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• W2984273087 abstract "A locally compact group G is Hermitian if the spectrum SpL1(G)(f)⊆R for every f∈L1(G) satisfying f=f⁎, and quasi-Hermitian if SpL1(G)(f)⊆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 PFp⁎(G) (1≤p≤∞) of Banach ⁎-algebras related to convolution operators that lie between L1(G) and Cr⁎(G), the reduced group C⁎-algebra of G. We show that if G is quasi-Hermitian, then PFp⁎(G) and Cr⁎(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 Jenkin's result in [23] 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 PFp⁎(G) is not “quasi-Hermitian relative to Cc(G)” unless p=2." @default.
• W2984273087 created "2019-11-22" @default.
• W2984273087 creator A5035140196 @default.
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• W2984273087 date "2020-01-01" @default.
• W2984273087 modified "2023-09-24" @default.
• W2984273087 title "Quasi-Hermitian locally compact groups are amenable" @default.
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• W2984273087 doi "https://doi.org/10.1016/j.aim.2019.106897" @default.
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