2018/05/21 by Samei, Ebrahim, Wiersma, Matthew
#FOS: Mathematics #Functional Analysis (math.FA) #Group Theory (math.GR) #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.1805.07908
A locally compact group G is called Hermitian if the spectrum SpL1(G)(f)⊆\mathbb R for every f∈ L1(G) satisfying f=f^*, and called quasi-Hermitian if SpL1(G)(f)⊆\mathbb R for every f∈ Cc(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 PFp^*(G) (1≤ p≤ ∞) of Banach *-algebras related to convolution operators that lie between L1(G) and C^*r(G), the reduced group C^*-algebra of G. We show that if G is quasi-Hermitian, then \rm PFp^*(G) and C^*r(G) have the same spectral radius on Hermitian elements in Cc(G) for p∈ (1,∞), 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 PFp^*(G) is not "quasi-Hermitian relative to Cc(G)" unless p=2.