Kadison–Kastler stability and its spatial boundaries

Source and verification

This article explains family 289 of the October 6, 2026 OpenAI collection. Research conclusions are attributed to its manuscripts. The review covers statements, introductions, and selected proof mechanisms; it is not an independent verification of the complete proofs.

Overview

How much of an operator algebra is determined by its position inside the bounded operators on a Hilbert space? This family separates three questions. Its first manuscript claims universal stability for two-sided perturbations of von Neumann algebras. The companions show why one-sided approximation and norm-closed algebras require different conclusions. The mechanisms are a correction of approximately multiplicative maps, an almost-commuting-matrix obstruction, and a sequence algebra that remembers how an inclusion sits inside its ambient algebra.

🏷️ Three perturbation questions

Throughout, algebras are complex, unital, and share the identity of their represented Hilbert space. A concrete $C^{\ast }$-algebra is an adjoint-closed subalgebra of $\mathcal{B}(H)$ closed in operator norm. A von Neumann algebra is also closed in the weak operator topology, meaning convergence of all coefficients $⟨T\xi ,\eta ⟩$. For a unital adjoint-closed algebra $A$, its weak closure is its double commutant $A^{\prime \prime }$, where $A^{\prime }$ consists of the operators commuting with every element of $A$.

Write $A_1$ for the closed unit ball. Define the directed unit-ball distance and the symmetric Kadison–Kastler distance by

$$\begin{array}{c}\alpha (A,B)=\underset{a\in A_1}{\sup }\mathrm{dist}(a,B_1),\\ \begin{array}{rl}d(A,B)& =\max \{\alpha (A,B),\\ & \alpha (B,A)\}.\end{array}\end{array}$$

The one-sided near-inclusion gap instead is

$$\gamma (A,B)=\underset{a\in A_1}{\sup }\mathrm{dist}(a,B).$$

Here the approximant in $B$ need not be a contraction. Rescaling it gives $\gamma (A,B)\le \alpha (A,B)\le 2\gamma (A,B)$; there is no bound on the reverse gap from this alone.

A spatial conjugacy is an equality $uA{u}^{\ast }=B$ for a unitary on the given $H$. A spatial embedding replaces equality by inclusion. Both retain information about the representation beyond abstract algebraic isomorphism. Moreover,

$$\Vert ua{u}^{\ast }-a\Vert \le 2\Vert u-I\Vert \Vert a\Vert .$$

Thus a small implementing unitary supplies small distances. The question is whether distances force such a unitary.

Setting	Smallness hypothesis	Manuscript conclusion
Von Neumann algebras	Two-sided $d(M,N)$	Uniform near-identity conjugacy
Von Neumann algebras	One-sided $\gamma (M,N)$	Embeddings can exist with every implementer bounded away from $I$
Norm-separable $C^{\ast }$-algebras	Two-sided $d(A,B)$	Arbitrarily close examples with no spatial conjugacy

The positive statement has the quantifiers

$$\begin{array}{c}\forall \epsilon >0\exists \delta >0\forall H,M,N:\\ d(M,N)<\delta ⟹\exists u\in \mathcal{U}(H),\\ uM{u}^{\ast }=N,\Vert u-I\Vert <\epsilon .\end{array}$$

The tolerance depends only on $\epsilon$, before the algebras, their types, representations, or Hilbert-space cardinalities are chosen. The first manuscript claims this statement without separability assumptions. (OpenAI, 2026)

🏷️ Matrix control and correction of a positive map

For a linear map $F$, its amplification $F_n$ acts entrywise on operator matrices. The completely bounded norm is $\Vert F{\Vert }_{\mathrm{cb}}={\sup }_n\Vert F_n\Vert$. A unital completely positive map, abbreviated UCP, preserves the identity and positivity at every matrix size. Such maps are complete contractions, but need not preserve products.

For von Neumann algebras $P,Q$, the positive proof repeatedly constructs a UCP map $F:P\to Q$ close to a faithful normal representation $\rho :P\to \mathcal{B}(H)$. Here normal means ultraweakly continuous. Its correction criterion assumes

$$\begin{array}{c}\Vert F-\rho {\Vert }_{\mathrm{cb}}\le e<10^{-3},\\ \underset{y\in Q_1}{\sup }\mathrm{dist}(y,\rho (P{)}_1)\le g,\\ g+e<1.\end{array}$$

Why is the reverse gap present? Complete isometry of $\rho$ gives $\Vert F_n(X)\Vert \ge (1-e)\Vert X\Vert$, so $F$ is injective with closed range $V$. Every contraction of $Q$ has distance at most $g+e<1$ from $V$. A proper closed linear subspace would have a quotient map of norm one. Therefore $V=Q$.

Now transport multiplication through the linear bijection:

$$m(x,y)=F^{-1}(F(x)F(y)).$$

This is an associative product. At all rectangular matrix sizes, inserting the product-preserving map $\rho$ gives

$$\begin{array}{rl}& \Vert F(XY)-F(X)F(Y)\Vert \\ & \le 3e\Vert X\Vert \Vert Y\Vert ,\\ & \Vert m-m_P{\Vert }_{\mathrm{cb}}\le \frac{3e}{1-e}.\end{array}$$

For a bilinear map, the displayed norm controls sums of products obtained by multiplying arbitrary rectangular operator matrices; it is the Haagerup product norm used by the manuscript. An imported multiplication-perturbation theorem corrects $m$ by a linear change of coordinates preserving adjoints. Composing with $F$ produces a normal $\ast$-isomorphism $\theta :P\to Q$ satisfying $\Vert \theta -\rho {\Vert }_{\mathrm{cb}}\le 32e$.

To implement it on the original Hilbert space, place $\rho$ and $\theta$ diagonally on $H\oplus H$. The switch $(\xi ,\eta )\mapsto (\eta ,\xi )$ has commutator controlled by $\Vert \rho -\theta {\Vert }_{\mathrm{cb}}$. A commutant-distance estimate provides an off-diagonal intertwiner close to the identity. Its unitary polar part gives

$$\begin{array}{c}u\rho (x)u^{\ast }=\theta (x),\\ \Vert u-I\Vert \le 128e.\end{array}$$

The original UCP map need not be normal. This matters when it is obtained by taking weak limits. (OpenAI, 2026)

🏷️ The positive stability architecture

A factor is a von Neumann algebra whose center consists of scalar operators. A ${\mathrm{II}}_1$ factor has a faithful normal tracial state and no minimal nonzero projections. Its standard trace representation is on $L^2(M,\tau )$, with left multiplication and cyclic vector $\Omega =1$; right multiplication supplies the commuting action.

For finite factors, the source combines two separate inputs. An absolute commutator-amplification estimate converts ordinary norm control into matrix control. A boundary argument constructs the approximately identity isomorphism to which that estimate can be applied. A derivation estimate alone does not choose a multiplicative matching between nearby unit balls.

The factors are first moved into common tracial position, sharing $\Omega$ and the left and right actions of an irreducible hyperfinite subfactor $R$. Hyperfinite here means generated in the weak topology by increasing finite matrix algebras; irreducible means $R^{\prime }\cap M=\mathbb{C}I$.

Choose a countable strongly dense subgroup $G\subset \mathcal{U}(M)$ and nearby unitaries $v_g\in N$. Their multiplication defects are small. The source corrects them over an amenable nonsingular action on a probability space $D$, obtaining measurable fields with

$$W_g(x)W_h(g^{-1}x)=W_{gh}(x).$$

This is a cocycle over the action. It does not assert that the possibly nonamenable group $G$ admits the same averaging on a point.

Two boundary comparisons are used. One intertwines the separate left and right actions on $D^2$; another produces actual unitary trace vectors inside $N$. Double ergodicity forces their combined equivariant Hilbert-space vector field to be constant. At a suitable boundary pair, the resulting Hilbert-space unitary sends each $g\Omega$ to $b_g\Omega$, with $b_g\in \mathcal{U}(N)$ uniformly close to $g$.

Linearity and the separating property of $\Omega$ extend this assignment to a unital linear map $\vartheta :M\to N$. Applying it to $e^{itx}$ for self-adjoint $x$, then comparing the first two Taylor coefficients of the unitary identity, shows preservation of adjoints and squares. Thus it is a Jordan map. The reverse gap proves surjectivity. The factor target forces either multiplication preservation or multiplication reversal. The latter would make all commutators small, contradicting two anticommuting symmetries in a copy of $M_2(\mathbb{C})$.

Transporting this isomorphism back gives ordinary norm closeness on the original $H$. Applying commutator amplification to its paired representation upgrades that bound to complete boundedness; the switch-and-polar argument gives the desired small unitary.

For infinite factors, the manuscript uses modular theory. A cyclic separating vector defines the closed antilinear operator $x\Omega \mapsto x^{\ast }\Omega$; its polar decomposition yields a positive modular operator $\Delta$ and automorphisms implemented by ${\Delta }^{it}$. After an auxiliary tensor preparation, a small change of position makes both algebras invariant under one modular action.

Their crossed products, generated by the algebras and translations implementing that action, can then be compared. One is a semifinite factor with finite corners. The finite-factor result conjugates corresponding corners and hence the crossed products. Averaging gives a UCP map back to the base algebras; slicing the auxiliary tensor factor gives another such descent. The correction criterion supplies the spatial conjugacies.

Finally, center decomposition assembles the factor results. For nonseparable $H$, finite collections of operators and vectors are placed in suitable separable reducing subspaces. Local small unitaries are extended by the identity. The proof takes a pointwise weak limit of their conjugation maps, which remains UCP and close at all matrix levels. Finite coefficient tests put its range in $N$. It does not take a weak limit of unitaries and declare that limit unitary. The UCP criterion finishes the argument with tolerances chosen before all finite tests. (OpenAI, 2026)

🏷️ One-sided gaps and almost commuting matrices

The first companion constructs separable examples with $\gamma (M_n,N_n)\to 0$, but an absolute ${\epsilon }_0>0$ such that every unitary implementing $u{M}_n{u}^{\ast }\subseteq N_n$ has $\Vert u-I\Vert \ge {\epsilon }_0$ for all sufficiently large $n$. An embedding does exist in every example. (OpenAI, 2026a)

The finite-dimensional obstruction begins with clock and shift matrices for $n\ge 3$:

$$\begin{array}{c}{U}_n{e}_j={\omega }_n^j{e}_j,\\ V_n{e}_j=e_{j+1\mathrm{mod}n},\\ {\omega }_n=e^{2\pi i/n},\\ U_n{V}_n={\omega }_n{V}_n{U}_n.\end{array}$$

Their commutator norm tends to zero. Yet they cannot approach commuting unitary pairs. For a pair whose multiplicative commutator avoids $-1$, define

$$\kappa (U,V)=\frac{1}{2\pi i}\mathrm{Tr}\log (UV{U}^{\ast }{V}^{\ast }).$$

The trace is unnormalized and the logarithm principal. The determinant of a multiplicative commutator is one, so $\kappa$ is an integer and remains constant along paths avoiding the cut. For the clock–shift pair it equals one; for a commuting pair it equals zero. Hypothetical close commuting approximants can be joined by short unitary logarithm paths whose commutators stay near the identity, giving the contradiction.

To turn this into an algebra perturbation, the source constructs circle-indexed representations of the countably generated free group $F$ on $K={\ell }^2(F)$:

$$\begin{array}{c}{\pi }_z(g)=\sum _{k\in \mathbb{Z}}{z}^k{A}_k(g),\\ \underset{g}{\sup }\Vert A_k(g)\Vert \le c_k,\\ \sum _k(1+|k|)c_k<\infty .\end{array}$$

Along distinct generators, ${\pi }_z(s_j)$ converges weakly to $\frac{1}{2}z{p}_e$, where $p_e$ projects onto the identity basis vector. Substituting $U$ and $V$ into the circle variable gives representations $a,b$ whose weak limits retain those matrices.

For $G=F\times F$, define $f(x)=a(x_1)b(x_2)$ and the diagonal unitary $W({\delta }_x\otimes v)={\delta }_x\otimes f(x)v$. Let $\lambda ,\rho$ be the commuting left and right regular representations of $G$. Set

$$\begin{array}{c}M=W(\lambda (G{)}^{\prime \prime }\otimes I)W^{\ast },\\ N=(\rho (G)\otimes I{)}^{\prime }.\end{array}$$

The map $f$ need not be a representation, but $W$ is still unitary, and $W^{\ast }MW\subseteq N$.

The essential estimate holds on the whole source unit ball. Reordering four Fourier expansions yields a bounded operator in $N$. Telescoping powers gives $\Vert U^a{V}^b-V^b{U}^a\Vert \le |a||b|\Vert UV-VU\Vert$, while summability of the coefficient first moments gives

$$\gamma (M,N)\le C\Vert UV-VU\Vert ,$$

with $C$ independent of matrix dimension. Bounds merely on generators would not establish this near inclusion.

A hypothetical sequence of embeddings tending to the identity would produce a representation of $F\times F$ by multiplying the conjugated left action with the right action. Its two factor algebras commute. Their weak limits recover operators close to supported copies of $U_n,V_n$.

Compression to an arbitrary finite subspace could destroy commutation. The proof first replaces the intended support projections by nearby spectral projections in the two commuting algebras. Their product is a common finite-dimensional projection. A small polar unitary identifies its range with the original matrix space; the compressed operators commute across adjoints, so their polar parts are commuting unitaries close to $U_n,V_n$. This contradicts $\kappa$. The construction supplies no reverse near-inclusion bound. (OpenAI, 2026a)

🏷️ Close norm-closed algebras with no spatial conjugacy

The second companion strengthens the negative conclusion while changing the class of algebras: for every $\epsilon >0$ it constructs norm-separable unital $C^{\ast }$-algebras $A,B$ on a common separable $H$ with $d(A,B)<\epsilon$ and $A^{\prime \prime }=B^{\prime \prime }$, but no unitary conjugating $A$ onto $B$. It does not assert abstract nonisomorphism. (OpenAI, 2026a)

Let $P,Q\subset D\subset \mathcal{B}(L)$, where $D$ is norm-separable and has scalar center. On ${⨁}_{n\ge 1}L$, form

$$\begin{array}{c}{\mathcal{E}}_D(P)=\{(x_n):x_n\in D,\\ x_n\to p\in P\text{ in norm}\}.\end{array}$$

This is $c_0(\mathbb{N},D)$ plus the constant $P$-valued sequences. Keeping a finite prefix and replacing its tail by an approximating constant proves one direction of

$$d({\mathcal{E}}_D(P),{\mathcal{E}}_D(Q))=d(P,Q).$$

Testing constant sequences proves the reverse direction. Both sequence algebras have commutant the bounded diagonal $D^{\prime }$-valued sequences, so their weak closures agree.

Their centers are the convergent scalar sequences. Thus any spatial conjugacy must permute the intrinsic minimal central coordinate projections. On each coordinate it is a unitary $U_n$ normalizing $D$. Pulling back a constant $b\in Q$ forces one sequence of these normalizers to satisfy

$$\Vert b-U_n{p}_b{U}_n^{\ast }\Vert ⟶0(p_b\in P).$$

The same $U_n$ works for every $b$, hence for each finite list simultaneously.

Now test the relative tensor identity

$$\begin{array}{c}\Vert \sum _i{p}_i{z}_i\Vert =\Vert \sum _i{p}_i\otimes z_i\Vert ,\\ p_i\in P,z_i\in D^{\prime }.\end{array}$$

Both sides are concrete operator norms, on $L$ and $L\otimes L$. The identity survives normalizers of $D$ and simultaneous norm limits: both error sums are bounded by ${\sum }_i\Vert p_i-q_i\Vert \Vert z_i\Vert$. If it holds for $P$ and fails for $Q$, the sequence algebras cannot be spatially conjugate.

The source realizes this with a tree deformation ${\pi }_t$ of the regular representation $\lambda$ of $F$, satisfying

$$\begin{array}{c}⟨{\pi }_t(s_i){\delta }_e,{\delta }_e⟩=t,\\ \underset{g}{\sup }\Vert {\pi }_t(g)-\lambda (g)\Vert \le c(t)\to 0.\end{array}$$

The construction uses unit vectors ${\xi }_e={\delta }_e$ and ${\xi }_x=t{\xi }_{p(x)}+\sqrt{1-t^2}{\delta }_x$, where $p(x)$ deletes the last letter. Their Gram matrix is $t^{d(x,y)}$, so left translation defines a unitary representation. The norm comparison reduces to the finite path from $e$ to $g$, with a geometric-series bound independent of path length. A globally bounded change of basis ${\delta }_x\mapsto {\xi }_x$ is not needed.

Put $P_t=C^{\ast }(\lambda (g)\otimes {\pi }_t(g):g\in F)$ and use ${\pi }_0=\lambda$. The common algebra $D$ is generated by $\lambda (g)\otimes I$, both second-factor representations, and all second-factor matrix units. Its commutant is $\lambda (F{)}^{\prime }\otimes I$, and its center is scalar. Absorption gives a unitary within $c(t)$ of $I$ conjugating $P_0$ to $P_t$.

For $P_0$, a tensor-coordinate change puts it on the second factor and $D^{\prime }$ on the first, proving the tensor identity. For $P_t$, choose $p_i=\lambda (s_i)\otimes {\pi }_t(s_i)$ and $z_i=\rho (s_i)\otimes I$. The identity vector gives

$$\begin{array}{c}\Vert \sum _{i=1}^m{p}_i{z}_i\Vert \ge mt,\\ \Vert \sum _{i=1}^m{p}_i\otimes z_i\Vert \le 2\sqrt{m}.\end{array}$$

The second estimate uses absorption and the creation/cancellation split of free shifts: orthogonal output ranges control the creation part by $\sqrt{m}$, and orthogonal input domains control the cancellation part by another $\sqrt{m}$. Taking $m>4/t^2$ violates the identity.

Consequently ${\mathcal{E}}_D(P_0)$ and ${\mathcal{E}}_D(P_t)$ have distance at most $2c(t)\to 0$ but no spatial conjugacy. The small unitary conjugating the limit subalgebras does not normalize $D$; the sequence construction is what makes their relative position an obstruction to every implementing unitary. (OpenAI, 2026a)

🏷️ Review scope

All three READMEs and main files were read, together with the complete principal sections of the near-inclusion paper and the sequence-algebra construction. The positive stability review covered its statements, foundations, selected finite-factor proof, modular and crossed-product interfaces, final finite-test argument, and comparison of uniformities. Its full free-word amplification, boundary, modular alignment, and measurable-selection proofs, and the external perturbation theorems, were not independently verified.

🏷️ Key steps and proof outlines

The reverse gap is used to prove surjectivity

Suppose the positive argument has constructed $F:P\to Q$ with $\Vert F-\rho {\Vert }_{\mathrm{cb}}\le e<1$, where $\rho$ is a faithful representation. Then $\Vert F(x)\Vert \ge (1-e)\Vert x\Vert$, so the range $V$ is closed. If $Q$ were larger than $V$, its quotient map $Q\to Q/V$ would have norm one: this is the definition of the quotient norm, or follows by scaling elements nearly attaining it. But the reverse gap says every element of $Q_1$ is within $g+e<1$ of $V$, making the quotient map’s norm strictly less than one. Thus $F$ is onto. (OpenAI, 2026c)

This is the exact logical point where a two-sided hypothesis is needed. Once $F^{-1}$ exists, associativity of the transported product is automatic, and the multiplication-perturbation input can correct it. A one-sided inclusion estimate does not provide this quotient contradiction.

Recover commuting matrices in a genuinely common corner

The near-inclusion companion obtains operators $T_i$ in two commuting von Neumann algebras, close to supported unitaries $E_i{\tilde{R}}_i$, where $E_1,E_2$ commute and $E_1{E}_2$ has the desired finite-dimensional range. Put ${\delta }_i=\Vert T_i-E_i{\tilde{R}}_i\Vert$. Then

$$\Vert T_i^{\ast }{T}_i-E_i\Vert \le {\delta }_i(2+{\delta }_i)=d_i.$$

If $d_i$ is small, the spectrum of $T_i^{\ast }{T}_i$ lies near zero or one. Its spectral projection $Q_i=1_{(1/2,\infty )}(T_i^{\ast }{T}_i)$ belongs to the corresponding algebra and satisfies $\Vert Q_i-E_i\Vert \le 2d_i$. Because the two algebras commute, $P=Q_1{Q}_2$ is a projection close to $E=E_1{E}_2$. (OpenAI, 2026b)

The operator

$$S=PE+(1-P)(1-E)=1+(P-E)(2E-1)$$

is close to one and intertwines $E$ with $P$. Its unitary polar part $J$ satisfies $JE{J}^{\ast }=P$ and remains close to one. This identifies the perturbed finite-dimensional corner with the original one without changing its dimension.

Why do the compressions commute if $Q_i$ need not commute with $T_i$? Write $A=Q_1{T}_1{Q}_1$ and $B=Q_2{T}_2{Q}_2$. The two compressed operators are $A{Q}_2$ and $Q_{1}B$; both orders of their product equal $AB$, and the same calculation works across adjoints. After transporting by $J$, their polar parts therefore remain in commuting $C^{\ast }$-algebras. They are commuting unitaries close to the original clock and shift matrices, contradicting the integer winding invariant. Arbitrary compression to the original range of $E$ would not have preserved this property.

Sequence algebras retain distance but expose spatial rigidity

For the norm-closed companion, let ${\mathcal{E}}_D(P)$ consist of $D$-valued sequences converging in norm to an element of $P$. Its directed distance to ${\mathcal{E}}_D(Q)$ equals that of $P$ to $Q$: for the upper bound retain a finite prefix and replace only the tail by an approximating $Q$-element; for the lower bound test constant $P$-sequences and take the limit of any proposed approximant. (OpenAI, 2026a)

If $Z(D)=\mathbb{C}1$, the center of this sequence algebra consists exactly of convergent scalar sequences, because commuting with every finitely supported $D$-sequence forces each coordinate to be scalar. Its minimal central projections are precisely the coordinate projections. A spatial conjugacy must therefore permute coordinates and supply coordinate unitaries $U_n$ normalizing $D$.

Pulling back a constant sequence $(b,b,\dots )$, with $b\in Q$, forces $U_n^{\ast }b{U}_n$ to converge in norm to some $p_b\in P$. This is a much stronger restriction than the original metric proximity. The source uses the tensor-norm identity

$$\Vert \sum _i{p}_i{z}_i\Vert =\Vert \sum _i{p}_i\otimes z_i\Vert ,z_i\in D^{\prime },$$

as an obstruction: normalizers preserve it, and pointwise norm limits preserve it for every finite family. The constructed $P$ has this identity and $Q$ does not. The sequence construction thus turns that asymmetric invariant into failure of every spatial conjugacy, while preserving arbitrarily small two-sided distance and the same weak closure. These arguments complement the positive correction mechanism rather than contradicting its von Neumann hypotheses.

See Also

• Kadison similarity and uniform commutators explains why scalar-to-matrix commutator bounds matter, while the present positive proof also needs a multiplicative matching of the two algebras.
• Reduced projections and traces develops the distinction between norm-closed group algebras, weak closures, and spectral functional calculus.

🏷️ Manuscripts and verification scope

• Universal strong Kadison–Kastler stability — pinned manuscript and companion source.
• Near Inclusions of von Neumann Algebras Without Small Spatial Embeddings — pinned manuscript and companion source.
• Close Separable C*-Algebras Without Spatial Conjugacy — pinned manuscript and companion source.

The family scope page describes selected formalization coverage. The documented selection covers universal near-identity unitary conjugacy for close unital von Neumann algebras, with a tolerance uniform over algebras, representations and Hilbert spaces. The two counterexample manuscripts are not listed in that scope page. No Lean build, axiom audit, or manuscript-to-formal-statement equivalence check was performed for these notes.

OpenAI collection provides prerequisites and the reading sequence; coverage register records completed and pending families.

🐻  OpenAI 2026a. Close Separable C^*-Algebras Without Spatial Conjugacy.

🐻  OpenAI 2026b. Near Inclusions of von Neumann Algebras Without Small Spatial Embeddings.

🐻  OpenAI 2026c. Universal strong Kadison–Kastler stability.