Hermitian Metric and the Infinite Dihedral Group

Abstract

For a tuple A = (A₁,A₂,…, A_n) of elements in a unital Banach algebra B, the associated multiparameter pencil is A(z) = z₁A₁ + z₂A₂ + … + z_nA_n. The projective spectrum P(A) is the collection of z ∈ ℂⁿ such that A(z) is not invertible. Using the fundamental form Ω_A = −ω_A^* ∧ ω_A, where ω_A(z) = A⁻¹(z) dA(z) is the Maurer–Cartan form, R. Douglas and the second author defined and studied a natural Hermitian metric on the resolvent set P^c(A) = ℂⁿ \ P(A). This paper examines that metric in the case of the infinite dihedral group, D_∞ = <a, t | a² = t² = 1>, with respect to the left regular representation λ. For the non-homogeneous pencil R(z) = I + z₁λ(a) + z₂λ(t), we explicitly compute the metric on P^c(R) and show that the completion of P^c(R) under the metric is ℂ² \ {(±1, 0), (0, ±1)}, which rediscovers the classical spectra σ(λ(a)) = σ(λ(t)) = {± 1}. This paper is a follow-up of the papers by R. G. Douglas and R. Yang (2018) and R. Grigorchuk and R. Yang (2017).