Recent zbMATH articles in MSC 43A65 https://www.zbmath.org/atom/cc/43A65 2022-01-14T13:23:02.489162Z Werkzeug Groups with irreducibly unfaithful subsets for unitary representations https://www.zbmath.org/1475.22012 2022-01-14T13:23:02.489162Z "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.pierre Summary: 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 groups https://www.zbmath.org/1475.43008 2022-01-14T13:23:02.489162Z "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.victoria This 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)