# Mackey-Borel structure

A Borel structure (i.e., a Borel system of subsets) on the spectrum $\widehat{A}$ of a separable $C^{*}$-algebra $A$ (cf. also Spectrum of a $C^{*}$-algebra), defined as follows. Let $\mathcal{H}_{n}$, where $n \in \mathbb{N}$, be a Hilbert space of dimension $n$, and let ${\operatorname{Irr}_{n}}(A)$ denote the set of non-zero irreducible representations (cf. Irreducible representation) of $A$ on $\mathcal{H}_{n}$ equipped with the topology of pointwise convergence in the weak topology. Let on ${\operatorname{Irr}_{n}}(A)$ be given the Borel structure generated by its topology (i.e., the smallest Borel structure relative to which all mappings $\pi \mapsto \langle [\pi(x)](\xi),\eta \rangle$ — where $x \in A$, $\xi,\eta \in \mathcal{H}_{n}$ and $\pi \in {\operatorname{Irr}_{n}}(A)$ — are Borel functions), and let $\operatorname{Irr}(A)$ denote the union of the sub-spaces ${\operatorname{Irr}_{n}}(A)$, $n \in \mathbb{N}$, provided with the Borel structure such that a subset of $\operatorname{Irr}(A)$ is a Borel set if and only if its intersection with each ${\operatorname{Irr}_{n}}(A)$ belongs to the Borel structure on the latter. Let $\phi$ denote the mapping of the Borel space $\operatorname{Irr}(A)$ into the spectrum $\widehat{A}$ of $A$ that maps a representation to its unitary equivalence class. The Borel structure on $\widehat{A}$ generated by the sets whose inverse images under $\phi$ are Borel sets in $\operatorname{Irr}(A)$ is called the Mackey–Borel structure on $\widehat{A}$. The Mackey–Borel structure contains all sets of the Borel structure generated by the topology of $\widehat{A}$; each point of $\widehat{A}$ is a Borel set in the Mackey–Borel structure. The following four conditions are equivalent:
2. The Mackey–Borel structure coincides with the Borel structure generated by the topology on $\widehat{A}$.
3. The Mackey–Borel structure on $\widehat{A}$ is countably separated.
4. If $A$ is a $\mathsf{GCR}$-algebra, then a Mackey–Borel structure can also be introduced on the quasi-spectrum of a separable $C^{*}$-algebra.