1 Introduction
Let M and N be von Neumann algebras acting on a complex Hilbert space H, both with identity I_H. Their one-sided near-inclusion gap is
The approximating element y is not required to be contractive. A small value of \gamma(M,N) says that every contraction in M is close to some element of N; it does not require these choices of approximants to preserve algebraic operations. If a unitary u satisfies uMu^*\subseteq N, then \gamma(M,N)\leq 2\|u-I_H\|. The converse question asks whether a small near inclusion can always be obtained in this way.
The underlying perturbation program began with Kadison and Kastler [8]. They measured the distance between operator algebras by the Hausdorff distance between their unit balls in the operator norm, now called the Kadison–Kastler distance, and proposed that sufficiently close von Neumann algebras should be related by a unitary close to the identity. Christensen, Johnson, and Raeburn–Taylor established the amenable case in the 1970s [2, 7, 10]. Christensen subsequently introduced near inclusions and proved a spatial embedding theorem for an amenable source algebra and an arbitrary target [3, Definition 2.1 and Theorem 4.3]. In particular, sufficiently small near inclusions in this setting admit implementing unitaries whose distance from the identity is bounded linearly by the near-inclusion tolerance.
We use the following formulation of the unrestricted one-sided question posed by Cameron, Christensen, Sinclair, Smith, White, and Wiggins in the paragraph preceding [1, Theorem 2]:
For every \varepsilon>0, is there a \delta>0 such that, for every complex Hilbert space H and every pair of von Neumann algebras M,N\subseteq\mathcal B(H) with common identity I_H, \gamma(M,N)<\delta implies the existence of a unitary u\in\mathcal B(H) satisfying
Here \delta must be independent of the algebras and their representations. We answer this question negatively.
Theorem 1.1. There exist \varepsilon_0>0, separable complex Hilbert spaces H_n, and von Neumann algebras M_n,N_n\subseteq\mathcal B(H_n) with common identity I_{H_n} such that
but, for all sufficiently large n, every unitary u\in\mathcal B(H_n) satisfying uM_nu^*\subseteq N_n obeys
In these examples a unitary embedding of M_n into N_n does exist; the obstruction concerns its distance from the identity. The theorem therefore separates normwise approximation from small spatial implementation even on separable Hilbert spaces. Its hypothesis is one-sided. The construction supplies no reverse near inclusion, and hence does not give a counterexample to the two-sided Kadison–Kastler conjecture. The positive results for nonamenable factors in [1], which concern two-sided perturbations under additional structural hypotheses, address a different assertion.
The proof transfers a classical finite-dimensional obstruction into the near-inclusion problem. Voiculescu’s clock and shift matrices U_n,V_n\in\mathcal B(\mathbb C^n) satisfy \|U_nV_n-V_nU_n\|\to0, but cannot be approximated in norm by commuting unitary pairs [11]. Exel and Loring gave an elementary winding-number proof of this phenomenon [5]; we use its trace-logarithm form [4, Lemma 3.1]. We recall the short obstruction argument in Section 4.
To place these matrices in operator algebras, we use the free group \F on countably many generators. The starting representation is the real-parameter Pytlik–Szwarc representation [9, §2.3, Equation (5)], whose cyclic coefficient is Haagerup’s positive-definite function g\mapsto t^{|g|}, 0<t<1, where |g| is reduced word length [6, Lemma 1.2]. Section 2 gives a self-contained tree realization at t=1/2 and adds a circle parameter z. The norms of its Fourier coefficients are bounded, uniformly over the group, by a sequence with summable first moment. Along successive distinct free generators, the values of the representations converge in the weak operator topology to tz times the projection onto the identity vector in \ell^2(\F). Substituting U_n and V_n for the circle parameter therefore produces two unitary representations of \F whose weak limits retain the original matrices.
Two further steps connect this representation family to the theorem. In Section 3, a diagonal unitary built from the two representations conjugates a copy of the group von Neumann algebra of \F\times\F. The target N_n is the commutant of an amplified right regular representation of this product group. Reordering the Fourier expansions gives an approximant in N_n and an error bounded by a constant times \|U_nV_n-V_nU_n\|. Crucially, this estimate holds on the entire unit ball of the source algebra. In Section 4, a hypothetical sequence of spatial embeddings tending to the identity places the conjugated left regular representation in N_n, where it commutes with the right regular representation. Their pointwise product is then a representation of \F\times\F, so the von Neumann algebras generated by its two factors commute. Weak operator limits in these algebras recover operators close to copies of U_n and V_n supported on projections. Perturbing those projections, compressing to a common finite-dimensional corner, and taking polar parts would yield commuting unitary matrices close to U_n,V_n, contradicting the winding-number obstruction.
2 A circle family with uniform Fourier bounds
We first construct representations in which a scalar circle parameter can later be replaced by a unitary matrix. Two properties are needed: Fourier coefficients with a summable first moment, uniformly over the group, and a weak operator limit that retains the parameter.
Let F be the free group on generators s_j, j\geq 1, with identity e. Put K=\ell^2(F), write (\delta_x)_{x\in F} for its standard orthonormal basis, and let p_e be the projection onto \mathbb C\delta_e. We use the left regular representation l(g)\delta_x=\delta_{gx}. Throughout the construction,
Proposition 2.1. There are unitary representations \pi_z:F\to\B(K), indexed by z\in\mathbb T, and operators A_k(g)\in\B(K), indexed by k\in\mathbb Z and g\in F, such that
The series in (2.1) converges absolutely in operator norm, uniformly in g and z, and
Proof. The underlying representation is the real-parameter Pytlik–Szwarc representation [9, Section 2.3, Equation (5)]; its cyclic coefficient is the positive-definite function g\mapsto t^{|g|} of Haagerup [6, Lemma 1.2]. We give the tree construction before introducing the circle parameter.
Use the Cayley tree whose edges join x to xs_j^{\pm1}, rooted at e, and write p(x) for the parent of x\ne e. Define
All these vectors have norm one. If v is the last common vertex of the paths from e to x and y, the two tails beyond v are orthogonal to each other and to \xi_v. Thus
where d is the tree distance. Their linear span contains every \delta_x, by (2.4). Left multiplication preserves d, so the rule
extends to a unitary on K. These unitaries form a representation.
The difference from the regular representation is confined to a single finite path. To see this precisely, put T_g=\pi(g)l(g)^*. For y\ne g, (2.4) gives
The vertex g p(g^{-1}y) is the first step from y toward g. Let e=h_0,h_1,\ldots,h_d=g be the path from e to g. For y outside this path, the first steps toward e and g coincide; hence T_g\delta_y=\delta_y. On the path,
In particular, the path subspace and its orthogonal complement reduce T_g. Expanding the vectors along the path gives
By (2.6), the subdiagonal entries are -t; every other nonzero entry in row a, column b is t^{|a-b|} times 1, r, or r^2. Thus the matrix entries on this subspace satisfy
This also includes g=e, when T_e=I.
Let q:F\to\mathbb Z be the homomorphism with q(s_j)=1 for every j, and define D_z\delta_x=z^{-q(x)}\delta_x. The desired representation is
The first expression proves that it is a unitary representation. The second follows from D_zl(g)D_z^*=z^{-q(g)}l(g) and supplies its Fourier bounds.
On the path subspace, the entry in row a, column b acquires the Laurent factor z^{q(h_b)-q(h_a)}. Each edge changes q by 1 or -1, so an entry of exponent k can occur only when |a-b|\geq|k|. For each row or column there are at most two entries at any given distance m. By (2.7), the absolute row and column sums of the coefficient of z^k are therefore at most
Outside the path the operator is the identity, which contributes only to the coefficient at k=0. The row and column sum bound for operator norm, followed by multiplication by l(g), proves (2.1). For each fixed g this is a finite Laurent expansion; the common bounds c_k give the asserted uniform absolute convergence and (2.3).
For any fixed x\ne e, all sufficiently large j satisfy s_jx\notin\{e,s_j\}, and then \pi_z(s_j)\delta_x=\delta_{s_jx}. These basis-vector calculations, together with the uniform norm bound \|\pi_z(s_j)\|=1, prove (2.2).
3 From a unitary pair to a near inclusion
We now use Proposition 2.1 to encode two unitaries U,V on a nonzero finite-dimensional Hilbert space D. The resulting von Neumann algebras will have one-sided gap bounded by a constant times \|UV-VU\|.
3.1 Matrix substitution and the algebras
On L=K\otimes K\otimes D, define
The series converge absolutely in norm. A spectral decomposition U=\sum_z zP_z rewrites the first expression as \sum_z\pi_z(g)\otimes I_K\otimes P_z, so a is a unitary representation. Diagonalizing V gives the same conclusion for b. There is no requirement that their ranges commute. The weak limits from Proposition 2.1 become
Only finite spectral sums are involved, so these limits follow directly from (2.2).
Put G=F\times F, S=\ell^2(G), and H=S\otimes L. Our conventions for the commuting left and right regular representations on S are
For x=(x_1,x_2)\in G, put f(x)=a(x_1)b(x_2) and define the diagonal unitary W by
Although a and b are representations, f need not be one. Define
Primes denote commutants. Here \lambda(G)''\otimes I_L means the algebra of all X\otimes I_L with X\in\lambda(G)''; it is weak operator closed, as is seen by taking matrix blocks relative to an orthonormal basis of L. Thus M,N are von Neumann algebras with common identity I_H. Notice also that W^*MW\subset N, so these examples always admit a spatial embedding.
The exactly commuting case explains the choice of W. If U and V commute, then so do the ranges of a and b, and f is a representation of G. In that case
so M\subset N. When U and V only nearly commute, we must control the error on the entire unit ball of M; estimates for the group elements alone would not suffice.
3.2 A norm estimate on the whole algebra
Proposition 3.1. For the algebras in (3.3),
Moreover, for every g\in G,
The constant is independent of D,U,V.
Proof. Write \nu=\|UV-VU\|. Fix X\in\lambda(G)'' and denote its scalar matrix entries by X_{x,y}, for x,y\in G. Since X commutes with \rho(G), these entries are invariant under common right translation:
The operator Z=W(X\otimes I_L)W^* has \B(L)-valued blocks
We seek an approximant Y with the reordered blocks
The last equality uses the representation laws for a and b separately. Since (xq)(yq)^{-1}=xy^{-1}, these blocks are right-translation invariant by (3.6). Thus, once boundedness is established, Y commutes with \rho(G)\otimes I_L and belongs to N.
The Fourier bounds establish boundedness and control the reordering in operator norm. On S\otimes K\otimes K, let D_{k,m} be the diagonal operator whose block at x\in G is A_k(x_1)\otimes A_m(x_2). Then \|D_{k,m}\|\leq c_kc_m. Set
The A_k coefficients of a and b act on separate copies of K; only their matrix factors U^k,V^m act on the same space. In the decomposition H=(S\otimes K\otimes K)\otimes D, expansion of the four factors therefore gives
Both sums converge absolutely because
Their blocks are precisely the expressions above, which also justifies the definition of Y.
For integers a,b, telescoping gives
For positive powers, expand a commutator one factor at a time; negative powers follow from [U^{-1},V]=-U^{-1}[U,V]U^{-1} and its analogue for V^{-1}, where [A,B]=AB-BA. Consequently the two matrix factors in (3.8) and (3.9) differ by at most |m-p|\,|h|\nu. Hence
Since Y\in N and \|Z\|=\|X\|, this proves (3.4); no bound of 1 on \|Y\| is needed.
If X=\lambda(g), its nonzero entries have x=gy. The representation laws for a and b turn (3.7) into Y=\lambda(g)\otimes a(g_1)b(g_2)=\lambda(g)\otimes f(g). Thus (3.5) is the same estimate for this choice of X.
4 Recovering the matrix obstruction
The preceding construction turns a small commutator into a small one-sided gap. We now show that a spatial embedding close to the identity would turn the original matrix pair into a nearby commuting pair. Choosing the clock and shift matrices will make this impossible.
4.1 The winding obstruction
The matrices below are Voiculescu’s almost commuting unitary pair [11]. We include the logarithmic winding argument, in the form developed in [5, 4].
Lemma 4.1. For n\geq3, let U_n,V_n be the unitaries on \mathbb C^n given by
Then \nu_n:=\|U_nV_n-V_nU_n\|=|\omega_n-1|\to0. There is no infinite subsequence along which commuting unitaries \widehat U_n,\widehat V_n on \mathbb C^n satisfy
Proof. The identity U_nV_n=\omega_nV_nU_n proves the first assertion. For any unitary matrices U,V whose commutator UVU^*V^* has no eigenvalue -1, define
using the principal logarithm and the unnormalized trace. The determinant identity
shows that \kappa(U,V) is an integer. It is continuous, and hence constant, along any path on which the logarithm remains defined. For the pair in (4.1), its value is 1, since U_nV_nU_n^*V_n^*=\omega_n I and 0<2\pi/n<\pi. For a commuting pair its value is 0.
Suppose the asserted approximants existed along a subsequence. For all sufficiently large n on that subsequence, write U_n^*\widehat U_n=e^{iA_n} and V_n^*\widehat V_n=e^{iB_n}, with A_n,B_n self-adjoint and spectra in (-\pi,\pi). The paths
stay within the respective endpoint distances from U_n,V_n. Indeed, for |\theta|<\pi and 0\leq s\leq1, |e^{is\theta}-1|\leq|e^{i\theta}-1|. A four-factor telescoping estimate gives, uniformly in s,
The principal logarithm is therefore defined along the whole path for all sufficiently large n. Its integer changes from 1 to 0, a contradiction.
4.2 A finite-dimensional corner
Compression of commuting operators to a prescribed subspace need not preserve commutation. The next lemma avoids this difficulty by replacing each support projection by a nearby projection in its corresponding algebra and then using their common range.
Lemma 4.2. For each n, let E_{1n},E_{2n} be commuting projections on a complex Hilbert space H_n, with D_n=E_nH_n nonzero and finite-dimensional, where E_n=E_{1n}E_{2n}. Let \widetilde R_{1n},\widetilde R_{2n} be unitaries commuting with both projections, and write R_{in}=\widetilde R_{in}|_{D_n}. Suppose \mathcal A_{1n},\mathcal A_{2n}\subset\B(H_n) are commuting unital von Neumann algebras and there are T_{in}\in\mathcal A_{in} such that
Then, for all sufficiently large n, there are commuting unitaries \widehat R_{1n},\widehat R_{2n} on D_n with
Proof. We first replace the intended projections by nearby projections in the commuting algebras. Set \delta_{in}=\|T_{in}-E_{in}\widetilde R_{in}\|. Since (E_{in}\widetilde R_{in})^*(E_{in}\widetilde R_{in})=E_{in},
For sufficiently large n, take the spectral projection
The spectrum of T_{in}^*T_{in} lies in [0,d_{in}]\cup[1-d_{in},1+d_{in}]. This follows from the resolvent bound for a self-adjoint operator within d_{in} of a projection. Functional calculus therefore gives
Since the two algebras commute, P_n=Q_{1n}Q_{2n} is a projection, and
We identify this perturbed corner with D_n by a small unitary. The operator
tends to the identity in norm and satisfies S_nE_n=P_nS_n. It is invertible for large n. Its absolute value commutes with E_n, so its unitary polar part J_n=S_n(S_n^*S_n)^{-1/2} satisfies
In particular, P_nH_n has exactly the dimension of D_n.
The compressions P_nT_{1n}P_n and P_nT_{2n}P_n commute, including across adjoints. This needs a check because Q_{in} need not commute with T_{in}. Suppress n and put A=Q_1T_1Q_1\in\mathcal A_1 and B=Q_2T_2Q_2\in\mathcal A_2. The two compressions are AQ_2 and Q_1B. Since AQ_1=Q_1A=A, BQ_2=Q_2B=B, and every element of \mathcal A_1 commutes with every element of \mathcal A_2, both orders of their product equal AB. Replacing A or B by its adjoint proves the other commutations.
Transfer these compressions to D_n using (4.3):
The pair still commutes across adjoints. Moreover,
by comparing T_{in} with E_{in}\widetilde R_{in} before conjugation. Thus C_{in} is invertible for large n and \|C_{in}^*C_{in}-I_{D_n}\|\to0. Its unitary polar part
remains close to R_{in}. The unital C^*-algebras generated by C_{1n} and C_{2n} commute; continuous functional calculus puts each polar part in its respective algebra. Hence the two polar unitaries commute, as required.
4.3 Proof of the main theorem
Apply the construction of Section 3 to D=\mathbb C^n and the clock–shift pair of Lemma 4.1. Use subscripts n for the resulting objects L_n,H_n,a_n,b_n,f_n,W_n, M_n,N_n. All these Hilbert spaces are separable. By Proposition 3.1,
Proof of Theorem 1.1. We show that no infinite subsequence admits unitaries u_n with
Suppose otherwise and work on that subsequence. Define, for g\in G,
The first factor in the expression for \Sigma_n(g) lies in N_n, by (4.5). Thus the two representations being multiplied have commuting ranges, and \Sigma_n is a unitary representation of G. Proposition 3.1 gives the uniform estimate
Let \mathcal A_{1n} and \mathcal A_{2n} be the von Neumann algebras generated by the restrictions of \Sigma_n to F\times\{e\} and \{e\}\times F, respectively. These algebras commute.
We will apply Lemma 4.2 by taking weak limits of the generators in each factor. Under S=K\otimes K, write
Let E_{1n} use p_e in the first and third slots, and let E_{2n} use p_e in the second and fourth slots, with identities in all other slots. These projections commute, and E_n=E_{1n}E_{2n} has range naturally identified with \mathbb C^n. Let \widetilde R_{1n} and \widetilde R_{2n} act as U_n,V_n, respectively, on the final slot and as the identity elsewhere.
Set g_{1j}=(s_j,e) and g_{2j}=(e,s_j). On the i-th factor of S, the operator \lambda(g_{ij})\rho(g_{ij}) sends \delta_x to \delta_{s_jxs_j^{-1}}. For each fixed nonidentity word x, this conjugate leaves every finite subset of F as j\to\infty: for large j, the letter s_j occurs neither in x nor in the finitely many words under consideration. Hence the conjugation operators converge weakly to p_e in that slot. Combining this observation with (3.2) gives, for each fixed n and i,
Here tensor products of bounded weakly convergent sequences on separate slots converge weakly, as follows first on elementary tensors and then by density. The finite spectral decompositions used in (3.2) justify the matrix-valued factor.
For each i,n, choose a weak operator cluster point B_{in} of \Sigma_n(g_{ij}) as j\to\infty. Such a point exists by weak operator compactness of the closed unit ball of \B(H_n), and belongs to \mathcal A_{in}. Norm balls are weak operator closed, so (4.6) and (4.7) imply
Thus T_{in}=t^{-1}B_{in}\in\mathcal A_{in} satisfies (4.2). No product of weak limits is used: commutation comes from membership in the two commuting algebras. Lemma 4.2 now produces commuting unitary matrices on \mathbb C^n approaching U_n,V_n in norm. This contradicts Lemma 4.1, proving that (4.5) cannot hold on any infinite subsequence.
It follows that some \epsilon_0>0 bounds \|u-I_{H_n}\| from below for every implementing unitary and all sufficiently large n. Otherwise one could choose increasing n_j and implementing unitaries with distance less than 1/j, contrary to what we have just proved. Together with (4.4), this is Theorem 1.1.
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