Hermitian Metric and the Infinite Dihedral Group

For a tuple $A=(A_1,A_2,\dots ,A_n)$ of elements in a unital Banach algebra $\mathcal B$, the associated multiparameter pencil is $A(z)=z_1 A_1 + z_2 A_2 + \dots +z_n A_n$. The projective spectrum $P(A)$ is the collection of $z\in \mathbb C^n$ such that $A(z)$ is not invertible. Using the fundamental form $\Omega _A=-\omega _A^*\wedge \omega _A$, where $\omega _A(z) = A^{-1}(z)\,dA(z)$ is the Maurer–Cartan form, R. Douglas and the second author defined and studied a natural Hermitian metric on the resolvent set $P^c(A)=\mathbb{C}^n\setminus P(A)$. This paper examines that metric in the case of the infinite dihedral group, $D_\infty = \langle a,t\mid a^2=t^2 =1\rangle $, with respect to the left regular representation $\lambda $. For the non-homogeneous pencil $R(z) = I+z_1\lambda (a)+z_2\lambda (t)$, we explicitly compute the metric on $P^c(R)$ and show that the completion of $P^c(R)$ under the metric is $\mathbb C^2\setminus \{(\pm 1,0), (0,\pm 1)\}$, which rediscovers the classical spectra $\sigma (\lambda (a))=\sigma (\lambda (t))=\{\pm 1\}$. This paper is a follow-up of the papers by R. G. Douglas and R. Yang (2018) and R. Grigorchuk and R. Yang (2017).