Trace-preserving finite models

Written by GPT-6.1 Sol (OpenAI), Ultra, October 2026. Self-checked by the writing AI. New original text: public domain (CC0).

Semidiscreteness supplies finite completely positive models, but their matrix states can have unequal weights. We will change the reconstruction map slightly, replace its state by a matrix trace, and take an ordinary trace adjoint. Both resulting maps preserve the prescribed traces exactly. Matrix contractions can then be replaced by unitaries while controlling their images.

Throughout, $M$ has a faithful normal tracial state $\tau$, and $\tau_m=\operatorname{Tr}/m$. No factor or separability assumption is needed here. The inputs are the weighted recording theorem and its Hilbert bound, the rational-density channel and its commutator bound, the ucp Schwarz inequality, and the injectivity–semidiscreteness equivalence. In its finite specialization, semidiscreteness gives normal ucp recording maps and ucp reconstruction maps whose composites approximate any finite list in $2$-norm. The scalar defect correction used in the properly infinite proof makes both maps unital before the following argument.

1. Rationalizing a density without losing positivity

Lemma 1.1. If $T:M_m\to M$ is ucp, then for every $\varepsilon>0$ there is a ucp map $T'$ with $\|T'-T\|<\varepsilon$ such that the density of $\tau T'$ has strictly positive rational eigenvalues. In particular $T'$ is faithful on positive elements.

Proof. Write

where the $e_i$ are rank-one orthogonal projections summing to $1$. Choose $0<\delta<1$. For each positive $\lambda_i$, choose a rational number

put $q_i=0$ when $\lambda_i=0$. Define

Thus $(1-\delta)1\le b\le1$. Set

Both summands are cp, and their values at $1$ add to $1$. For $\|x\|\le1$, contractivity of $T$ gives

Here $\|1-b^{1/2}\|\le\delta$ follows from $1-\sqrt t\le1-t$ on $[0,1]$.

Since $b$ commutes with $h$, the new density is

The number $c$ is rational, and so are all $q_i+c$. Strict positivity also fills every original zero eigenspace. The density remains normalized, because $\tau_m(h')=1$. Choose $3\delta<\varepsilon$. If $x\ge0$ and $T'(x)=0$, then $\tau_m(h'x)=0$; invertibility of $h'$ gives $x=0$. $\square$

The correction in (3) changes the actual cp map as well as its density. Simply replacing $h$ by a rational matrix would not specify a compatible positive reconstruction map.

A state-preparation construction

There is also a state-preparation proof with a different error budget. Choose $0<\alpha<\min(1,\varepsilon/2)$ and positive rational numbers $r_i$ such that

Such a choice is possible because the vector $((1-\alpha)\lambda_i+\alpha)_i$ lies in this open simplex and has sum $m$. Rational points in the sum-$m$ hyperplane are dense: approximate the first $m-1$ coordinates rationally and define the last by subtraction; sufficiently small errors retain all strict inequalities. For $m=1$, take $r_1=1$.

Put $g=\sum_i r_i e_i$ and $k=(g-(1-\alpha)h)/\alpha$. Then $k>0$ and $\tau_m(k)=1$, so $\psi(x)=\tau_m(kx)$ is a state. The map

is ucp, has density $g$ under $\tau$, and satisfies $\|\widetilde T-T\|\le2\alpha<\varepsilon$. Thus mixing reconstruction with state preparation produces the same full conclusion, including faithfulness, while changing the actual map and filling zero eigenspaces. The first proof retains the congruence formula needed for the worked diagnostics.

2. Reconstruction maps preserving the matrix trace

Theorem 2.1. Suppose $M$ is injective. Given unitaries $u_1,\ldots,u_n\in M$ and $\varepsilon>0$, there are a ucp map $T:M_q\to M$ and contractions $y_k\in M_q$ such that

Proof. Fix a small $\eta>0$. Semidiscreteness gives ucp maps $S_1:M\to M_m$, $T_1:M_m\to M$ with $\|T_1S_1(u_k)-u_k\|_2<\eta/2$. Put $x_k=S_1(u_k)$, so $\|x_k\|\le1$. Lemma 1.1 gives a faithful ucp $T_2$ with rational strictly positive density $h$ and $\|T_2-T_1\|<\eta/2$. Therefore

Write $\varphi=\tau T_2=\tau_m(h\,\cdot)$. The improved weighted Hilbert bound in the preceding lesson implies, for every contraction $x$,

The two cross traces agree by cyclicity; they are real and nonnegative. From (7), $\|T_2(x_k)\|_2>1-\eta$ and $\|T_2(x_k)\|_2\le1$. In particular

Apply the rational density theorem to obtain ucp maps $R:M_m\to M_q$ and $L:M_q\to M_m$ satisfying

For arbitrary $a$, Schwarz and traciality of the target give

Set $y_k=R(x_k)$ and $T=T_2L$. These are contractions and a ucp map. Equations (7), (10) and (11) give

Choosing $\eta+2\sqrt\eta<\varepsilon$ proves the result. $\square$

One common error budget $\eta$ controls the perturbed reconstruction throughout (7)–(12). The original perturbation and the semidiscrete error have already been added in (7).

3. An ordinary adjoint preserves both traces

Theorem 3.1. For every finite list $a_1,\ldots,a_N\in M$ in an injective faithfully tracial algebra and every $\varepsilon>0$, there are ucp maps

such that $S$ is normal and

Proof. First treat a finite list of unitaries. Choose $T,y_k$ from Theorem 2.1 with error less than $\eta$. Its ordinary trace adjoint $S=T^\sharp$ is normal and cp by the preceding lesson. Since $\tau T=\tau_q$, its density is $1$, so $S(1)=1$. Pairing with $1$ gives $\tau_qS=\tau$. Both maps are $L^2$ contractions by Schwarz and their trace identities.

The adjoint identity gives the complex estimate

Consequently its real part exceeds $1-\eta$. As both $\|S(u_k)\|_2$ and $\|y_k\|_2$ are at most $1$,

Choose the small budget before applying Theorem 2.1.

For a general finite list, use an exact finite linear combination of unitaries for each element. Indeed a self-adjoint contraction $b$ is $(v+v^*)/2$, where $v=b+i(1-b^2)^{1/2}$ is unitary. Applying this to the real and imaginary parts, with their norm bounds, gives at most four unitary summands for any element. Collect their finite list and take a unitary error smaller than $\varepsilon$ divided by the maximum sum of absolute coefficients. Zero elements need no tests. Linearity proves (13). $\square$

In (14) the difference of the complex pairing from $1$ is controlled directly. A lower bound on its absolute value alone would allow an incorrect phase and would not justify (15).

4. Replacing matrix contractions by unitaries

Lemma 4.1. Let $T:M_m\to M$ be any ucp map, let $u\in M$ be unitary and let $y\in M_m$ be a contraction. If $\|T(y)-u\|_2<\eta$, there is a unitary $v\in M_m$ with

Trace preservation of $T$ is unnecessary.

Proof. Extend the polar partial isometry of the square matrix $y$ to a unitary $v$, so $y=v|y|$. Put $\varphi=\tau T$. Schwarz gives

The scalar inequality $(1-t)^2\le1-t^2$ holds for $0\le t\le1$. For the final bound, put $d=\|T(y)-u\|_2<\eta$. Contractivity and the reverse triangle inequality give $0\le1-\|T(y)\|_2\le d$; hence $1-\|T(y)\|_2^2\le2d<2\eta$. This works for every positive $\eta$. The triangle inequality proves (16). $\square$

Corollary 4.2. Given finitely many unitaries in an injective faithfully tracial $M$, they can be approximated in $2$-norm by $T(v_k)$, where $T:M_m\to M$ is ucp, $\tau T=\tau_m$, and every $v_k$ is unitary. This follows from Theorem 2.1 and Lemma 4.1 with a sufficiently small initial budget. Their reconstruction images still need to be moved into an actual finite-dimensional subalgebra; the next lessons do that in a factor.

5. Problems with complete solutions

Exercise 1. In Lemma 1.1 take $h=\operatorname{diag}(2,0)$, $q_1=19/10$, $q_2=0$. Compute $b,c,h'$.

Solution. Equations (2) and (5) give $b=\operatorname{diag}(19/20,1)$, $c=1/20$, and $h'=\operatorname{diag}(39/20,1/20)$. Its normalized trace is $1$, its eigenvalues are positive rationals, and its previously zero eigenvalue is now positive. For any $\delta>1/20$ the required strict rational inequalities hold.

Exercise 2. Explain why the scalar correction in (3) is completely positive.

Solution. A positive scalar functional is cp. At any matrix level, $[\tau_m(x_{ij})]\ge0$ when $[x_{ij}]\ge0$. Tensoring this scalar positive matrix with the positive operator $T(1-b)$ gives the positive block matrix $[\tau_m(x_{ij})T(1-b)]$. This is exactly the amplification of the correction.

Exercise 3. Check that $\tau_m(h')=1$ even when $h$ has a kernel.

Solution. Equation (5) gives $\tau_m(h')=m^{-1}\sum_iq_i+c=1$. The coordinates with zero eigenvalues contribute $q_i=0$, and the positive constant $c$ occurs in every coordinate. No division by a zero eigenvalue occurs in (2).

Exercise 4. Derive the cross term in (8) from the square of the commutator.

Solution. Expand $(h^{1/2}x-xh^{1/2})^*(h^{1/2}x-xh^{1/2})$ and take $\tau_m$. The first two terms become $\tau_m(hxx^*)$ and $\tau_m(hx^*x)$. Cyclicity makes both negative cross terms $\tau_m(h^{1/2}x^*h^{1/2}x)$. This equals $\|h^{1/4}xh^{1/4}\|_{2,\tau_m}^2$, so it is real and nonnegative.

Exercise 5. Give an explicit positive $\eta$ making the bound in (12) smaller than a prescribed $\varepsilon>0$.

Solution. Take $\eta=\min(\varepsilon/4,\varepsilon^2/64,1/4)$. Then $\eta\le\varepsilon/4$, and $2\sqrt\eta\le\varepsilon/4$. Their sum is at most $\varepsilon/2<\varepsilon$. The extra bound $\eta\le1/4$ keeps the near-unitary lower bound positive.

Exercise 6. Why do the two trace identities in (13) imply $L^2$ contractivity?

Solution. Schwarz gives $T(x)^*T(x)\le T(x^*x)$. Applying $\tau$ gives $\|T(x)\|_2^2\le\tau_m(x^*x)$. The same argument for $S$, applying $\tau_m$ and using $\tau_mS=\tau$, proves its contraction inequality. Bounded elements are dense in their trace Hilbert spaces, so both maps extend contractively.

Exercise 7. Why is $|z|>1-\eta$ insufficient for (15), whereas $|z-1|<\eta$ suffices?

Solution. The number $z=-1$ has absolute value $1$, but real part $-1$; a squared-distance formula involving $2\operatorname{Re}z$ would then have the wrong bound. If $|z-1|<\eta$, then $\operatorname{Re}z>1-\eta$, which is the estimate actually used in (15).

Exercise 8. Show that the state in (17) need not be $\tau_m$.

Solution. Take $M=\mathbb C$, $m=2$, and $T(x)=x_{11}$. This is ucp, but $\tau T(e_{11})=1$ while $\tau_2(e_{11})=1/2$. The proof therefore uses $\varphi=\tau T$ rather than the normalized matrix trace. Schwarz and the scalar functional calculus inequality remain valid for this state.

Exercise 9. Explain how to extend the polar partial isometry of a square matrix to a unitary.

Solution. Its initial and final projections have the same rank, equal to the rank of the matrix. Their complements therefore have the same dimension. Choose an isometric bijection between those complements and add it to the polar partial isometry. The orthogonal initial and final partitions make the sum unitary, and the extra term annihilates $|y|$, so $y=v|y|$ remains true.

Exercise 10. Suppose the error in Theorem 3.1 is at most $\gamma$ on all unitary summands of $a=\sum_j\alpha_j u_j$. Bound its error on $a$.

Solution. Linearity and the triangle inequality give $\|TS(a)-a\|_2\le\gamma\sum_j|\alpha_j|$. A finite list has a finite maximum of these coefficient sums. Choosing its unitary budget below the desired error divided by that maximum gives the claimed general finite-set approximation. If every element is zero, any unital trace-preserving scalar model suffices.

References and proof scope

Uffe Haagerup, A new proof of the equivalence of injectivity and hyperfiniteness for factors on a separable Hilbert space, Journal of Functional Analysis 62 (1985), 160–201. The rational perturbation and trace-preserving factorization follow that method, with a common error budget and an explicit real-part estimate. The state-preparation proof above gives an alternative perturbation with bound $2\alpha$. Every faithfully tracial injective algebra is retained; factoriality and separable predual enter the later subalgebra conclusions. These finite maps prove no AFD conclusion by themselves.

For the trace-preserving expectation prerequisite, Matthew Daws's Conditional Expectations, Section 4, Theorem 4.1 (source label thm:main), Proof 1, gives an accessible Hilbert-space compression proof: the trace on the ambient algebra is normal, semifinite and faithful, and its restriction to the subalgebra must also be semifinite. Editable author source at commit a2d5477. The repository notice licenses those notes under CC BY-NC-SA 4.0. This supplementary reading complements the exact OA-MOD prerequisite specified above; that internal proof route and this lesson's finite-model proofs are retained.