Recent zbMATH articles in MSC 43A65https://www.zbmath.org/atom/cc/43A652022-01-14T13:23:02.489162ZWerkzeugGroups with irreducibly unfaithful subsets for unitary representationshttps://www.zbmath.org/1475.220122022-01-14T13:23:02.489162Z"Caprace, Pierre-Emmanuel"https://www.zbmath.org/authors/?q=ai:caprace.pierre-emmanuel"de la Harpe, Pierre"https://www.zbmath.org/authors/?q=ai:de-la-harpe.pierreSummary: Let \(G\) be a group. A subset \(F\subset G\) is called irreducibly faithful if there exists an irreducible unitary representation \(\pi\) of \(G\) such that \(\pi(x)\neq\text{id}\) for all \(x\in F\setminus\{e\}\). Otherwise \(F\) is called irreducibly unfaithful. Given a positive integer \(n\), we say that \(G\) has Property \(\mathcal{P}(n)\) if every subset of size \(n\) is irreducibly faithful. Every group has \(\mathcal{P}(1)\), by a classical result of Gelfand and Raikov. Walter proved that every group has \(\mathcal{P}(2)\). It is easy to see that some groups do not have \(\mathcal{P}(3)\).
We provide a complete description of the irreducibly unfaithful subsets of size \(n\) in a countable group \(G\) (finite or infinite) with Property \(\mathcal{P}(n-1)\): it turns out that such a subset is contained in a finite elementary abelian normal subgroup of \(G\) of a particular kind. We deduce a characterization of Property \(\mathcal{P}(n)\) purely in terms of the group structure. It follows that, if a countable group \(G\) has \(\mathcal{P}(n-1)\) and does not have \(\mathcal{P}(n)\), then \(n\) is the cardinality of a projective space over a finite field.
A group \(G\) has Property \(\mathcal{Q}(n)\) if, for every subset \(F\subset G\) of size at most \(n\), there exists an irreducible unitary representation \(\pi\) of \(G\) such that \(\pi (x)\neq\pi (y)\) for any distinct \(x,y\) in \(F\). Every group has \(\mathcal{Q}(2)\). For countable groups, it is shown that Property \(\mathcal{Q}(3)\) is equivalent to \(\mathcal{P}(3)\), Property \(\mathcal{Q}(4)\) to \(\mathcal{P}(6)\), and Property \(\mathcal{Q}(5)\) to \(\mathcal{P}(9)\). For \(m,n\geq 4\), the relation between Properties \(\mathcal{P}(m)\) and \(\mathcal{Q}(n)\) is closely related to a well-documented open problem in additive combinatorics.Spaces invariant under unitary representations of discrete groupshttps://www.zbmath.org/1475.430082022-01-14T13:23:02.489162Z"Barbieri, Davide"https://www.zbmath.org/authors/?q=ai:barbieri.davide-francesco"Hernández, Eugenio"https://www.zbmath.org/authors/?q=ai:hernandez.eugenio"Paternostro, Victoria"https://www.zbmath.org/authors/?q=ai:paternostro.victoriaThis paper introduces general tools for the study of closed subspaces of a Hilbert space that are invariant under a unitary representation of a countable group on that Hilbert space. Let \(\Gamma\) be a countable group and \(\Pi\) a unitary representation of \(\Gamma\) on a separable Hilbert space \(\mathcal{H}\). There are a variety of known results concerning \(\Pi\)-invariant closed subspaces of \(\mathcal{H}\) when \(\Gamma\) is abelian. The authors formulate and prove generalizations of many of these results to non-abelian \(\Gamma\). In these generalizations, \(\mathcal{R}(\Gamma)\) plays the role of the dual of \(\Gamma\), where \(\mathcal{R}(\Gamma)\) is the von Neumann algebra generated by the right regular representation \(\rho\) of \(\Gamma\). The usual Lebesgue spaces on the dual of \(\Gamma\) are replaced by the noncommutative \(L^p\)-spaces formed using the canonical finite trace \(\tau\) on \(\mathcal{R}(\Gamma)\).
The significant results in this paper assume that \(\Pi\) satisfies the square-integrability condition that there exists a dense subspace \(\mathcal{D}\) of \(\mathcal{H}\) such that \(x\mapsto \langle\varphi,\Pi(x)\psi\rangle\) is in \(\ell^2(\Gamma)\), for all \(\varphi\in\mathcal{H}\) and \(\psi\in\mathcal{D}\). In that case, there exists a sesquilinear map \([\cdot,\cdot]: \mathcal{H}\times\mathcal{H}\to L^1\big(\mathcal{R}(\Gamma)\big)\) which satisfies \(\langle\varphi,\Pi(x)\psi\rangle= \tau\big([\varphi,\psi]\rho(x)\big)\), for all \(\varphi,\psi\in\mathcal{H}\) and \(x\in\Gamma\). The authors refer to any triple \((\Gamma,\Pi,\mathcal{H})\) as being dual integrable if such a bracket map \([\cdot,\cdot]\) exists. A core theorem is that \((\Gamma,\Pi,\mathcal{H})\) is dual integrable if and only if there exists a \(\sigma\)-finite measure space \((\mathcal{M},\nu)\) and an isometry \(\mathcal{J}:\mathcal{H}\to L^2\big((\mathcal{M},\nu),L^2(\mathcal{R}(\Gamma))\big)\) that satisfies \(\mathcal{J}\big(\Pi(x)\varphi\big)= \mathcal{J}(\varphi)\rho(x)^*\), for all \(x\in\Gamma\) and \(\varphi\in\mathcal{H}\). This map \(\mathcal{J}\) is called a Helson map as it directly generalizes the Helson map used in characterizing shift-invariant closed subspaces of \(L^2(\mathbb{R})\). This generalized Helson map is the main tool in generalizing results on invariant subspaces to nonabelian \(\Gamma\).
Reviewer: Keith Taylor (Halifax)