Abstract:We study the two $\ell^{2}$-Dirichlet structures on a countable group $G$ arising from the left and right regular actions on $\mathbb{R}^{G}$. Although the two regular representations are unitarily equivalent, their $\ell^{2}$-Dirichlet subspaces of $\mathbb{R}^{G}$ need not coincide. Our main result gives a complete classification of this asymmetry for countable amenable groups: $$\mathcal{D}_{2}\left(G,\lambda\right)=\mathcal{D}_{2}\left(G,\rho\right)\quad\Longleftrightarrow\quad G \text{ is an FC-group}.$$
The proof is based on a skewed Følner-geometric mechanism, called a left scheme, combining summability of left boundaries with displacement under a right translation. We develop this mechanism generally, and demonstrate it concretely in the Heisenberg group and amenable wreath products over $\mathbb{Z}$.
We also show that this mechanism has a dynamical counterpart in the theory of nonsingular Bernoulli shifts: every countable amenable group that is not an FC-group admits Bernoulli schemes whose left shift is nonsingular, conservative and weakly mixing, whereas the right shift by some element is singular.
From: Nachi Avraham-Re'em [view email]
[v1]
Tue, 12 May 2026 16:32:36 UTC (28 KB)
[v2]
Fri, 15 May 2026 09:00:28 UTC (29 KB)
[v3]
Tue, 23 Jun 2026 06:47:47 UTC (21 KB)
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