Quasi-Hermitian locally compact groups are amenable

A locally compact group $G$ is called Hermitian if the spectrum $\text{Sp}_{L^1(G)}(f)\subseteq\mathbb R$ for every $f\in L^1(G)$ satisfying $f=f^*$, and called quasi-Hermitian if $\text{Sp}_{L^1(G)}(f)\subseteq\mathbb R$ for every $f\in C_c(G)$ satisfying $f=f^*$. We show that every quasi-Hermitian locally compact group is amenable. This, in particular, confirms the long-standing conjecture that every Hermitian locally compact group is amenable, a problem that has remained open since the 1960s. Our approach involves introducing the theory of "spectral interpolation of triple Banach $*$-algebras" and applying it to a family ${\rm PF}_p^*(G)$ ($1\leq p\leq \infty$) of Banach $*$-algebras related to convolution operators that lie between $L^1(G)$ and $C^*_r(G)$, the reduced group C$^*$-algebra of $G$. We show that if $G$ is quasi-Hermitian, then ${\rm PF}_p^*(G)$ and $C^*_r(G)$ have the same spectral radius on Hermitian elements in $C_c(G)$ for $p\in (1,\infty)$, and then deduce that $G$ must be amenable. We also give an alternative proof to Jenkins' result that a discrete group containing a free sub-semigroup on two generators is not quasi-Hermitian. This, in particular, provides a dichotomy on discrete elementary amenable groups: either they are non quasi-Hermitian or they have subexponential growth. Finally, for a non-amenable group $G$ with either rapid decay or Kunze-Stein property, we prove the stronger statement that ${\rm PF}_p^*(G)$ is not "quasi-Hermitian relative to $C_c(G)$" unless $p=2$.