# Null sequences and hypercentral commutants

*Original exposition by OpenAI Codex, GPT-6 Astra, Ultra. New text and figure: CC0-1.0.*

There are two commutation conditions for a bounded sequence in a von Neumann algebra. One tests fixed operators in the strong* topology; the other tests normal functionals in their norm. We keep these conditions distinct and determine exactly what an additional hypercentrality test gives. The finite tracial case admits a direct equivalence proof. A weighted tensor product shows why its proof cannot be transferred to a general state by changing notation.

Let \(M\) have a faithful normal state, let \(\omega\) be a free ultrafilter, and write \(\mathcal L=\ell^\infty(\mathbb N,M)\). Strong* limits below are along \(\omega\). We use the proved normal-predual duality and Hahn–Banach argument from Concrete preduals, the product construction in Infinite products and modular time, and the quotient conventions of [Multiplier ultraproducts and normal embeddings](https://github.com/KokunoYumeto/open-math-courses-public/blob/main/docs/courses/OA-APPROX/src/multiplier-ultraproducts-and-normal-embeddings.md). The normal-functional decomposition and faithful-state topology used below are proved in [Central-sequence foundations, NT1–NT2](https://github.com/KokunoYumeto/open-math-courses-public/blob/main/docs/courses/OA-APPROX/components/central-sequence-foundations.md). For modules our convention is

\[
(a\psi)(b)=\psi(ba),\qquad(\psi a)(b)=\psi(ab).
\tag{NHC.1}
\]


## Null sequences and the two commutation tests

Define

\[
\begin{aligned}
I_\omega&=\{z\in\mathcal L:z_n\longrightarrow0\text{ strong*}\},\\
C_\omega&=\{x\in\mathcal L:\|x_n\psi-\psi x_n\|\longrightarrow0\quad(\psi\in M_*)\},\\
S_\omega&=\{x\in\mathcal L:[x_n,a]\longrightarrow0\text{ strong*}\quad(a\in M)\},\\
N_\omega&=\{x\in\mathcal L:xI_\omega\subset I_\omega,\ I_\omega x\subset I_\omega\}.
\end{aligned}
\tag{NHC.2}
\]

The letter \(S\) denotes the operator test, and \(C\) denotes the normal-functional test.

**Lemma.** \(I_\omega\subset C_\omega\subset S_\omega\cap N_\omega\).

**Proof.** For a positive normal functional \(\psi\), Cauchy–Schwarz gives

\[
\|z\psi\|\le\psi(1)^{1/2}\psi(z^*z)^{1/2},\qquad
\|\psi z\|\le\psi(1)^{1/2}\psi(zz^*)^{1/2}.
\tag{NHC.3}
\]

Both sides tend to zero for a strong*-null bounded sequence. Every normal functional is a linear combination of four positive normal functionals, by the normal-functional decomposition proved in the predual construction. Thus \(I_\omega\subset C_\omega\).

Take \(x\in C_\omega\), with \(\|x_n\|\le K\). For a fixed \(a\in M\), the module identities are

\[
\begin{aligned}
[x_n,a]\psi&=[x_n,a\psi]-a[x_n,\psi],\\
\psi[x_n,a]&=[x_n,\psi a]-[x_n,\psi]a.
\end{aligned}
\tag{NHC.4}
\]

Every functional on the right is fixed before taking the limit. Consequently both left sides tend to zero in norm. With \(b_n=[x_n,a]\), the estimates

\[
\psi(b_n^*b_n)\le\|b_n\|\,\|b_n\psi\|,\qquad
\psi(b_nb_n^*)\le\|b_n\|\,\|\psi b_n\|
\]

and \(\|b_n\|\le2K\|a\|\) give strong* convergence to zero. This proves \(C_\omega\subset S_\omega\).

For the multiplier assertion, let \(z\in I_\omega\), \(\|z_n\|\le L\), and let \(\psi\) be positive and normal. One square is immediately bounded by

\[
\psi((x_nz_n)^*(x_nz_n))\le K^2\psi(z_n^*z_n).
\]

For the other square, moving one \(x_n\) through the functional and applying Cauchy–Schwarz gives

\[
\begin{aligned}
0\le\psi(x_nz_nz_n^*x_n^*)
&\le KL^2\|[x_n,\psi]\|
  +|\psi(z_nz_n^*x_n^*x_n)|\\
&\le KL^2\|[x_n,\psi]\|
  +LK^2\psi(1)^{1/2}\psi(z_nz_n^*)^{1/2}.
\end{aligned}
\tag{NHC.5}
\]

Indeed the difference in the first line is \((\psi x_n-x_n\psi)(z_nz_n^*x_n^*)\); also \((z_nz_n^*)^2\le L^2z_nz_n^*\) and \((x_n^*x_n)^2\le K^4 1\). Both terms vanish. Thus \(xz\in I_\omega\). The adjoint sequence \(x^*\) is again centralizing: for positive \(\psi\), its commutator has the same norm as that of \(x\), and the general case follows by decomposition. Apply the left-product result to \(x^*,z^*\), then take adjoints, to obtain \(zx\in I_\omega\). Hence \(x\in N_\omega\). \(\square\)

These computations also justify the algebraic quotients used below. The module product identity makes \(C_\omega\) a norm-closed star algebra. The two ideal-preservation conditions make \(N_\omega\) a unital norm-closed star algebra. Bounded strong*-null sequences form a norm-closed star algebra, and \(I_\omega\) is an ideal of both \(C_\omega\) and \(N_\omega\). Thus

\[
M_\omega=C_\omega/I_\omega\ \subset\ M^\omega=N_\omega/I_\omega
\tag{NHC.6}
\]

is an injective inclusion of C* algebras. Constant sequences embed \(M\) in \(M^\omega\), and (NHC.4) shows that \(M_\omega\subset M'\cap M^\omega\). The earlier quotient construction proves their von Neumann algebra structure; the next argument only needs these quotients and their exact kernel.

## The null-sequence test forces the multiplier condition

**Lemma.** For any bounded sequence \(x\), the following are equivalent:

\[
x\in N_\omega
\quad\Longleftrightarrow\quad
[x,z]\in I_\omega\quad\text{for every }z\in I_\omega.
\tag{NHC.7}
\]


**Proof.** The forward implication is the definition of \(N_\omega\). For the converse, fix \(z\in I_\omega\). In a faithful normal representation, boundedness of \(x_n\) and strong convergence of \(z_n\) imply \(x_nz_n\to0\) strongly. Similarly \((z_nx_n)^*=x_n^*z_n^*\to0\) strongly. Put \(w_n=x_nz_n-z_nx_n\), which tends to zero strong* by assumption. The identity \(z_nx_n=x_nz_n-w_n\) supplies strong convergence of \(z_nx_n\). Taking adjoints in \(x_nz_n=z_nx_n+w_n\) supplies strong convergence of \((x_nz_n)^*\). Both products are therefore strong*-null. \(\square\)

Define the weaker-input hypercentral set by

\[
H_\omega^{\rm op}
=\{x\in S_\omega:[x,y]\in I_\omega\text{ for every }y\in C_\omega\},
\qquad
H_\omega^{\rm pre}=H_\omega^{\rm op}\cap C_\omega.
\tag{NHC.8}
\]

Let \(q:N_\omega\to M^\omega\) denote the quotient map and put

\[
A=M'\cap M^\omega,\qquad F=M_\omega.
\]


**Theorem.** The first set in (NHC.8) is contained in \(N_\omega\), and

\[
q(H_\omega^{\rm op})=F'\cap A,
\qquad
q(H_\omega^{\rm pre})=Z(F).
\tag{NHC.9}
\]

In particular, the equality of the two input conventions is equivalent to

\[
F'\cap A=Z(F).
\tag{NHC.10}
\]


**Proof.** By Section 01, \(I_\omega\subset C_\omega\). An element of \(H_\omega^{\rm op}\) therefore passes every test in (NHC.7), and hence belongs to \(N_\omega\). Its commutators with constants vanish in the quotient because it lies in \(S_\omega\), and its commutators with all elements of \(F=q(C_\omega)\) vanish by (NHC.8). Thus its image belongs to \(F'\cap A\).

Conversely, lift an element of \(F'\cap A\) to any bounded \(x\in N_\omega\). Commutation with constants means exactly \([x_n,a]\in I_\omega\) for every \(a\in M\). Commutation with \(F\) means exactly \([x,y]\in I_\omega\) for every \(y\in C_\omega\). This is (NHC.8). Restricting to lifts in \(C_\omega\) gives precisely the center of \(F\), proving both formulas.

Finally, the kernel \(I_\omega\) lies in both hypercentral sets: it commutes modulo itself with constants and with \(C_\omega\subset N_\omega\). If (NHC.10) holds and \(x\in H_\omega^{\rm op}\), choose \(c\in C_\omega\) with \(q(c)=q(x)\). Then \(x-c\in I_\omega\subset C_\omega\), so \(x\in C_\omega\). The converse follows at once by applying \(q\). This proves the exact equivalence without presuming it in the definition. \(\square\)

![The operator and predual tests descend to different relative commutants; the finite trace argument identifies them.](assets/hypercentral-quotient-tests.svg)

## Equality for a finite tracial algebra

**Theorem.** Suppose \(M\) has a faithful normal tracial state \(\tau\). Then \(S_\omega=C_\omega\), every bounded sequence is in \(N_\omega\), and

\[
M'\cap M^\omega=M_\omega,
\qquad H_\omega^{\rm op}=H_\omega^{\rm pre}.
\tag{NHC.11}
\]

No factor assumption is needed.

**Proof.** Write \(\|a\|_2=\tau(a^*a)^{1/2}\). On bounded sets this norm detects strong* convergence: the faithful-state topology theorem applies, and traciality makes the two square terms equal. Both multiplication estimates

\[
\|ab\|_2\le\|a\|\|b\|_2,\qquad
\|ba\|_2\le\|a\|\|b\|_2
\]

show \(N_\omega=\mathcal L\).

The functionals \(\psi_b(a)=\tau(ba)\), \(b\in M\), are norm dense in \(M_*\). Indeed, a nonzero annihilator of their norm closure would, by Hahn–Banach and \((M_*)^*=M\), give a nonzero \(v\in M\) with \(\tau(bv)=0\) for every \(b\). Choosing \(b=v^*\) contradicts faithfulness.

For \(x\in S_\omega\), traciality and Cauchy–Schwarz give

\[
\|[x_n,\psi_b]\|\le\|[x_n,b]\|_2\longrightarrow_\omega0.
\tag{NHC.12}
\]

If \(\|x_n\|\le K\), replacement of \(\psi_b\) by \(\psi\) changes this norm by at most \(2K\|\psi-\psi_b\|\). First approximate \(\psi\) in norm, then take the ultralimit. Thus \(x\in C_\omega\). Section 01 proves the reverse inclusion. The quotient and hypercentral conclusions follow from their definitions, or from (NHC.9). \(\square\)

## A central operator sequence detected by a centralizing projection

Fix \(0<\lambda<1\), and put

\[
p=\frac1{1+\lambda},\qquad q=\frac{\lambda}{1+\lambda},\qquad
\rho=\begin{pmatrix}p&0\\0&q\end{pmatrix}.
\]

Use the infinite product von Neumann algebra and its faithful normal product state

\[
(M,\varphi)=\overline{\bigotimes_{j\ge1}}(M_2(\mathbb C),\operatorname{Tr}(\rho\,\cdot)).
\tag{NHC.13}
\]

The bar denotes the faithful product GNS construction. Finite tensor elements are strong*-dense on bounded sets, and splitting off any one leg gives the corresponding two-by-two matrix algebra over the other legs.

Let \(u_n\) be \(e_{12}\) in leg \(n\), with identity in all other legs; let \(d_n\) be \(e_{11}\) in that leg. Then

\[
u\in S_\omega\cap N_\omega,\quad u\notin C_\omega,
\qquad d\in C_\omega,
\qquad [u,d]\notin I_\omega.
\tag{NHC.14}
\]

Thus this example separates the two input conditions but does not refute their equivalence after the hypercentrality test.

**Proof.** In the split-off leg, write \(a=[a_{ij}]\), with product state \(\varphi_0\) on the remaining legs. Direct multiplication gives

\[
\begin{aligned}
\varphi(a^*a)&=\sum_{i,j}w_j\varphi_0(a_{ij}^*a_{ij}),\\
\varphi(aa^*)&=\sum_{i,j}w_i\varphi_0(a_{ij}a_{ij}^*),
\end{aligned}
\qquad w_1=p,\quad w_2=q.
\tag{NHC.15}
\]

Consequently left or right multiplication by \(e_{12}\) or \(e_{21}\) has norm at most \(\lambda^{-1/2}\) for
\(\|a\|_{\varphi,\#}^2=(\varphi(a^*a)+\varphi(aa^*))/2\). The constant is independent of the chosen leg. Multiplication by \(e_{11}\) is contractive for this seminorm. In particular, \(u,d\in N_\omega\).

Every fixed finite tensor element commutes with \(u_n,d_n\) for sufficiently large \(n\). Approximate any fixed \(a\in M\) by a finite tensor element \(a_0\) in the symmetric state seminorm. The uniform bounds just proved give

\[
\|[u_n,a]\|_{\varphi,\#}
\le2\lambda^{-1/2}\|a-a_0\|_{\varphi,\#}
\]

once that leg is disjoint from the support of \(a_0\); the analogous bound for \(d_n\) has constant \(2\). This proves \(u,d\in S_\omega\), in fact with ordinary convergence.

In the same matrix coordinates,

\[
[u_n,\varphi](a)=\varphi(au_n-u_na)=(q-p)\varphi_0(a_{21}).
\]

The slice of the \((2,1)\)-entry is contractive, and \(a=e_{21}\) realizes its norm. Hence

\[
\|[u_n,\varphi]\|=p-q=\frac{1-\lambda}{1+\lambda}>0.
\tag{NHC.16}
\]

So \(u\notin C_\omega\).

On the other hand, \([d_n,\varphi]=0\) exactly, because \(d_n\) commutes with the diagonal density on its leg. The functionals \(\varphi b:a\mapsto\varphi(ba)\) are norm dense in \(M_*\): the Hahn–Banach argument in Section 03 works with any faithful state in place of the trace. The module identity now gives

\[
[d_n,\varphi b]=\varphi[d_n,b],\qquad
\|[d_n,\varphi b]\|\le\varphi([d_n,b][d_n,b]^*)^{1/2}\longrightarrow0.
\]

Uniform boundedness and density extend the convergence to every normal functional. Therefore \(d\in C_\omega\). Finally,

\[
[u_n,d_n]=-u_n,\qquad
\|[u_n,d_n]\|_{\varphi,\#}^2=\frac{p+q}{2}=\frac12.
\tag{NHC.17}
\]

This particular centralizing projection sequence fails the hypercentrality test for \(u\), as claimed. \(\square\)

**Reading.** The ultrafilter clause of Takesaki, *Theory of Operator Algebras III*, XIV.4, Definition 4.4, p.117, starts with the fixed-operator condition, while the ordinary clause starts with the normal-functional condition. Formula (NHC.9) records the precise consequence of the former convention. Ando and Haagerup, [*Ultraproducts of von Neumann algebras*](https://arxiv.org/abs/1212.5457), Sections 3.1 and 4.4 and Example 5.1, distinguish the ambient ultrapower from the asymptotic centralizer. The matrix calculation above uses normalized weights throughout. Formula (NHC.10) remains a separate general assertion; the theorem of Section 03 establishes it under its stated tracial hypothesis.