Unitary dual and full group C star algebra of the integers

Given: AC; the discrete group $\mathbb{Z}$ with counting Haar measure; its unitary dual $\widehat{\mathbb{Z}}$; the left regular representation $\lambda$ on ${\mathrm{\ell }}^2(\mathbb{Z})$; the element ${\delta }_1\in L^1(\mathbb{Z})$.

The Pontryagin dual consists of continuous characters into $\mathbb{T}$ with the compact-open topology; for discrete $\mathbb{Z}$ this is the topology of pointwise convergence (The Pontryagin dual with the compact-open topology, On a discrete domain the compact-open topology is the topology of pointwise convergence). Every character $\gamma$ is determined by $z=\gamma (1)\in \mathbb{T}$ and then $\gamma (n)=z^n$; conversely each $z\in \mathbb{T}$ gives a continuous character because $\mathbb{Z}$ is discrete. The map $z\mapsto (z^n{)}_{n\in \mathbb{Z}}$ is a group homomorphism by $(zw{)}^n=z^n{w}^n$, and it is continuous into ${\mathbb{T}}^{\mathbb{Z}}$ because each power map is continuous; its inverse is the continuous evaluation at $1$. Thus it is a topological group isomorphism (The multiplicative unit circle is a compact metrizable topological abelian group, Exponent laws in a group: $g^{m+n}=g^m{g}^n$ and $(g^m{)}^n=g^{mn}$ for all $m,n\in \mathbb{Z}$, and $(gh{)}^n=g^n{h}^n$ when $g$ and $h$ commute). For an irreducible unitary representation $\pi$ of $\mathbb{Z}$, $\pi (1)$ commutes with every $\pi (n)$ and is scalar by Schur's lemma; then all $\pi (n)$ are scalar and irreducibility forces the representation space to be one-dimensional. Conversely, every continuous unitary character is irreducible; thus the same parameters index the unitary dual (The unitary dual of a locally compact group, Schur lemma for complex unitary representations).

For an LCH abelian group $G$, $C^{\ast }(G)$ is commutative with Gelfand transform an isometric $\ast$-isomorphism $C^{\ast }(G)\cong C_0(\widehat{G})$, and the character attached to $\gamma \in \widehat{G}$ sends $f\in L^1(G)$ to ${\int }_{G}f(g)\gamma (g)dg$; in particular $C^{\ast }(\mathbb{Z})\cong C(\mathbb{T})$ with ${\delta }_1$ sent to $z\mapsto {\chi }_z(1)=z$ (The abelian group C star algebra recovers Pontryagin duality, The full (maximal) group C star algebra).

The left regular representation $\lambda$ of $\mathbb{Z}$ on ${\mathrm{\ell }}^2(\mathbb{Z})$ satisfies $(\lambda (n)\xi )(k)=\xi (k-n)$, so $U:=\lambda (1)$ is the bilateral shift $(U\xi )(k)=\xi (k-1)$; $\lambda$ is a strongly continuous unitary representation and $C_r^{\ast }(\mathbb{Z})$ is the norm closure of $\{\lambda (f):f\in L^1(\mathbb{Z})\}$, where $\lambda (f)={\sum }_{n}f(n)\lambda (n)$ is the integrated form (Left and right regular unitary representations of an LCH group, The regular representations are unitary, strongly continuous, and the left one is faithful, The integrated form of a unitary representation, The reduced group C star algebra).

If $b$ is an element of a unital Banach algebra with $\mathrm{\parallel }b\mathrm{\parallel }<1$, then $1-b$ is invertible with inverse ${\sum }_{n\ge 0}{b}^n$ (Neumann series); the spectrum $\sigma (T)$ is the set of $\lambda \in \mathbb{C}$ for which $T-\lambda I$ is not invertible (Spectrum and resolvent of a bounded operator).

A unitary operator is normal, and for a bounded normal operator $T$ the continuous functional calculus is a unique isometric unital $\ast$-isomorphism $C(\sigma (T))\to C^{\ast }(I,T)$ sending the coordinate function to $T$ (Continuous functional calculus for bounded normal operators, Self-adjoint, positive, unitary and normal operators).