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Uniqueness in the imprimitivity theorem

Statement

Assume AC and keep the hypotheses of the imprimitivity theorem. If $σ : H \to U (K)$ and $σ^{'} : H \to U (K^{'})$ are strongly continuous unitary representations, then the canonical transitive systems of ${Ind}_{H}^{G} σ$ and ${Ind}_{H}^{G} σ^{'}$ on $G / H$ are unitarily equivalent if and only if $σ$ and $σ^{'}$ are unitarily equivalent. Consequently the map of the imprimitivity theorem is a bijection between unitary equivalence classes of transitive systems on $G / H$ and unitary equivalence classes of strongly continuous unitary representations of $H$ .

Facts & Assumptions

Given: AC, the second-countable LCH group $G$ and closed subgroup $H$ , and strongly continuous unitary representations $σ : H \to U (K)$ , $σ^{'} : H \to U (K^{'})$ on separable spaces.

[F1]

The canonical system of $σ$ is the induced representation on its covariant completion together with the multiplication PVM $P (E) = M_{1_{E}}$ ; the induced action in section coordinates is $D_{g} (x)^{1 / 2} σ (s (x)^{- 1} g s (g^{- 1} x)) f (g^{- 1} x)$ (An induced representation carries a canonical system of imprimitivity on $G / H$ , The imprimitivity reconstruction map is isometric and intertwining, Unitary induction from a closed subgroup).

[F2]

A system equivalence between two multiplicity-normalized models intertwines the diagonal multiplications, so it is decomposable with unitary fibres almost everywhere, and the fibre dimensions agree a.e.; equivalently, over a fixed base the unitary intertwiners of two models are precisely the decomposable unitaries (Unitary intertwiners preserve fibre multiplicity over a standard Borel base, Decomposable operators are the commutant of diagonal multiplication, Spectral multiplicity model of a transitive system of imprimitivity).

[F3]

The cocycle fields of the canonical model of $σ$ factor through a trivialization $b_{σ}$ : writing $φ_{g}^{σ} (x) = σ (s (g x)^{- 1} g s (x))$ at the source variable, the Haar regularization uniqueness argument shows that if two trivializations of the same cocycle differ by a gauge $A$ , then $b_{σ^{'}} (t)^{- 1} A (q (t)) b_{σ} (t)$ is left-translation invariant for a.e. $t$ , hence a constant unitary $T$ , and right- $H$ covariance gives $σ^{'} (h) T = T σ (h)$ for all $h$ (Haar regularization of transitive unitary cocycles, Measurable cocycle fields for a multiplicity-normalized system, The stabilizer acts unitarily on an imprimitivity fibre).

[F4]

The imprimitivity theorem gives the forward and inverse constructions and the zero cases: zero fibres induce exactly the zero system, and a nonzero fibre induces a nonzero space because the quotient measure has full support and nonzero square-integrable sections exist (Mackey's imprimitivity theorem, Transitive systems of imprimitivity and their normalized measure class, Unitary equivalence of systems of imprimitivity and of the induced representations).

[F5]

AC is the standing hypothesis (The Axiom of Choice, Strongly continuous unitary representations, invariant linear subspaces and intertwiners, Borel cross-sections for closed subgroups of second-countable locally compact Hausdorff groups).

Proof

technique · direct

Given: AC, the two representations $σ, σ^{'}$ and their canonical systems.

1.1F1

If $σ$ and $σ^{'}$ are unitarily equivalent via $T : K \to K^{'}$ , define $\hat{T}$ on the covariant completion of $Ind σ$ pointwise, $(\hat{T} F) (x) = T (F (x))$ . Then $\hat{T}$ is unitary, preserves covariance ( $T (F (x h)) = T (σ (h)^{- 1} F (x)) = σ^{'} (h)^{- 1} (T F) (x)$ ), and intertwines the induced actions and the multiplication PVM: $\hat{T} Π_{σ} (g) {\hat{T}}^{- 1} = Π_{σ^{'}} (g)$ and $\hat{T} P (E) {\hat{T}}^{- 1} = P^{'} (E)$ . Hence the canonical systems are unitarily equivalent.

2.1F2F3step 1.1

Conversely, suppose the canonical systems are unitarily equivalent by $W_{0}$ . Then $W_{0}$ intertwines all multiplications by indicators, and by [F2] it is multiplication by a Borel unitary field $A (x)$ between the two constant fibres, whose dimensions agree. Fix unitary identifications of the fibres and use [F3]: the two cocycle fields of the canonical models are related by the gauge $A$ , and lifting the gauge to $G$ produces a constant unitary $T$ with $σ^{'} (h) T = T σ (h)$ for every $h \in H$ . Thus $σ$ and $σ^{'}$ are unitarily equivalent.

3.1F4step 2.1

Zero cases: if $K = 0$ then the canonical system is the zero system and $H$ acts trivially; two zero systems are unitarily equivalent, and the zero representation of $H$ is unitarily equivalent only to the zero representation; if both $K, K^{'}$ are nonzero the argument [step 2.1] applies verbatim, and a nonzero fibre induces a nonzero system by [F4], so the zero and nonzero classes do not mix.

4.1step 1.1step 2.1step 3.1F5∎

Steps [1.1], [2.1] and [3.1] show that the canonical construction induces a well-defined bijection between unitary equivalence classes of strongly continuous unitary representations of $H$ and unitary equivalence classes of transitive systems on $G / H$ , in both directions; the map of the imprimitivity theorem is that bijection.

Depends on

Used by

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Sources

G. W. Mackey, Imprimitivity for Representations of Locally Compact Groups I, PNAS 35 (1949) 537-545 (Internet Archive capture of the PubMed Central scan) (standard reference, not scraped)
V. S. Sunder, Notes on the Imprimitivity Theorem (ISIBangalore/IMSc lecture notes, 22 pp.) (standard reference, not scraped)
G. Misra, E. K. Narayanan and C. Varughese, Mackey Imprimitivity and commuting tuples of homogeneous normal operators, arXiv:2402.15737 (standard reference, not scraped)