Bader–Furman–Gelander–Monod Question 4.1 on property (T_X)

For every $1<p<\infty$ , every measure space $(\Omega,\mu)$ , and every real Banach space $X$ that is either a closed subspace of $L_p(\Omega,\mu;\mathbb{R})$ or a quotient of such a space, does every locally compact group $G$ with Kazhdan's property $(T)$ have property $(T_X)$ ? Here property $(T_X)$ means that every continuous isometric representation $\pi:G\to\operatorname{Isom}(X)$ with almost invariant unit vectors, namely such that for every compact set $K\subseteq G$ and every $\varepsilon>0$ there exists $x\in X$ with $\lVert x\rVert=1$ and $\sup_{g\in K}\lVert\pi(g)x-x\rVert<\varepsilon$ , has a nonzero invariant vector: $X^\pi\neq\{0\}$ .

References

Primary source

arXiv

Additional references

Kazhdan's Property $(T)$ for Subspaces and Quotients of $L_p$ -Spaces — arXiv

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

A September 2026 unrefereed preprint claims an affirmative answer for all the subspaces and quotients in the question, with independent verification still absent.

Bader, Furman, Gelander, and Monod posed in 2005 whether property $(T)$ implies property $(T_B)$ for every closed subspace and quotient $B$ of $L^p(\mu)$ , with $1<p<\infty$ .

Known results

Full spaces $B=L^p(\mu)$ : affirmative for $1\le p<\infty$ .
Closed subspaces: affirmative for $1<p<\infty$ , excluding $p=4,6,8,\ldots$ .
Quotients: affirmative for $1<p<\infty$ , excluding $p=\frac{4}{3},\frac{6}{5},\frac{8}{7},\ldots$ . These restrictions arise from Hardin’s extension theorem and duality.

September 2026 claimed solution

A preprint, “Kazhdan's Property $(T)$ for Subspaces and Quotients of $L_p$ -Spaces,” claims that property $(T)$ implies property $(T_X)$ throughout the stated subspace and quotient classes, removing the earlier exclusions. The affirmative resolution is unrefereed and was not corroborated by the other retrieved sources.

Current status (as of September 2026): The unrestricted implication from property $(T)$ to property $(T_X)$ is claimed in a new preprint but remains unverified; the classical partial cases are settled.

Sources

No solutions have been posted yet.