Haar amenability and modular averaging

Amenability admits several equivalent descriptions: an invariant mean, fixed points in compact convex sets, almost invariant probability densities, compact sets with small translation boundary, and almost invariant vectors in the regular representation. The passage between these descriptions must preserve uniformity on compact sets. Pointwise estimates alone are insufficient.

We prove these equivalences for arbitrary locally compact Hausdorff groups, without separability or unimodularity assumptions. We then average actions on a von Neumann algebra. A preserved faithful normal state makes the resulting expectation normal and identifies its Hilbert-space projection, modular operator and complete domains.

The earlier programme construction is Haar averages and compact translation control, §§0–5. Its proofs establish the locally completed Haar convention, smoothing, compact-uniform estimates and integrated representations used below. The full and reduced norm argument, §6 supplies the elementary state-extension calculation, which we also spell out. For the operator-algebra application we use the proved concrete predual, The concrete predual and its intrinsic norm, normality criterion, The positive-map equivalence, automorphism implementation, Canonical implementation of automorphisms, and closed modular transport, 6. Transporting an arbitrary semifinite weight.

1. Haar classes and invariant smoothing

Fix left Haar measure \(dt\) on \(G\). Use

Write \(\mathcal P(G)=\{k\in L^1(G):k\geq0,\int k=1\}\). A mean is a positive unital bounded functional; it need not be normal.

For groups that are not sigma-compact, Haar \(L^\infty(G)\) means the locally completed space \(L^1(G)^*\). To describe it, choose a compact identity neighbourhood \(V\); the subgroup generated by \(V\cup V^{-1}\) is open and sigma-compact. Its left cosets are disjoint clopen sigma-compact spaces, and any compact set meets finitely many of them. Haar \(L^1\) is the \(\ell^1\) sum of the coset spaces, and its dual is their bounded product. Each integrable function has nonzero components on at most countably many cosets. Radon approximation on those cosets proves density of \(C_c(G)\) in \(L^p(G)\), \(1\leq p<\infty\). For a compactly supported continuous function, nearby translates have a common compact support and converge uniformly there. Approximation proves norm continuity of translations on \(L^p\); right translations include their continuous Haar scaling factor. These are the full statements of the earlier Haar Lemma 0.1.

Let \(\mathrm{LUC}\) and \(\mathrm{RUC}\) be the bounded continuous functions for which \(L_sf\to f\), respectively \(R_sf\to f\), in supremum norm as \(s\to e\), and put \(\mathrm{UC}=\mathrm{LUC}\cap\mathrm{RUC}\). These are translation-invariant unital C-star algebras: products satisfy the usual two-term norm estimate, conjugation preserves each translation norm, and uniform limits preserve the defining continuity.

For \(h\in\mathcal P(G)\) define

The dual-pairing formula gives a continuous representative even for a locally completed Haar class. Change of variable and (AMH.1) give

Thus \(S_hf\in\mathrm{RUC}\), \(k*f\in\mathrm{LUC}\), and \(k*f\in\mathrm{UC}\) when \(f\in\mathrm{RUC}\), because right translations commute with left convolution. Convolution identities here follow first for compactly supported kernels, then by \(L^1\) approximation. On compact products, continuous scalar functions are product measurable by uniform approximation with sums of separated continuous functions. Consequently this argument does not assume a countable base or a globally Borel representative for every Haar class.

Extension theorem. A left invariant mean \(m\) on \(\mathrm{UC}(G)\) produces a left invariant mean \(M\) on \(L^\infty(G)\) satisfying

Here is the complete extension. A bounded positive left invariant functional on \(L^1(G)\) is a nonnegative multiple of the integral: its restriction to \(C_c(G)\) is a positive Radon measure dominated by a multiple of Haar measure; Haar uniqueness makes it a scalar multiple, and \(L^1\) density extends the identity. For \(f\in\mathrm{RUC}\), apply this observation to \(k\mapsto m(k*f)\), first for \(f\geq0\), then by linear decomposition. There is a scalar \(\widetilde m(f)\) such that

Fixing one probability density proves that \(\widetilde m\) is positive and unital. A positive approximate identity has \(k_i*f\to f\) uniformly for \(f\in\mathrm{UC}\), so \(\widetilde m\) extends \(m\).

It is left invariant. If \(k^s(t)\,dt\) is the pushforward of \(k(t)\,dt\) under \(t\mapsto ts\), direct substitution gives \(k*L_sf=k^s*f\), and \(\int k^s=\int k\). This notation retains the Haar modular factor without setting it equal to one. Equation (AMH.4) gives \(\widetilde m(L_sf)=\widetilde m(f)\).

Now fix \(h\in\mathcal P(G)\) and put \(M(f)=\widetilde m(S_hf)\). Right smoothing commutes with left translations, so \(M\) is an invariant mean. Since \(S_h(k*f)=k*S_hf\) belongs to \(\mathrm{UC}\), equations (AMH.4) and the extension property give (AMH.3). The final smoothing need not preserve a mean that was only left invariant; what it preserves is existence of an invariant mean with the stronger convolution identity.

A group is amenable when it has a left invariant mean on Haar \(L^\infty(G)\). Restriction to \(\mathrm{UC}\), together with the extension just proved, makes the two formulations equivalent.

2. Fixed points and two-sided means

Fixed-point criterion. Amenability is equivalent to the assertion that every separately continuous affine action on a nonempty compact convex subset \(K\) of a Hausdorff locally convex space has a fixed point. This includes the source's weakly compact convex sets in a Banach dual pair.

Suppose \(M\) is an invariant mean and fix \(x\in K\). The restriction of \(M\) to \(C_b(G)\) is a weak-star limit of finite convex combinations of point evaluations. Otherwise real Hahn–Banach separation would supply a bounded continuous real function whose value under \(M\) lies outside its infimum and supremum, contradicting positivity and \(M(1)=1\). For such finite probabilities \(\sum_j a_j\delta_{g_j}\), form \(\sum_j a_jg_jx\in K\). Compactness gives a convergent subnet, with limit \(\bar x\). Every continuous affine function \(\psi\) on \(K\) satisfies

For a continuous linear functional \(\ell\) on the ambient space and a fixed \(s\), both \(\ell\) and \(y\mapsto\ell(sy)\) are continuous affine functions on \(K\). Left invariance in (AMH.5) gives \(\ell(s\bar x)=\ell(\bar x)\). Such functionals separate points, so \(\bar x\) is fixed. This constructs a point inside the specified compact set; a bounded bilinear form alone would not justify an operator taking values in a nonreflexive Banach space.

Conversely, the state space of \(\mathrm{UC}(G)\) is a nonempty weak-star compact convex set. Let \(s\cdot m\) be given by

The inverse makes this a left action. It is separately continuous because the translations of \(f\in\mathrm{UC}\) are norm continuous. A fixed point is an invariant mean on \(\mathrm{UC}\); AMH-01 extends it to Haar classes.

Two-sided means. Every amenable group has a mean invariant under both left and right translations, even when it is nonunimodular. Inversion sends Haar-null classes to Haar-null classes and converts a left mean into a right mean; the inverse Haar measure has a strictly positive continuous Haar modular density. Let these means be \(m_L,m_R\). For \(f\in\mathrm{UC}\), put

The outer function is continuous: its change from \(g\) to \(sg\) is bounded by \(\|L_{s^{-1}}f-f\|_\infty\). Outer left invariance and inner right invariance show that \(m_0\) is two-sided. This continuity check is essential before applying an outer Haar mean.

Extend \(m_0\) to \(\widetilde m\) on \(\mathrm{RUC}\) by (AMH.4). It remains right invariant since \(k*R_sf=R_s(k*f)\). For \(f\geq0\), the functional \(h\mapsto\widetilde m(S_hf)\) on \(L^1\) is positive and left invariant by (AMH.2). Hence it is a scalar multiple of \(\int h\). Linear decomposition gives the same independence of \(h\in\mathcal P(G)\) for arbitrary bounded \(f\). Since \(S_h(R_sf)=S_{h^s}f\), where \(h^s\,dt\) is the right-translation pushforward of \(h\,dt\), the mean \(M(f)=\widetilde m(S_hf)\) is right invariant. Its left invariance was proved in AMH-01.

3. Reiter densities with compact-uniform control

The following five conditions are equivalent:

1. \(G\) is amenable.
2. There is a net \(p_i\in\mathcal P(G)\) with \(\langle f,k*p_i-p_i\rangle\to0\) for every \(k\in\mathcal P(G)\) and \(f\in L^\infty(G)\).
3. There is a net \(p_i\in\mathcal P(G)\) with \(\|k*p_i-p_i\|_1\to0\) for every \(k\in\mathcal P(G)\).
4. For every compact \(C\subset G\) and \(\varepsilon>0\), some \(p\in\mathcal P(G)\) satisfies \(\sup_{s\in C}\|L_sp-p\|_1<\varepsilon\).
5. The density in condition 4 can be chosen in \(C_c(G)\).

The source labels these “four” conditions but actually lists five; all five are included here.

To prove \(1\Rightarrow2\), choose \(M\) satisfying (AMH.3). Normal states, identified with \(\mathcal P(G)\), are weak-star dense in the state space of \(L^\infty(G)\). Indeed the supremum of a real bounded function over these states is its essential supremum: choose a finite positive-measure subset where the function is near that supremum and normalize its indicator. Real separation then proves density, including for the locally completed space described in AMH-01.

The adjoint of convolution by \(k\) sends \(f\) to

where \(k^\#\,ds\) is the inversion pushforward of \(k\,ds\), a probability density. Thus (AMH.3) gives \(M(T_kf)=M(f)\), and normal states tending to \(M\) yield condition 2.

For a finite family \(k_1,\ldots,k_n\), condition 2 puts zero in the weak closure of

This set is convex. Hahn–Banach separation makes its weak and norm closures agree, so choose \(p\) with the sum of these \(L^1\) errors below any prescribed tolerance. Direct the choices by finite families and decreasing tolerance. This proves condition 3; the implication \(3\Rightarrow2\) is immediate.

For \(3\Rightarrow4\), fix \(h\in\mathcal P(G)\). The map \(s\mapsto L_sh\) is norm continuous, so its image on \(C\) is compact. Choose \(s_1,\ldots,s_n\) whose translates approximate all those in \(C\) within \(\varepsilon/3\). Choose \(p\) so that the convolution errors for \(h,L_{s_1}h,\ldots,L_{s_n}h\) are each below \(\varepsilon/3\), and put \(q=h*p\). For an appropriate \(j\),

It is the compact family of translates of the fixed smoothing density \(h\) that makes the estimate uniform.

Positive \(C_c\) probability densities are \(L^1\)-dense in \(\mathcal P(G)\): approximate a positive integrable function by nonnegative compactly supported continuous functions, then normalize. Changing a density by \(L^1\) distance \(\eta\) changes every translation error by at most \(2\eta\), uniformly in the translation. This proves \(4\Rightarrow5\), while \(5\Rightarrow4\) is immediate.

Finally direct condition 4 by compact sets and decreasing tolerances and take a weak-star cluster point of the corresponding normal states on \(L^\infty(G)\). For each fixed \(s\), the translation error tends to zero, so the cluster point is invariant. This proves \(4\Rightarrow1\). Nets, rather than sequences, retain the full locally compact scope.

4. One compact set for every translation

Compact Følner theorem. Amenability is equivalent to the following statement: for every compact \(C\subset G\) and \(\varepsilon>0\), there is a compact \(F\subset G\) such that

The reverse implication follows by taking \(p=1_F/|F|\) in AMH-03.

For the forward implication, enlarge \(C\) to a compact \(D\) of positive Haar measure, and put \(Q=D\cup D^{-1}D\). For every \(s\in D\),

If a Borel \(B\subset Q\) satisfies \(|Q\setminus B|<|D|/2\), deleting the two missing pieces from \(D\subset Q\cap sQ\) shows \(|B\cap sB|>0\). Hence

This overlap observation will turn a large-measure set of good translations into every translation in \(C\).

Choose a positive continuous compactly supported probability density \(p\) with \(\sup_{s\in Q}\|L_sp-p\|_1<\delta\), where

For \(a>0\) let \(F_a=\{p\geq a\}\), a compact set. Integrating indicators of intervals in the value variable gives

At levels of positive measure set

Set \(B_a=Q\) when \(|F_a|=0\). Fubini and the defining inequality of its complement give

The weight \(|F_a|\,da\) has mass one. Therefore some positive-measure level satisfies \(|Q\setminus B_a|<|D|/2\). Product measurability requires no metrizability: for \(s\in Q\), the relevant \(t\)'s lie in the fixed compact set \(\operatorname{supp}p\cup Q\operatorname{supp}p\). The continuous functions \(p(t),p(s^{-1}t)\) are product measurable there by AMH-01. Comparison with the real variable \(a\), followed by integration in \(t\), proves measurability of all the quantities in (AMH.11).

By (AMH.9), write each \(s\in C\) as \(uv^{-1}\) with \(u,v\in B_a\). Left invariance and the triangle inequality give

Take \(F=F_a\). The source's intermediate large-measure condition \((FC^*)\) has a printed \(s\in K\) where the preceding argument only proves \(s\in K_\delta\). The overlap step above, and the source's following page, supply the required upgrade to every \(s\in C\). Selecting a good level separately for each translation would not suffice.

5. Regular coefficients and the full group norm

Let \(\lambda_s\xi(t)=\xi(s^{-1}t)\) on \(L^2(G)\), with inner products linear in the first variable. A Reiter density \(p\) gives the unit vector \(\xi=\sqrt p\) and

Conversely a unit vector gives \(p=|\xi|^2\) and, by Cauchy–Schwarz,

Thus amenability is equivalent to compact-uniform almost invariant unit vectors.

For the exact coefficient convention in the source put

Cauchy–Schwarz defines this coefficient even if inversion does not preserve the Haar \(L^2\) norm. Almost invariant unit vectors give \(c_{\xi_i}\to1\) uniformly on compact sets. Conversely, if \(\xi_i\) are merely in the unit ball and their coefficients tend compact-uniformly to one, evaluation at \(e\) gives \(\|\xi_i\|_2^2\to1\). Normalize the nonzero tail to \(\eta_i\). Its coefficients still tend to one, and

This proves the complete unit-ball coefficient formulation, including normalization.

Define \(C^*(G)\) by the supremum norm over integrated strongly continuous unitary representations of \(L^1(G)\), and \(C_r^*(G)\) by its regular representation. We now prove

Suppose almost invariant unit vectors \(\xi_i\) exist. For any strongly continuous unitary representation \(\pi\) on \(K\), the unitary

on \(L^2(G;K)\) conjugates \(\lambda_s\otimes\pi(s)\) to \(\lambda_s\otimes1\). Indeed \(\pi(t)^{-1}\pi(s)\pi(s^{-1}t)=1\). No separability of \(K\) is needed: begin with finite sums of compactly supported scalar functions times fixed vectors. On compact sets the strongly continuous vector orbits have separable metric image and are Bochner measurable. The isometric formula and density extend the unitary.

Compress by \(J_iv=\xi_i\otimes v\). With \(d_i(s)=\langle\lambda_s\xi_i,\xi_i\rangle\), the integrated compression differs from \(\pi(f)\) by at most

Compact-uniform convergence treats a compact part and \(|d_i-1|\leq2\) treats the \(L^1\) tail. Each compression has norm at most \(\|\lambda(f)\|\), so \(\|\pi(f)\|\leq\|\lambda(f)\|\). Taking the supremum proves equality of the norms and faithfulness.

Conversely, faithfulness lets the trivial representation \(\epsilon(f)=\int f\) define a state on \(C_r^*(G)\). Extend it, through unitization if necessary, to a norm-one functional \(\psi\) on \(B(L^2(G))\) with \(\psi(1)=1\). It is positive. For self-adjoint \(A\),

for both signs of \(t\) forces \(\psi(A)\) to be real. If \(0\leq A\leq1\), the bound \(|\psi(1-A)|\leq1\) then gives \(\psi(A)\geq0\).

Take a positive contractive approximate identity \(e_j\) in \(C_r^*(G)\). We have \(\psi(e_j)=\epsilon(e_j)\to1\). Although \(\lambda_s\) need not belong to the nonunital algebra, \(\lambda_se_j\) does and \(\epsilon(\lambda_se_j)=\epsilon(e_j)\). These statements follow first on integrated \(L^1\) elements by translation and \(\int L_sf=\int f\), then by norm density. Moreover

Thus \(\psi(\lambda_s)=1\). The variance of \(\lambda_s-1\) and of its adjoint is zero; positivity and Cauchy–Schwarz imply \(\psi(B(\lambda_s-1))=\psi((\lambda_s-1)B)=0\). It follows that \(\psi(\lambda_sB\lambda_s^*)=\psi(B)\) for every \(B\). Restrict to multiplication operators: \(\lambda_sM_f\lambda_s^*=M_{L_sf}\). The restriction \(f\mapsto\psi(M_f)\) is a left invariant mean, proving (AMH.13). The multiplier step treats general locally compact groups, including the nonunital reduced algebra.

6. Averaging a von Neumann algebra action

Let \(G\) be amenable and let \(\alpha:G\to\operatorname{Aut}(M)\) be continuous in the pointwise predual-norm topology specified in SF-12. Let \(N=M^\alpha\). The algebra \(N\) is a von Neumann subalgebra: it is the intersection of the ultraweakly closed fixed spaces of the normal automorphisms \(\alpha_g\).

Choose a two-sided Haar mean \(m\). For \(x\in M\), define

The coefficient is bounded continuous, and its mean has absolute value at most \(\|\omega\|\|x\|\). The exact duality \(M=(M_*)^*\), proved in CP-06, therefore gives a unique element \(E_mx\in M\). This proves existence on the whole algebra, not merely on a dense subalgebra.

Linearity, preservation of the identity and positivity follow from (AMH.15). Complete positivity follows by applying the same argument to every positive normal functional on a matrix algebra \(M_n(M)\): its evaluation on the entrywise mean is the mean of its nonnegative evaluations on \([\alpha_g(x_{ij})]\). Positive normal functionals separate the positive cone, so that matrix is positive.

Left invariance gives \(\alpha_sE_mx=E_mx\), and right invariance gives \(E_m\alpha_sx=E_mx\). If \(b,c\in N\), then \(\alpha_g(bxc)=b\alpha_g(x)c\); evaluating normal functionals in (AMH.15) gives

Thus \(E_m\) is a unital completely positive \(N\)-bimodule projection onto \(N\). Such an averaging projection need not be normal.

Now assume a faithful normal state \(\varphi\) satisfies \(\varphi\alpha_g=\varphi\) for every \(g\). Equation (AMH.15) gives \(\varphi E_m=\varphi\). If \(0\leq x_i\uparrow x\) is a bounded net, put \(y=\sup_iE_mx_i\leq E_mx\). Normality of \(\varphi\) gives

Faithfulness forces \(E_mx=y\). By NP-04, \(E_m\) is normal. It is faithful as well: \(E_mx=0\), \(x\geq0\), implies \(\varphi(x)=0\) and hence \(x=0\).

This is the unique \(\varphi\)-preserving conditional expectation onto \(N\). In the GNS space let \(e_N\) be the projection onto \(\overline{\Lambda_\varphi(N)}\). For \(a\in N\), (AMH.16) gives

As \(\Lambda_\varphi(E_mx)\) belongs to that closed subspace, we obtain

Any other state-preserving bimodule projection satisfies the same identity. Faithfulness makes the GNS map injective, proving uniqueness. Consequently the expectation is independent of the chosen two-sided mean.

7. Strong averages and complete modular domains

Under the invariant faithful normal state hypothesis, write \(H=H_\varphi\), \(\Omega=\Lambda_\varphi(1)\), and

State invariance makes this unitary, with inverse \(U_{g^{-1}}\), and \(U_gxU_g^*=\alpha_g(x)\). Strong continuity follows on the dense GNS range from

at the identity, and then on every vector by uniform boundedness.

The fixed-vector space of \(U\) is exactly \(e_NH\). One inclusion follows because \(U\) fixes \(\Lambda_\varphi(N)\). If \(\eta\) is fixed, the vector functional \(x\mapsto\langle x\Omega,\eta\rangle\) is normal in the faithful normal GNS representation. Equation (AMH.15) gives

Using (AMH.17) and density, \(\eta=e_N\eta\).

Let \(p_i\) be a left Reiter net from AMH-03. To retain the direction of that invariance, average the inverse action:

The first integral is the strong vector integral. The second is the ultraweak integral, equivalently the adjoint of the Bochner integral of the predual maps. Norm continuity of predual orbits and the sigma-compact support of an \(L^1\) density make that integral legitimate. Each \(A_i\) is a normal unital completely positive map preserving \(\varphi\). For all \(h\),

Indeed substitute \(g=hk\) in the first term; left Haar measure is unchanged. The closed span of the vectors \((U_h-1)\xi\) is the orthogonal complement of the fixed vectors, since being orthogonal to every such vector is exactly being fixed by every \(U_h^*\). The contractions \(B_i\) vanish asymptotically on that span and equal the identity on fixed vectors. Hence

The GNS identity behind the second assertion follows by testing normal vector functionals in the ultraweak integral. Using \(\int p_i(g)U_g\) with only a left Reiter condition would not justify the coboundary estimate (AMH.19).

There is also exact modular compatibility, beyond this \(L^2\) limit. The initial Tomita map sends \(x\Omega\) to \(x^*\Omega\). Each \(U_g\) preserves that core and commutes with its involution. Graph closure, adjoints and uniqueness of polar decomposition give

with equality of domains, as in ANC-06. Every spectral projection of \(\Delta_\varphi\) commutes with every \(U_g\); it and its complement therefore preserve the fixed-vector space. Consequently \(e_N\) commutes with that spectral measure and with \(J_\varphi\), and

For clarity, this is spectral reduction: the domain is the set of vectors with finite integral of \(\lambda^{2\operatorname{Re}z}\) against their spectral measure. Orthogonal projection to a reducing subspace decreases that integral. No boundedness of \(\Delta_\varphi\) or of its powers is assumed.

Put \(\psi=\varphi|_N\). Its GNS space identifies with \(e_NH\), and \(\Omega\) remains cyclic and separating there. The projection \(e_N\) commutes with the closed \(S_\varphi\) and maps its graph core \(M\Omega\) into \(N\Omega\) by (AMH.17). Projecting any graph-core approximating sequence shows that \(N\Omega\) is a graph core for \(S_\varphi|_{e_NH}\). Therefore

This proves equality of the full closed operators and their maximal domains.

Finally the same unitary calculation gives \(\alpha_g\sigma_t^\varphi=\sigma_t^\varphi\alpha_g\). The normality of \(\sigma_t^\varphi\) allows it to be moved through the coefficient definition (AMH.15), yielding

The restriction identity follows also directly from (AMH.22) on \(e_NH\). This is the modular meaning of the expectation constructed by amenable averaging.

8. Examples that separate the hypotheses

A mean can destroy normality. Let \(\mathbb Z\) translate \(\ell^\infty(\mathbb Z)\). Normalized indicators of \([-n,n]\) have translation boundary tending to zero for every finite set of translations, so the group is amenable by AMH-03. Its fixed algebra is the constants, and (AMH.15) is \(E(f)=m(f)1\). Every singleton has the same mean value \(c\geq0\); finite disjoint unions force \(kc\leq1\) for every \(k\), hence \(c=0\). Thus

This expectation is not normal. There is no invariant normal state: its point masses would all equal the same \(c\), whereas countable additivity requires their sum to be one. This explains precisely why the preserved normal state in AMH-06 changes the conclusion.

A state-preserving average need not be tracial. In \(M_2(\mathbb C)\), set

Let \(\alpha_k=\operatorname{Ad}(u^k)\). Since \(uD=Du\), the state is invariant, and the fixed algebra is the diagonal algebra. Its expectation is

Identify GNS vectors with \(xD^{1/2}\) in Hilbert–Schmidt space. The involution is \(S\xi=D^{-1/2}\xi^*D^{1/2}\); its polar factors are

Hence the eigenvalues on \(E_{12},E_{21}\) are \(1/2,2\); both diagonal matrix units have eigenvalue one. The Hilbert projection \(e_N\) deletes the off-diagonal coordinates, commutes with \(\Delta\), and leaves the restricted modular operator equal to one. All domains here are the whole finite-dimensional spaces, in agreement with (AMH.22), while the modular operator on \(M\) is nontrivial.

The inverse-action averages in (AMH.18) with \(p_n=1_{[-n,n]}/(2n+1)\) multiply each off-diagonal entry by one of the conjugate scalars

Thus

This gives an exact norm test for the averaging direction, the unequal state weights and the fixed-space projection.

Compact-uniform estimates are stronger than separate level choices. In AMH-04, the layer identity controls an integral over levels for each translation. It does not directly select a single level working for an uncountable compact set. The positive-measure good set \(B_a\), followed by \(D\subset B_aB_a^{-1}\), is the step that makes one compact Følner set work for all the prescribed translations. Neither a countable dense subset of \(G\) nor an intersection of uncountably many full-measure sets appears in that argument.

References

M. Takesaki, Theory of Operator Algebras III, XIII §4, opening convolution conventions, Proposition 4.1, Definition 4.2, Proposition 4.3, Proposition 4.5 and Theorems 4.6–4.7, pp. 60–70. The proofs above retain all five Reiter formulations, the unit-ball coefficient condition and the full locally compact Følner conclusion.

The programme proofs linked in the introduction give the Haar and operator-algebra foundations used at each step. The averaging and modular-domain arguments are written above for a continuous action preserving a faithful normal state; the example without such a state shows the limitation of a general invariant mean.