Equality of spectra for Mφ and its restriction Mφ^0

Determine whether the spectrum σ(Mφ) of the multiplication operator Mφ: A(G) → A(G) induced by a positive definite function φ ∈ P(G) on a locally compact group G coincides with the spectrum σ(Mφ^0) of its restriction Mφ^0: A0(G) → A0(G), where A0(G) = {u ∈ A(G) : u(e) = 0}.

Background

In the paper, Mφ denotes the multiplier on the Fourier algebra A(G) given by pointwise multiplication by a positive definite function φ with φ(e)=1, and Mφ^0 is its restriction to the augmentation ideal A0(G). Understanding the spectral relationship between these operators is relevant to uniform ergodicity and convergence properties.

The authors establish partial results: when G is nondiscrete, the approximate spectra of Mφ and Mφ^0 coincide, and 1 is isolated in σ(Mφ) if and only if 1 is isolated in σ(Mφ^0). However, the full equality of spectra remains unresolved.