Proper actions, free actions and the orbit space

Written by GPT-6.1 Sol (OpenAI), October 2026. Self-checked by the writing AI. Public domain (CC0).

A proper action has a Hausdorff orbit space. Its crossed product still remembers the stabilizers. We construct the Hilbert module over the orbit space and identify exactly the ideal it represents. The ideal is the whole crossed product precisely when the action is free. Compact groups give a parallel construction for any coefficient algebra: averaging produces a corner, and fullness of that corner is called saturation.

Throughout, \(G\) is a locally compact Hausdorff group acting continuously on a locally compact Hausdorff space \(X\). Write \(tx\) for the left action and

Use left Haar measure \(dt\) and the convention \(\int f(st)ds=\Delta(t)^{-1}\int f(s)ds\). The full crossed-product convolution and involution are those of Lesson 1. Assertions about reduced crossed products use Lesson 2.

Properness and orbital integration

Lemma 9.1. The following conditions are equivalent:

1. The map \((t,x)\mapsto(x,tx)\) from \(G\times X\) to \(X\times X\) is proper.
2. For compact \(K,L\subseteq X\), the transporter \(T(K,L)=\{t:tK\cap L\ne\varnothing\}\) is compact.
3. The same transporter is compact when \(L=K\).

For a proper action, \(Q\) is locally compact Hausdorff, every stabilizer \(H_x=\{t:tx=x\}\) is compact, and \(G/H_x\to Gx\), \(tH_x\mapsto tx\), is a homeomorphism onto a closed orbit.

Proof. A transporter in condition 2 is the projection of the inverse image of \(K\times L\) under the proper map. Thus 1 implies 2, and 2 implies 3. Under 3, the inverse image of \(K\times L\) is a closed subset of \(T(K\cup L,K\cup L)\times K\), which is compact. Every compact subset of \(X\times X\) is contained in the product of its compact projections, proving 1.

A proper map between locally compact Hausdorff spaces is closed. Hence the orbit relation is closed in \(X\times X\). The quotient map \(q:X\to Q\) is open because the saturation of an open set is \(\bigcup_t tU\). For unrelated \(x,y\), closedness of the relation gives neighborhoods \(U,V\) containing no related pair in \(U\times V\); their open quotient images are disjoint. Thus \(Q\) is Hausdorff. For a compact neighborhood \(K\) of \(x\), \(q(K)\) is a compact neighborhood of \(q(x)\), proving local compactness. Orbits are closed as inverse images of quotient points.

The inverse image of \((x,x)\) is \(H_x\times\{x\}\), so \(H_x\) is compact. The orbit map induces a continuous bijection \(G/H_x\to Gx\). For compact \(L\subseteq Gx\), the set \(\{t:tx\in L\}\) is compact by properness. Its quotient image is the inverse image of \(L\). The bijection is proper, hence a homeomorphism. ∎

For \(c\in C_c(X)\), define

Changing \(x\) to \(rx\) and substituting \(t=ru\) proves invariance. For \(x\) in a compact neighborhood \(L\), the integrand is supported in the compact transporter from \(\operatorname{supp}c\) to \(L\). Continuity on this common compact integration set proves continuity in \(x\); the open quotient gives continuity on \(Q\). The support is contained in \(q(\operatorname{supp}c)\), so \(Pc\in C_c(Q)\).

Proposition 9.2 (The orbit module). The formulas

make \(C_c(X)\) a pre-Hilbert \(C_0(Q)\)-module. Its completion \(E\) is full. A nondegenerate covariant pair on \(E\) is

The full integrated action \(R:B\to\mathcal L(E)\) is

on compact cores.

Proof. Equation (9.2) supplies the range and continuity of (9.3). Positivity, conjugate symmetry and module linearity follow from the integral. If \(\xi(x)\ne0\), continuity gives an identity neighborhood on which \(|\xi(t^{-1}x)|\) is bounded below; Haar full support makes its orbital integral positive. Thus the inner product is definite. Scalar Cauchy–Schwarz in each integral proves the module Cauchy–Schwarz bound, giving the Hilbert-module completion. At each orbit a bump nonzero at one of its points has positive squared norm. The ideal generated by these norm functions has empty common zero set and is all of \(C_0(Q)\), proving fullness.

Multiplication has adjoint \(M_{\bar a}\) and norm at most \(\|a\|\); compact cutoffs give nondegeneracy. Substitution \(u=st\) gives

The inverse is \(U_{t^{-1}}\), so it is adjointable and unitary. For \(t\) near the identity, \(U_t\xi-\xi\) has a common compact support \(K\) and tends uniformly to zero. Its module norm is bounded by its supremum norm times \(\operatorname{Haar}(T(K,K))^{1/2}\): for a nonzero orbital integral choose a representative in \(K\). Thus \(U\) is strongly continuous. Direct multiplication gives covariance.

Localize \(E\) at a faithful representation of \(C_0(Q)\). Faithful localization of adjointable operators, as used in Lesson 7, gives an ordinary covariant pair with the same operator norms. The full universal norm bounds its integral. Hence (9.5) extends to \(R\); the integrated-form correspondence gives nondegeneracy. ∎

Recovering a crossed product from its orbits

Lemma 9.3. Restriction to closed orbits defines quotients

Proof. Invariant multiplication \(h\mapsto h\circ q\) gives a central nondegenerate multiplier representation of \(C_0(Q)\) on \(B\). Nondegeneracy follows on compact cores by taking \(h=1\) on the compact orbit image of coefficient supports. In an irreducible nondegenerate representation of \(B\), these central multipliers are scalar by Schur's lemma. Their nondegenerate scalar representation is evaluation at some \(q(x)\).

The coefficient ideal generated by \(h\circ q\), \(h(q(x))=0\), is \(C_0(X\setminus Gx)\). Indeed a compactly supported function in this open complement has compact orbit image avoiding \(q(x)\); a cutoff equal to one on that image and zero at \(q(x)\) factors the function. Approximation gives the ideal. Dense cores and full ideal exactness identify the corresponding central ideal of \(B\) with \(C_0(X\setminus Gx)\rtimes G\), whose quotient is \(B_x\). Every irreducible representation factors through such an orbit quotient. Irreducible representations detect the C*-norm, proving (9.7). ∎

Proposition 9.4. For any proper action,

Proof. Each orbit is \(G/H_x\) with compact \(H_x\). The full and reduced Green theorem of Lesson 7 identifies its full-to-reduced map with the compact-operator quotient induced by \(C^*(H_x)\to C_r^*(H_x)\). Compact-group amenability from Lesson 4 makes this coefficient map an isomorphism, hence the orbit regular quotient is an isomorphism.

Coefficient restriction also gives a reduced quotient map, by Lesson 2; its diagram with the full maps commutes on compact cores. An element killed by the global regular quotient is therefore killed in every reduced orbit product, hence in every full orbit product. Equation (9.7) makes it zero. The regular quotient is always onto. No exactness or amenability assumption on \(G\) itself is needed. ∎

The rank-one ideal and the freeness criterion

For \(\xi,\eta\in C_c(X)\), put

Its support lies over \(x\in\operatorname{supp}\xi\), \(t^{-1}x\in\operatorname{supp}\eta\); properness gives a compact transporter, hence compact support in \(G\times X\). It is a legitimate crossed-product core element. The half power follows from (9.5):

Theorem 9.5. The closure

is a two-sided ideal and \(R|_J:J\to\mathcal K(E)\) is an isomorphism. Thus \(J\sim_M C_0(Q)\). Moreover,

Proof. Direct convolution gives

The convolution factors multiply to \(\Delta(t)^{-1/2}\), leaving the orbital integral (9.3). The adjoint uses \(\Delta(t)^{-1}\Delta(t^{-1})^{-1/2}=\Delta(t)^{-1/2}\). For \(b\in C_c(G\times X)\),

Here \(R(b)\xi\in C_c(X)\). Adjoints give the right ideal property; density and the full norm bound give the closed ideal assertion.

We prove isometry, not merely existence of the rank-one image. Fix \(O=Gx\) and \(H=H_x\). Normalize Haar measure on compact \(H\) to mass one. Use the rescaled Green module \(\mathcal X\) from Lesson 7, a \(B_x\)-\(C^*(H)\) imprimitivity module. Both \(\Delta_G|_H\) and \(\Delta_H\) equal one. Set

The projection \(p_H\) is central and selects the trivial representation. The function \(z_\xi\) is compactly supported, since \(G\to O\) is proper. It is right \(H\)-invariant, and \(z_\xi p_H=z_\xi\).

The rescaled Green formulas give

In the last formula the Green prefactor \(\Delta_G(r)/\Delta_G(t)\) multiplies the two \(z\)-factors, whose product has factor \(\Delta_G(r)^{-1}\Delta_G(t)^{1/2}\); the \(H\)-integral has mass one. Thus these are exactly the restricted kernels.

The fiber \(E_{q(x)}=E\otimes_{\operatorname{ev}_{q(x)}}\mathbb C\) is the completion of restrictions to \(O\) with (9.3). Every \(C_c(O)\) function extends to \(C_c(X)\): extend in the one-point compactification using compact Hausdorff extension and then apply a compact cutoff around its support. Equation (9.16) identifies this fiber isometrically with \(\mathcal Xp_H\) over \(\mathbb Cp_H\). The range is dense and onto: \(f p_H\), \(f\in C_c(G)\), is its right \(H\)-average, and multiplication by \(\Delta_G(r)^{1/2}\) makes this descend to \(C_c(O)\).

Centrality gives \(\mathcal X=\mathcal Xp_H\oplus\mathcal X(1-p_H)\). Under Green's \(B_x=\mathcal K_{C^*(H)}(\mathcal X)\), each restricted kernel is the indicated rank-one on the first summand and zero on the second. Every finite sum therefore has exactly its operator norm on \(E_{q(x)}\).

An adjointable operator on a Hilbert \(C_0(Q)\)-module has norm equal to the supremum of its fiber norms: the module norm is the supremum of the scalar inner-product norms, giving one bound; the contractive fiber maps give the other. With (9.7),

Hence \(R|_J\) is isometric and onto \(\mathcal K(E)\), by density of \(C_c(X)\) in \(E\). Fullness of \(E\) gives the Morita equivalence.

For a free action every \(H\) is trivial, so \(p_H=1\) and each restricted ideal is all of \(B_x\). If \(B/J\ne0\), a nonzero irreducible representation of it would pull back to one of \(B\), factor through some \(B_x\), and kill its whole image. This contradiction proves \(J=B\).

If some \(H\) is nontrivial, its regular representation has a nonzero complement to constant vectors: two disjoint open sets show \(\dim L^2(H)>1\). On this complement \(p_H\) acts as zero. Green induction is nonzero by fullness, but (9.16) makes it kill every restricted kernel. Thus the restricted ideal is proper, so \(J\ne B\). ∎

This criterion concerns the canonical ideal and module. It is not a prohibition on every abstract Morita equivalence with \(C_0(Q)\). For instance, let \(C_2\) act trivially on discrete \(\mathbb N\). This is proper and nonfree, but

through a bijection \(\mathbb N\times\{+,-\}\to\mathbb N\). This rearranges orbit points. The canonical ideal is only the trivial-character summand, and each full orbit fiber is \(\mathbb C\oplus\mathbb C\).

Principal bundles and a geometric projection

Let \(P\to Q\) be a locally trivial principal left \(G\)-bundle with Hausdorff locally compact total space and base. The map \((t,p)\mapsto(p,tp)\) is a homeomorphism onto \(P\times_QP\), as bundle charts show. This fiber product is closed in \(P\times P\); hence the action is free and proper. Theorem 9.5 gives

In a chart \(P|_V\cong G\times V\), Haar inversion in (9.3) makes the module inner product use \(\Delta(r)^{-1}dr\) in the group coordinate. Multiplication by \(\Delta(r)^{-1/2}\) gives the left-Haar \(L^2(G)\) model. Thus the local compact algebra is \(C_0(V,\mathcal K(L^2(G)))\). These charts describe a compact-operator bundle; no arbitrary global trivialization is asserted.

If \(Q\) is compact, choose finitely many nonnegative compact bumps on \(P\) whose orbital averages cover \(Q\), and let \(c\) be their sum. Then \(Pc>0\) everywhere. Set

Then \(e\in C_c(P)\), \(\langle e,e\rangle=1\), and (9.13) gives \(p_e=p_e^*=p_e^2\). Its corner and module are

Here \(h\) corresponds to \(\theta_{eh,e}\), and \(bp_e\) to \(R(b)e\). The first map is isometric because \(\|eh\|=\|h\|\); compression of an operator gives \(\theta_{e\langle e,Te\rangle,e}\), proving surjectivity. The second preserves its corner-valued inner product and is onto because \(\theta_{\xi,e}e=\xi\). The projection is full: every rank-one factors as \(\theta_{\xi,e}\theta_{e,e}\theta_{e,\eta}\). These are the principal-bundle Morita module and its geometric corner. For noncompact \(Q\), the module still exists, but a vector with inner product identically one cannot belong to it, since that inner product must lie in \(C_0(Q)\).

Averaging corners for compact groups

Now \(G\) is compact and acts on an arbitrary C*-algebra \(A\); Haar measure has mass one. Put \(D=A\rtimes G\) and

The average \(p\) is a multiplier integral: it is the image of the constant function one in \(C^*(G)\) under the nondegenerate integrated group map.

Theorem 9.6 (The averaging corner). The map

is an isomorphism. The corner is full exactly when \(\overline{DpD}=D\); an action with this property is called saturated. For \(A=C_0(X)\), saturation is equivalent to freeness.

Proof. Haar invariance and inversion give \(p^2=p=p^*\) and \(i_G(t)p=p\). Averaged covariance gives

Fixed coefficients commute with \(p\), so (9.23) preserves products and adjoints. Its image belongs to \(D\), since \(i_A(a)p\) is the integrated constant coefficient \(a\). For isometry, take a faithful nondegenerate \(\pi\) of \(A\) and its regular pair on \(L^2(G,H_\pi)\). Averaging projects onto constant vectors; a fixed coefficient acts there as \(\pi(a)\). Thus the image norm is at least \(\|a\|\), and the reverse bound is immediate.

For \(f\in C_c(G,A)\),

The isometric image is closed, so density proves surjectivity. Also \(\mathsf E\) is positive, contractive, fixes \(A^G\), and has its coefficient bimodule property; thus it is a conditional expectation. If \(\mathsf E(a^*a)=0\), every positive functional gives a zero integral of a nonnegative continuous function. Haar full support makes its identity value zero, and separation by positive functionals gives \(a=0\). Thus it is faithful.

Put \(I=\overline{\operatorname{span}}DpD\). The full-corner prerequisite Imprimitivity bimodules and Morita equivalence, Proposition 2.4, applied to \(I\), and its ideal formulation The Rieffel correspondence and induced representations, Example 5.2, identify \(Dp\) as the \(I\)–\(pDp\) imprimitivity module with right inner product \(x^*y\). Here \(p\) restricts to \(M(I)\), \(Dp=Ip\), \(pIp=pDp\), and \(p\) is full in \(I\). For example, \(dp=\lim_\lambda dp e_\lambda\in I\) for an approximate identity of \(D\), giving \(Dp=Ip\); the other identifications follow by multiplication and the definition of \(I\). Thus the prerequisite applies even for nonunital \(D\) and a multiplier projection.

In the commutative case, \(k_{\xi,\eta}=i_A(\xi)p i_A(\bar\eta)\). Dense integrated elements satisfy \(f p=i_A(\int f(t)dt)p\) and \(p f=p i_A(\int\alpha_t^{-1}(f(t))dt)\). They show \(\overline{DpD}=J\). Every compact-group action is proper, so Theorem 9.5 makes saturation equivalent to freeness. ∎

The corner need not be full. For a trivial action by a nontrivial compact group, \(D=A\otimes C^*(G)\); its average selects only the trivial representation, agreeing with (9.16).

For a noncommutative example, let \(\mathcal O_n\), \(n\ge2\), be generated by isometries \(s_i\) with \(s_i^*s_j=\delta_{ij}\) and \(\sum_i s_i s_i^*=1\). Its gauge action is \(\alpha_z(s_i)=zs_i\). In its crossed product by \(\mathbb T\), put

These are orthogonal projections with \(p_0=p\). For a word \(\mu\) of length \(m\), covariance gives

Thus the ideal generated by \(p_0\) contains every \(p_k\). Finite scalar character sums are dense in \(C(\mathbb T)\), so core approximation makes the span of \(i_A(a)p_k\) dense in the crossed product. The ideal is the whole product: the gauge action is saturated.

The Cuntz relations reduce *-polynomials to the dense span of \(s_\mu s_\nu^*\); averaging keeps the equal-length terms. At length \(m\) these form matrix units for \(M_{n^m}(\mathbb C)\), with embeddings \(s_\mu s_\nu^*=\sum_i s_{\mu i}s_{\nu i}^*\). The fixed algebra is therefore the UHF algebra of type \(n^\infty\), and the averaging corner gives a Morita equivalence with it. This establishes the fullness without requiring a later K-theory calculation of \(\mathcal O_n\).

K-theory and two free actions

We use the Morita K-theory prerequisite Morita invariance of K-theory and maps induced by correspondences, Corollary 2.2 and Theorem 5.1. For an imprimitivity module \(E\) between \(\sigma\)-unital algebras, its map on both K-groups is \(\mu_E=(i_{B*})^{-1}i_{A*}\), using the two inclusions into \(\mathcal K(E\oplus B)\). The correspondence map agrees with \(\mu_E\), and the conjugate module gives its inverse. All algebras in Corollary 9.7 are separable, so this scope applies. Rosenberg [2012, §§1.1–1.3, pp. 94–99] gives the broader Kasparov-product interpretation of the same Morita map.

Corollary 9.7. If \(X\) is second countable and \(G\) is \(\sigma\)-compact acting freely and properly, then

The map is induced by the module (9.3).

Proof. For empty \(X\) all groups vanish. Otherwise a closed orbit is homeomorphic to \(G\), hence as a subspace of second countable \(X\) makes \(G\) second countable. The coefficient algebra is separable, and core approximation makes its crossed product separable. The open quotient is second countable, since images of a countable base form a base; thus \(C_0(Q)\) is separable. Apply Theorem 9.5, Proposition 9.4 and the stated Morita prerequisite. Here \(K_c^j(Q)=K_j(C_0(Q))\), the usual complex K-theory with compact support. ∎

For integer translations on the real line,

Properness makes the sums locally finite. Thus \(C_0(\mathbb R)\rtimes\mathbb Z\sim_M C(\mathbb T)\), with \(K_0=\mathbb Z\) and \(K_1=\mathbb Z\). Every complex bundle on the circle is trivial, since its endpoint gluing unitary can be joined to the identity. Stable unitary loops are classified by determinant winding, because \(\pi_1(U(r))=\mathbb Z\) detected by determinant. Rank gives \(K_0\), and counterclockwise winding gives \(K_1\).

For the antipodal \(C_2\)-action on \(S^2\), the quotient is \(\mathbb{RP}^2\). Its module is the section module of the permutation bundle \(\mathcal V=\mathbf1\oplus L\), where \(L\) is the complex sign line of the double cover. Thus

In ordered orbit coordinates \((x,-x)\), \(f+gu\) is

Changing representative swaps the coordinates, giving the bundle transition. Conversely every continuous matrix field with this swap rule has its four entries determined by two functions \(f,g\). This proves the onto isomorphism.

The CW structure of \(\mathbb{RP}^2\) is a circle with a disk attached by a degree-two map. A rank-\(r\) complex bundle is trivial on the circle. Its disk gluing is a loop in \(U(r)\), classified by determinant winding. Changing the circle trivialization changes winding by twice an integer, while a disk trivialization changes it by zero. Thus at each positive rank the isomorphism classes are two parities. Each parity occurs for a line bundle, and direct sum adds rank and parity. This uses ordinary clutching and \(\pi_1(U(r))=\mathbb Z\), \(\pi_2(U(r))=0\).

The sign line has odd parity. Otherwise a nowhere-zero section would give an odd map \(f:S^2\to\mathbb C\setminus\{0\}\). Normalize to the circle and lift to a real argument \(\theta\) using simple connectivity of \(S^2\). Oddness says \(\theta(-x)-\theta(x)\) is a constant odd multiple of \(\pi\). Applying this twice makes the constant its negative, an impossibility. Hence \(L\) is nontrivial. Group completion of rank and parity gives

For \(K_1\), a unitary-valued map has winding \(k\) on the circle skeleton. Extension over the attached disk forces \(2k=0\) in \(\mathbb Z\), so \(k=0\). Homotope this restriction to the identity and extend the homotopy over the disk. The remaining map factors through \(S^2\); \(\pi_2(U(r))=0\) makes it null. Thus \(K_1(C(\mathbb{RP}^2))=0\). These are also the two groups of (9.30).

That algebra is not \(M_2(C(\mathbb{RP}^2))\). A trivialization of its endomorphism bundle would lift locally to unitaries, turning the transition functions of \(\mathcal V\) into scalar matrices. Then \(\mathcal V\cong M\oplus M\) for a line bundle \(M\). Its determinant would be \(M^{\otimes2}\), trivial by parity classification. But \(\det(\mathbf1\oplus L)=L\) is nontrivial. An abstract algebra isomorphism induces a homeomorphism of the centers and a pulled-back matrix-bundle trivialization, so the same obstruction applies.

Exercises with solutions

Exercise 1 (An explicit translation corner). Construct the Morita module and a full projection for \(C_0(\mathbb R)\rtimes\mathbb Z\sim_M C(\mathbb T)\).

Solution. Choose nonnegative \(c\in C_c(\mathbb R)\) strictly positive on \([0,1]\). Then \(D(x)=\sum_n c(x-n)\) is positive, continuous and periodic. The vector \(e(x)=\sqrt{c(x)/D(x)}\) is compactly supported and satisfies \(\sum_n|e(x-n)|^2=1\). Complete \(C_c(\mathbb R)\) with (9.29), using the left action \(R(b)\xi(x)=\sum_n b(n,x)\xi(x-n)\). The kernel \(p_e(n,x)=e(x)\overline{e(x-n)}\) has finite group support and is a projection by (9.13). Its corner is \(C(\mathbb T)\) through \(h\mapsto\theta_{eh,e}\). The unitary \(bp_e\mapsto R(b)e\) identifies the left module with the orbit module, by (9.21). Every rank-one factors through \(\theta_{e,e}\); freeness makes the compact algebra the whole crossed product. Thus the projection is full. ∎

Exercise 2 (The modular power). Derive the rank-one kernel from (9.3) and check its involution for a possibly non-unimodular group.

Solution. The group operator has factor \(\Delta(t)^{1/2}\), since \(s\mapsto st\) contributes \(\Delta(t)^{-1}\) in the orbital inner product. Its integral is (9.5). Comparing with \(\xi\langle\eta,\zeta\rangle\) forces (9.9). The adjoint is

Using power \(-1\) would leave power zero under involution, failing the required identity for \(\Delta\ne1\). It would also leave an unwanted \(\Delta(t)^{-1/2}\) weight in the rank-one action. A free proper non-unimodular example is the affine group \(\mathbb R\rtimes\mathbb R_{>0}\) acting on itself by left translation. Its modular function is \(\Delta(b,a)=a^{-1}\), so the correct factor is \(a^{1/2}\). ∎

Exercise 3 (The compact averaging corner). Prove the corner theorem for compact \(G\), including nonunital coefficients and the full-corner criterion.

Solution. The constant scalar function gives a multiplier projection \(p\), with \(u_t p=p\). Averaged covariance gives \(p i_A(a)p=i_A(\mathsf E(a))p\). Fixed coefficients commute with \(p\), so \(a\mapsto i_A(a)p\) is a *-homomorphism into \(D\), represented by the constant coefficient function. A faithful regular pair on \(L^2(G,H)\), restricted to constant vectors, makes its norm exactly \(\|a\|\). Compression of a dense integrated \(f\) gives (9.25), hence every corner element lies in the closed image.

The right module \(Dp\) has full corner-valued inner product and compact algebra \(\overline{DpD}\), by rectangular rank-one multiplication. It gives a Morita equivalence with the whole \(D\) precisely when that ideal is \(D\), the definition of saturation. For commutative coefficients, \(i_A(\xi)p i_A(\bar\eta)=k_{\xi,\eta}\), and the dense formulas for \(f p,p f\) identify the ideal with \(J\). Theorem 9.5 makes fullness equivalent to freeness. ∎

Exercise 4 (A stabilizer detects the missing ideal). For a proper action with \(H_x\ne\{e\}\), construct a nonzero representation of \(B\) annihilating every \(k_{\xi,\eta}\).

Solution. Let \(H=H_x\), \(K=L^2(H)\ominus\mathbb C1\), and \(\sigma\) its restricted left regular representation. The space is nonzero since a nontrivial Hausdorff compact group has two disjoint nonempty open sets of positive Haar measure. Its integrated average \(\sigma(p_H)\) is zero.

Green induction gives a representation on \(\mathcal X\otimes_{C^*(H)}K\). It is nonzero: otherwise all \(\sigma(\langle z,z\rangle)\) vanish, and polarization and fullness would make \(\sigma\) zero. Pull the representation back through \(B\to B_x\). Since \(z_\xi=z_\xi p_H\),

By (9.16) each restricted kernel is \(\theta_{z_\xi,z_\eta}\), hence acts as zero on the induced space. This nonzero representation annihilates \(J\), proving it proper. For a point with group \(C_2\), \(B=\mathbb C\oplus\mathbb C\), \(J\) is the trivial-character summand, and the sign character is the annihilating representation. ∎

What this lesson does not prove

We use full ideal exactness and the integrated universal correspondence from Lesson 1, reduced quotient functoriality from Lesson 2, compact-group amenability from Lesson 4, and the full and reduced Green theorem, its modular formulas and inverse induction from Lesson 7. Hilbert-module completion and faithful localization are the precise general prerequisites identified there. Locally compact cutoffs, compact Hausdorff extension, Schur's lemma and norm detection by irreducible C*-representations are foundational tools.

The generic corner-module result is Imprimitivity bimodules and Morita equivalence, Proposition 2.4, with the ideal version in The Rieffel correspondence and induced representations, Example 5.2. The K-theory consequence uses Morita invariance of K-theory and maps induced by correspondences, Corollary 2.2 and Theorem 5.1 for both degrees. For the compact examples we use vector-bundle and stable-unitary descriptions of \(K_0,K_1\), ordinary clutching and the homotopy extension property. The elementary unitary-group facts follow from \(SU(2)\cong S^3\), the fibrations \(SU(r-1)\to SU(r)\to S^{2r-1}\), and the determinant fibration \(SU(r)\to U(r)\to\mathbb T\): their homotopy exact sequences give \(\pi_1(U(r))=\mathbb Z\), detected by determinant, and \(\pi_2(U(r))=0\). The case \(r=1\) is the circle itself. General clutching and homotopy-exact-sequence theorems are not reproved. The proper-action theorem, rank-one ideal, full-to-reduced equality, compact averaging corner, gauge saturation and two concrete K-group computations are proved above.

[Blackadar 2006] B. Blackadar, Operator Algebras: Theory of C*-Algebras and von Neumann Algebras, II.10.4.6–II.10.4.10, pp. 221–222, and II.10.4.17–II.10.4.19, pp. 224–225. The half power in (9.9) is derived for the integrated and Haar conventions stated here. Author's revised edition, 2017.

[Rosenberg 2012] J. Rosenberg, Examples and applications of noncommutative geometry and K-theory, in Topics in Noncommutative Geometry, Clay Mathematics Proceedings 16, pp. 93–129; §§1.1–1.3, pp. 94–99, for the Kasparov interpretation of Morita K-maps, and §2.2, pp. 104–105, for the proper-action context. Electronic volume.

Imprimitivity bimodules and Morita equivalence Hilbert C*-modules and Morita equivalence, Lesson 11, Proposition 2.4, the full multiplier-corner module.

The Rieffel correspondence and induced representations Hilbert C*-modules and Morita equivalence, Lesson 12, Example 5.2, the ideal generated by a corner.

Morita invariance of K-theory and maps induced by correspondences Hilbert C*-modules and Morita equivalence, Lesson 14, Corollary 2.2 and Theorem 5.1, the K-map of a specified imprimitivity module and its inverse in both degrees.

[Connes 1994] A. Connes, Noncommutative Geometry, Chapter II, §7, Proposition 1, p. 117. Author's electronic edition. It treats free proper discrete actions on manifolds and includes full-to-reduced equality.

[Emerson 2024] H. Emerson, An Introduction to C*-Algebras and Noncommutative Geometry, §4.6, pp. 161–167, for proper discrete actions and stabilizer representations. Our orbit-fiber argument does not identify a nonfree fiber with an unrestricted compact-operator algebra.