanence properties of \(C^{*}\)-unique groups
A Banach \(*\)-algebra \(A\) is called \(C^*\)-unique if it admits a unique \(C^*\)-norm. Equivalently, every non-trivial ideal of the enveloping \(C^*\)-algebra, \(C^*(A)\), intersects \(A\) non-trivially. A locally compact group \(G\) is called \(C^*\)-unique if its group algebra \(L^1(G)\) is \(C^*\)-unique. The question of which locally compact groups are \(C^*\)-unique, has been investigated by several authors. Evidently, every \(C^*\)-unique group is amenable. However, there exist solvable groups which fail to be \(C^*\)-unique. The present paper contains a number of interesting results. In particular, the authors are interested in permanence properties of \(C^*\)-uniqueness. Recall that if \(G\) and \(H\) are locally compact groups, then \(H\) is said to be a retract of \(G\) if there exist continuous homomorphisms \(i:H\to G\) and \(q :G\to H\) such that \(q\circ i = id_H\). One of the main results is that a (second countable) amenable locally compact group which can be continuously embedded into a linear group over a non-discrete locally compact field, is a retract of a \(C^*\)-unique group (Theorem 2.17). This theorem applies to an amenable connected Lie group \(G\) and its adjoint representation and yields that \(G/Z\), where \(Z\) is the centre of \(G\), is a retract of a \(C^*\)-unique group. Another achievement is Theorem 3.4: Let \(H\) be any discrete group, \(N\) the direct sum of copies \(N_h = \mathbb{Z}\), \(h\in H\), and form the semidirect product \(G = N\times_\alpha H\), where the action \(\alpha\) of \(H\) on \(N\) is defined through multiplication on \(H\). Then \(G\) is \(C^*\)-unique if and only if \(H\) is amenable. In particular, every amenable discrete group is a retract of a \(C^*\)-unique group. These results lead the authors to conjecture that every amenable locally compact group is a retract of a \(C^*\)-unique group. Both, Theorems 2.17 and 3.4, evolve from a certain type of extension result: If \(N\) is a \(C^*\)-unique closed normal subgroup of G and the canonical twisted action of \(G\) on the \(C^*\)-algebra \(C^*(N)\) is free in some sense, then \(G\) is \(C^*\)-unique.
scientific article; zbMATH DE number 2091724
scientific article