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A nontransitive system with two orbits is not classified by one stabilizer

Statement refuted

False claim: every system of imprimitivity for a group $G$ is classified, up to unitary equivalence, by a single closed subgroup $H \leq G$ and a strongly continuous unitary representation of $H$ ; that is, Mackey's imprimitivity classification needs no transitivity or ergodicity hypothesis.

Facts & Assumptions

Given: the discrete finite group $G = Z / 2$ , the three-point set $X = {a, b, c}$ , the permutation action with $s \cdot a = a$ , $s \cdot b = c$ , and the unitary $U (s)$ acting as the identity on $C δ_{a}$ and as the swap $δ_{b} \leftrightarrow δ_{c}$ on $C δ_{b} \oplus C δ_{c}$ .

[F1]

A system of imprimitivity is a pair $(U, P)$ with $U$ a strongly continuous unitary representation and $P$ a PVM satisfying $U_{g} P (E) U_{g}^{- 1} = P (g E)$ ; it is ergodic when every invariant $P (E)$ is $0$ or $I$ , and transitive when its base is equivariantly identified with some homogeneous space $G / H$ (Systems of imprimitivity for a Borel $G$ -space, Transitive systems of imprimitivity and their normalized measure class, Strongly continuous unitary representations, invariant linear subspaces and intertwiners, Projection valued measure).

[F2]

For a finite group, every homogeneous space $G / H$ is the set of left cosets of a subgroup, so $∣ Z / 2 / H ∣ \in {1, 2}$ ; an equivariant isomorphism of $G$ -sets preserves orbit cardinalities and the number of orbits (Left and right cosets $g H$ and $H g$ of a subgroup, Left group actions, transitive actions, and faithful actions, Equivariant maps and isomorphisms of group actions).

[F3]

The finite set $X$ with the discrete metric is Polish (every Cauchy sequence is eventually constant and the full set is dense), and its power-set $σ$ -algebra is standard Borel (Polish spaces are separable completely metrizable spaces, Standard Borel spaces).

Counterexample

The counterexample is the following finite model. Let $G = Z / 2 = {e, s}$ with the discrete topology act on $X = {a, b, c}$ by $s \cdot a = a$ , $s \cdot b = c$ , $s \cdot c = b$ , let $H = C^{3}$ be the direct sum of the trivial representation on $C δ_{a}$ and the regular representation on $C δ_{b} \oplus C δ_{c}$ , and let $P$ be the projection-valued measure with $P ({a}), P ({b}), P ({c})$ the three coordinate projections. Then $(U, P)$ is a system of imprimitivity on the standard Borel space $X$ ; the two orbits are ${a}$ and ${b, c}$ , the projections $P ({a})$ and $P ({b, c})$ are nontrivial and invariant, and the system is not transitive (nor ergodic). No closed subgroup $H \leq G$ with a strongly continuous unitary representation $σ$ classifies it: every homogeneous space $G / H$ has one or two points, so it cannot be equivariantly identified with the three-point base, and the theorem correctly decomposes the system as the direct sum of the transitive systems on the two orbits.

Proof technique: counterexample.

Given: the action and the pair $(U, P)$ described above.

1.1F1F3algebra

The operator $U (s)$ is a unitary swap with $U (s)^{2} = I$ , so it defines a unitary representation of the discrete group $G$ ; every orbit map from this discrete group is continuous. Covariance: $U (s)$ fixes $δ_{a}$ and swaps $δ_{b}, δ_{c}$ , so $U (s) P ({a}) U (s)^{- 1} = P ({a}) = P (s \cdot {a})$ , $U (s) P ({b}) U (s)^{- 1} = P ({c}) = P (s \cdot {b})$ , and $U (s) P ({c}) U (s)^{- 1} = P ({b}) = P (s \cdot {c})$ ; for the identity the identity is trivial, and covariance extends to all subsets since the three singletons generate the power set and both sides are PVM-valued. Hence $(U, P)$ is a system of imprimitivity; it is defined on the standard Borel three-point space of [F3].

2.1step 1.1

Invariant projections and non-ergodicity: $P ({a})$ and $P ({b, c}) = P ({b}) + P ({c})$ are nonzero and different from $I$ , and both are invariant under $U$ , since the orbits are ${a}$ and ${b, c}$ ; thus the system is not ergodic.

3.1F1F2step 2.1

Nontransitivity: the orbits of the action are the singleton ${a}$ and the two-point set ${b, c}$ , while a homogeneous space of $Z / 2$ has one or two points by [F2]; a transitive system on a homogeneous space is concentrated on a single orbit, so the three-point base with two orbits cannot be equivariantly identified with any $G / H$ . Hence the system is not transitive.

3.2step 2.1F1

Correct decomposition: $P ({a})$ and $P ({b, c})$ are complementary invariant projections, and on their ranges the system restricts to the transitive system on the single orbit ${a}$ (the one-point homogeneous space $G / G$ with the trivial representation) and to the transitive system on ${b, c}$ (the two-point homogeneous space $G / {e}$ with the regular representation), respectively. So $(U, P)$ is the direct sum of the two transitive systems, and the failure above is exactly the failure of a direct sum of transitive systems to be classified by one subgroup.

4.1F2step 3.1

Non-classification by one subgroup: the classification data $(H, σ)$ determine a system whose base is the homogeneous space $G / H$ , of one or two points by [F2], and whose imprimitivity measure is concentrated on the orbits of that base; no such data can reproduce the three-point base with two orbits, since equivariant Borel isomorphisms preserve cardinalities and orbit counts. Therefore the nontransitive system is not classified by a single closed subgroup and a representation of it.

5.1step 3.1step 4.1step 3.2∎

The explicit finite computation therefore exhibits a system of imprimitivity that is neither transitive nor ergodic and is not classified by one stabilizer subgroup; transitivity (or ergodicity) is essential to the one-subgroup form of Mackey's imprimitivity theorem.

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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)