Joint Projective Spectrum of $D_{\mathrm{\infty }h}$

We compute the joint spectrum of $D_{\mathrm{\infty }h}$ with respect to the left regular representation, and finds two generators of the De Rham cohomology group of joint resolvent set which is induced by different central linear functionals. Through action of $D_{\mathrm{\infty }h}$ on 4-ary trees, we get a self-similar realization of the group $C^{\ast }$ algebra of $D_{\mathrm{\infty }h}$.