Completely positive finite models

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

A finite-dimensional model of an algebra need not preserve products. It can still preserve positivity after we couple the algebra to any matrix system. This is the reason completely positive maps are useful for approximation. They retain the order information that survives compression to a finite-dimensional Hilbert space.

We will construct such models, explain precisely what convergence means, and prove that sufficiently good models force uniqueness of the C*-tensor norm. The arguments work for nonunital and nonseparable algebras. Finite sets, rather than countable lists, organize the approximation.

Prerequisites are the lessons Completely positive maps and AF-algebras, together with the construction of the minimal and maximal C*-tensor products. We use Stinespring's theorem in its following form: a completely positive map $\phi:A\to B(H)$ is $V^*\pi(\,\cdot\,)V$, where $\pi$ is a representation and $\|V\|^2=\|\phi\|$. No faithfulness of an additional commutant representation is assumed. We also use Hahn–Banach, continuous functional calculus, and positive contractive approximate identities. The extension, dilation and commutant methods are developed in [Arveson 1969]. The exact unital dilation and faithful scalar representation are written out locally; Section 4 below supplies the nonunital dilation step explicitly. These particular spans do not consume the lifting or extension theorem.

Unital algebras in the unital assertions are nonzero. The zero algebra has the approximation property by the zero maps.

1. Approximation is a finite-set question

Write $M_n=M_n(\mathbb C)$. A completely positive contraction, abbreviated cpc, is a completely positive linear map of norm at most one. A unital completely positive map is abbreviated ucp. For C*-algebras $A,D$, a cpc map $\theta:A\to D$ has finite-dimensional completely positive approximations if, for every finite set $F\subset A$ and every $\varepsilon>0$, there are cpc maps

For $\theta=\operatorname{id}_A$, this is the completely positive approximation property.

The maps $\alpha$ and $\beta$ play different roles. The first records finitely many matrix measurements. The second reconstructs an element of $D$. Neither is required to be multiplicative or injective.

Proposition 1.1. The finite-set definition is equivalent to the existence of a net of cpc factorizations $\theta_i=\beta_i\alpha_i$ through matrix algebras such that $\|\theta_i(a)-\theta(a)\|\to0$ for every $a\in A$.

Proof. Direct the pairs $(F,\varepsilon)$, with $F$ finite, by increasing $F$ and decreasing $\varepsilon$. Choose one approximation for each pair. Once the index contains a given $a$ and has tolerance less than a given positive number, all later indices meet that requirement. This gives pointwise norm convergence.

Conversely, for each $a$ in a finite $F$, convergence gives an index beyond which its error is less than $\varepsilon$. A common upper bound of these finitely many indices works for all of $F$. $\square$

Proposition 1.2. If $A$ is separable, a net in Proposition 1.1 can be replaced by a sequence.

Proof. Choose a norm-dense sequence $a_1,a_2,\ldots$ in $A$. At stage $k$, approximate $a_1,\ldots,a_k$ within $1/k$. For $a\in A$, choose $a_j$ close to $a$. Contractivity gives

First let $k\to\infty$ and then let $a_j\to a$. $\square$

Thus a sequence is available under separability. The definition itself does not require separability.

Finite-dimensional C*-algebras can replace the middle matrix algebra without changing the property. Indeed, write $F=\bigoplus_{j=1}^rM_{n_j}$, place $F$ as diagonal blocks in $M_N$, where $N=\sum_jn_j$, and let

The inclusion and $P$ are cpc, and $P$ fixes $F$. They turn a factorization through $F$ into one through $M_N$. Conversely a full matrix algebra is already finite-dimensional.

2. Order allows extensions

An operator system $S\subset C$ is a self-adjoint linear subspace of a unital C*-algebra containing $1_C$. Its matrix positive cones are inherited from $C$. We now prove the extension theorem needed to put finite-dimensional models on a larger algebra.

We first record the norm estimate used in the extension. If a positive operator block has diagonal entries bounded by $c1$, positivity of its quadratic form on vectors $(\xi,z\eta)$, minimized over $z\in\mathbb C$, gives

Thus its off-diagonal entry $Y$ has norm at most $c$, including the zero diagonal cases. For $x$ in an operator system with $\|x\|\le1$, the matrix $\begin{pmatrix}1&x\\x^*&1\end{pmatrix}$ is positive in the ambient algebra: it factors as $R^*R$, where

If $\phi$ is completely positive, applying it at size two and using the preceding block estimate proves $\|\phi(x)\|\le\|\phi(1)\|$. Consequently $\|\phi\|=\|\phi(1)\|$, whether or not the operator system is norm closed. A faithful representation of the target lets the same calculation treat any C*-algebra target.

We also make the compactness input explicit. For a normed space $E$, put its dual unit ball in

by recording all values of a functional. Each disc is compact, and the product is compact by the complete ultrafilter argument in CF, Section 4. The linearity equations are closed coordinate conditions, so this subset is compact. Its coordinate topology is exactly weak* convergence. Scaling proves the assertion for every bounded dual ball, including radius zero. For real normed spaces, use closed real intervals instead of discs; the same equations and proof apply. The same finite-coordinate argument proves compactness for products of these balls. In particular the actual onto concrete duality for $B(H)$ in CP1–CP6 makes its bounded balls ultraweakly compact, for arbitrary $H$.

We now translate a matrix-valued map into a scalar functional. We use column vectors and write quadratic forms as $\xi^*T\xi$.

Lemma 2.1. For a linear map $\phi:S\to M_n$, define

Then $\phi$ is completely positive if and only if $f_\phi$ is a positive functional on the operator system $M_n(S)$.

Proof. If $\phi$ is completely positive and $[s_{ij}]\ge0$, apply $\phi$ entrywise and evaluate its positive quadratic form on the vector whose $i$-th component is the $i$-th standard basis vector. The result is $f_\phi([s_{ij}])\ge0$.

Conversely, let $X=[s_{pq}]\in M_m(S)_+$ and let $\xi_1,\ldots,\xi_m\in\mathbb C^n$. Set $U_{pi}=(\xi_p)_i$. The scalar compression $U^*XU$ belongs to $M_n(S)_+$. Its entries are

Therefore

All quadratic forms of $\phi^{(m)}(X)$ are nonnegative. Hence $\phi$ is completely positive. $\square$

Lemma 2.2. A positive functional $f$ on an operator system $R\subset D$ extends to a positive functional $g$ on $D$ with $g(1)=f(1)$.

Proof. On the real vector space $D_{\rm sa}$, define

The set is nonempty because it contains $\|x\|1$. It is bounded below: $y\ge x\ge-\|x\|1$ implies $f(y)\ge-\|x\|f(1)$. The function $p$ is sublinear, and $p(y)=f(y)$ for $y\in R_{\rm sa}$. Real Hahn–Banach extends $f|_{R_{\rm sa}}$ to a real linear $g_0\le p$ on $D_{\rm sa}$.

If $x\ge0$, then $p(-x)\le f(0)=0$, so $g_0(x)\ge0$. Also $g_0(1)=f(1)$. Complexify $g_0$. The result is a positive linear functional $g$. Positivity gives $\|g\|=g(1)$, so it is bounded. $\square$

Theorem 2.3 (Arveson extension). If $S\subset C$ is an operator system and $\phi:S\to B(H)$ is completely positive, there is a completely positive extension $\Phi:C\to B(H)$. If $\phi$ is unital, $\Phi$ is unital.

Proof. If $H=0$, the zero map is the required extension. Otherwise suppose first that $H=\mathbb C^n$, with $n\ge1$. Extend $f_\phi$ from $M_n(S)$ to a positive functional $g$ on $M_n(C)$, using Lemma 2.2. Define

Then $f_\Phi=g$, so Lemma 2.1 makes $\Phi$ completely positive. For $s\in S$, the definition gives $\Phi(s)_{ij}=\phi(s)_{ij}$. In particular $\Phi(1)=\phi(1)$.

For arbitrary $H$, direct its finite-dimensional subspaces $K$ by inclusion. Compress $\phi$ to $B(K)$, extend it by the finite-dimensional case, and regard the extension $\Phi_K$ as a map into $B(H)$, zero on $K^\perp$. Each map is completely positive and

For every $c\in C$, the values lie in the ultraweakly compact ball of radius $\|\phi(1)\|\|c\|$. Compactness of the product of these balls supplies a subnet converging ultraweakly at every $c$. Its limit $\Phi$ is linear. Matrix positivity is preserved by this limit: for every positive matrix over $C$ and every finite tuple of Hilbert vectors, its image quadratic form is a finite sum of vector functionals, and hence is the limit of nonnegative numbers. Thus $\Phi$ is completely positive at every size.

For $s\in S$, we have $\Phi_K(s)=P_K\phi(s)P_K$, which converges strongly to $\phi(s)$. Its common norm bound makes this convergence ultraweak: each vector coefficient converges, and the square-summable vector series of CP1–CP6 has uniformly small tails by Cauchy–Schwarz. Split such a series into a finite part and its uniformly bounded tail, then let the finite part converge. Thus $\Phi$ extends $\phi$. Its value at $1$ is $\phi(1)$, proving the unital assertion. $\square$

No separability of $H$ was used. The extension need not be normal even if part of the original setting involves von Neumann algebras.

Corollary 2.4. If $F\subset A$ is a finite-dimensional C*-subalgebra, there is a cpc map $E:A\to F$ that fixes $F$.

Proof. Represent $F=\bigoplus_jM_{n_j}$. Let $p$ be its unit, which is a projection of $A$. In the unital algebra $pAp$, extend each coordinate map $F\to M_{n_j}$ by Theorem 2.3. Their direct sum is a ucp map $E_0:pAp\to F$. Put $E(a)=E_0(pap)$. Compression is cpc, and $E$ fixes $F$. $\square$

Proposition 2.5 (Normalize at the unit). Let $S\subset C$ be an operator system and $\phi:S\to M$ a completely positive map into a von Neumann algebra. Put $b=\phi(1)$. There is a ucp map $\phi_0:S\to M$ such that

Proof. Represent $M$ on its given Hilbert space. Put $e=P_{\overline{bH}}$, whose complementary space is $\ker b$ by the bounded adjoint identity. The contractions $r_\delta=b(b+\delta1)^{-1}$ vanish on $\ker b$, and

They tend strongly to the identity on $bH$, and their common bound one extends the convergence to $\overline{bH}$. Thus $r_\delta\to e$ strongly. All $r_\delta$ belong to $M$, so its weak operator closedness puts $e$ in $M$. This constructs the support projection without a spectral-measure theorem.

The other contractions $k_\delta=b^{1/2}(b+\delta1)^{-1/2}$ have the same strong limit. They also vanish on $\ker b$, and the scalar identity

gives $\|(1-k_\delta)b\|\le\delta/2$. Density in $\overline{bH}$ and the common contraction bound again prove $k_\delta\to e$ strongly. For $\delta>0$, set $\psi_\delta(x)=(b+\delta1)^{-1/2}\phi(x)(b+\delta1)^{-1/2}$. These maps are completely positive and have norm at most one, since their values at $1$ are $b(b+\delta1)^{-1}\le1$. Compactness of the product of ultraweakly compact balls in $M$ gives a pointwise ultraweak cluster map $\psi_0$, which is completely positive and satisfies $\psi_0(1)=e$.

For a positive $x\in S$, $0\le\phi(x)\le\|x\|b$. Its quadratic form vanishes on $\ker b$; positive-form Cauchy–Schwarz then makes every mixed coefficient there zero. Thus $\phi(x)$ annihilates $\ker b$, and self-adjointness gives $\phi(x)=e\phi(x)e$. The positive elements span $S$, so this holds for every $x$. For the contractions $k_\delta$ whose strong limit was proved above, bounded multiplication with the fixed $\phi(x)$ gives $k_\delta\phi(x)k_\delta\to e\phi(x)e=\phi(x)$ strongly. Therefore

strongly, and the ultraweak cluster limit gives $b^{1/2}\psi_0(x)b^{1/2}=\phi(x)$.

Choose a state $\omega$ of $C$ and put $\phi_0(x)=\psi_0(x)+\omega(x)(1-e)$. It is ucp, and the added term disappears after multiplication by $b^{1/2}$. The case $b=0$ is included: then $\phi=0$. $\square$

3. Finite models that we can see

Example 3.1 (Compact operators). Let $H$ be any Hilbert space. For a finite-dimensional subspace $K$, compression and inclusion give cpc maps

These approximate the identity in norm as $K$ increases. For a finite-rank $T$, a $K$ containing the ranges of $T$ and $T^*$ makes the compression exact. For a compact $T$, approximate by a finite-rank operator and use contractivity. This proves the completely positive approximation property without assuming $H$ separable.

Theorem 3.2 (Locally finite-dimensional algebras). Suppose that $A$ is the norm closure of an upward-directed family of finite-dimensional C*-subalgebras. Then $A$ has the completely positive approximation property.

Proof. Given finite $F\subset A$ and $\varepsilon>0$, choose one finite-dimensional subalgebra $D$ and, for each $a\in F$, an element $d_a\in D$ with $\|a-d_a\|<\varepsilon/2$. Use Corollary 2.4 for a cpc retraction $E:A\to D$. Inclusion $j:D\to A$ gives

Replace $D$ by a full matrix algebra as explained in Section 1. $\square$

In particular every AF-algebra has this property. The directed version also treats norm closures of noncountable families; no countable generating sequence is required.

Theorem 3.3 (Functions vanishing at infinity). For every locally compact Hausdorff space $X$, $C_0(X)$ has the completely positive approximation property.

Proof. If the finite test list is empty or every selected function is zero, use the zero maps through $M_1$. Otherwise write the list as $f_1,\ldots,f_m$, choose $0<\delta<\varepsilon$, and put

Each nonzero level set of a $C_0(X)$ function is compact, so $K$, a finite union of such level sets, is compact. If $K$ is empty, every selected function has norm at most $\delta/2$, and the zero maps already work. Otherwise its image under $z$ is compact. Choose finitely many $x_j\in K$ so that, for every $x\in K$, some $j$ has $\|z(x)-z(x_j)\|_\infty<\delta/8$. Define

and also set $g_0=w_0/W$. On $K$, at least one $w_j$ is greater than $\delta/8$; off $K$, $w_0>\delta/2$. Thus $W>0$ everywhere. All the weights are continuous, nonnegative, and satisfy $g_0+\sum_jg_j=1$.

For $j\ge1$, positivity of $w_j(x)$ implies $\|z(x)\|_\infty>\|z(x_j)\|_\infty-\delta/4\ge\delta/4$. Therefore the support of $g_j$ is contained in the compact level set $\{x:\|z(x)\|_\infty\ge\delta/4\}$; hence $g_j\in C_c(X)_+$. The weight $g_0$ represents the omitted zero sample and need not have compact support. Define

Evaluation $\alpha:C_0(X)\to\mathbb C^r$ is a contractive *-homomorphism. At every matrix size and every point of $X$, $\beta$ sends a positive matrix to a nonnegative sum of positive scalar matrices. Thus $\beta$ is completely positive, and $\|\beta\|=\|\sum_jg_j\|\le1$.

If $g_j(x)>0$ for $j\ge1$, then $|f_i(x_j)-f_i(x)|<\delta/4$. If $g_0(x)>0$, then $|f_i(x)|<\delta$. Consequently

This is uniform on all of $X$. Include $\mathbb C^r$ diagonally in $M_r$ and use the diagonal compression from Section 1 to obtain the required matrix factorization. All compact sets and samples came from the given finite list; no countability assumption or partition-of-unity theorem is used. $\square$

For $X=[0,1]$, one may take equally spaced sample points and the piecewise linear tent functions. If every $f$ in the finite set has Lipschitz constant at most $L$, a mesh of width $h$ gives error at most $Lh$. This finite model consists of samples and interpolation weights; it is not a finite-dimensional subalgebra of $C([0,1])$.

4. Why matrix models control tensor norms

For an algebraic tensor $z\in A\odot B$, the minimal norm is obtained in a tensor product of faithful representations. The maximal norm is the supremum over representations of $A$ and $B$ with commuting ranges on one Hilbert space. We always have $\|z\|_{\min}\le\|z\|_{\max}$.

Lemma 4.1. If $\phi:A\to D$ is cpc, then $\phi\odot\operatorname{id}_B$ extends to a contraction both from $A\otimes_{\min}B$ to $D\otimes_{\min}B$ and from $A\otimes_{\max}B$ to $D\otimes_{\max}B$.

Proof. For the minimal norm, represent $D$ faithfully on $H$ and $B$ faithfully on $L$, and dilate the resulting map $A\to B(H)$ as $V^*\pi(\,\cdot\,)V$, with $\|V\|\le1$. Then

Every tensor product of representations is bounded by the minimal norm, so this expression has norm at most $\|z\|_{\min}$.

For the maximal norm, fix commuting representations $\rho:D\to B(H)$ and $\sigma:B\to B(H)$, and put $\psi=\rho\phi$. We need a dilation of $\psi$ that also carries the $B$-action. Here is the construction. On $A\odot H$, use the positive form associated to $\psi$:

Positive-form Cauchy–Schwarz makes the zero vectors an orthogonal subspace; quotienting and completing gives a Hilbert space. Left multiplication by $c\in A$ is bounded by $\|c\|$. Indeed its squared norm defect is the positive form of $[a_i^*(\|c\|^2 1-c^*c)a_j]$; this is a Gram matrix after multiplying every $a_j$ by the positive square root of $\|c\|^2 1-c^*c$ in the unitization, and all Gram entries lie in $A$. Thus left multiplication preserves the nullspace, extends boundedly, and gives the representation $\pi$; its product and adjoint identities follow on algebraic vectors and then by continuity. Since $\sigma(b)$ commutes with every $\psi(a)$, the operation

is bounded by $\|b\|$ in this seminorm: the positive operator matrix $[\psi(a_i^*a_j)]$ commutes with the diagonal matrix whose entries are $\sigma(b)$. The operation therefore descends to the completed quotient. Its adjoint is the operation for $b^*$, and multiplication is preserved. Thus $\widehat\sigma$ is a representation commuting with $\pi$.

Here is the approximate-identity step when $A$ is nonunital. Let $(e_\lambda)$ be a positive contractive approximate identity. For $\xi\in H$, the quotient vectors $e_\lambda\otimes\xi$ have norms at most $\|\xi\|$, since

For every $a\otimes\eta$, their inner products converge, because $e_\lambda a\to a$ in norm and $\psi$ is bounded. Density and the uniform bound give a weak limit $V\xi$, with $\|V\|\le1$. Left multiplication satisfies

in the quotient norm: the squared error is at most $\|ae_\lambda-a\|^2\|\xi\|^2$. Taking weak limits proves $\pi(a)V\xi=a\otimes\xi$, and pairing with $V\eta$ gives $V^*\pi(a)V=\psi(a)$. The bounded operator $\widehat\sigma(b)$ preserves weak limits and takes $e_\lambda\otimes\xi$ to $e_\lambda\otimes\sigma(b)\xi$. Consequently $\widehat\sigma(b)V=V\sigma(b)$. In the unital case the same identities follow immediately from $V\xi=1\otimes\xi$. Neither construction requires $V$ or the commutant representation to be faithful. It follows that

Its norm is at most $\|\sum_i a_i\otimes b_i\|_{\max}$. Take the supremum over the commuting representations. $\square$

Lemma 4.2. For every $B$, the minimal and maximal norms on $M_n\odot B$ agree. The same holds for a finite direct sum of matrix algebras.

Proof. Let $\pi:M_n\to B(K)$ be any representation and put $e=\pi(1_n)$. Its action is zero on $(1-e)K$, so the product with a commuting representation contributes zero there. On $eK$, put $L=\pi(E_{11})K$ and define

The matrix-unit identities give

Thus $U$ is an isometry. For every $\zeta\in eK$, $\xi_i=\pi(E_{1i})\zeta$ belongs to $L$, and

Hence $U$ is onto, and $U^*\pi(x)U=x\otimes1_L$.

An operator commuting with every $E_{ii}\otimes1$ has zero off-diagonal blocks. Commuting also with every $E_{ij}\otimes1$ forces all diagonal blocks to be the same operator. Consequently any representation $\sigma:B\to B(K)$ commuting with $\pi(M_n)$ reduces $eK$, and on that subspace

where $\sigma_0:B\to B(L)$ is a representation: linearity, products and adjoints follow from the diagonal blocks. No faithfulness or nondegeneracy of $\sigma_0$ is needed. Every product representation defining the maximal tensor norm is therefore a spatial representation, and its norm on an algebraic tensor is bounded by the minimal tensor norm. The always valid reverse inequality gives equality.

For $F=\bigoplus_{j=1}^rM_{n_j}$, the images of its central summand units are mutually orthogonal projections. They split the active representation space into reducing summands for both commuting actions; on each, the preceding matrix-unit proof applies. The product norm is the largest of these finitely many summand norms, as is the spatial direct-sum norm. The complementary zero-action space again contributes zero. Thus the minimal and maximal norms on $F\odot B$ agree. $\square$

A C*-algebra $A$ is nuclear if these two norms agree on $A\odot B$ for every C*-algebra $B$.

Theorem 4.3. A C*-algebra with the completely positive approximation property is nuclear.

Proof. Fix $B$ and $z=\sum_{j=1}^ra_j\otimes b_j$. Choose cpc factorizations $\theta_i=\beta_i\alpha_i$ approximating $\operatorname{id}_A$. The cross-norm estimate gives

By Lemmas 4.1 and 4.2,

Pass to the limit. The reverse inequality always holds. $\square$

Combining this with Section 3 proves nuclearity of compact operators, commutative C*-algebras and AF-algebras. The converse, nuclearity implies completely positive approximation, requires an additional separation argument; it is proved in Tensor positivity and nuclearity.

5. Permanence from the maps

Proposition 5.1. If $A$ has the completely positive approximation property and $\theta:A\to D$ is cpc, then $\theta$ has finite-dimensional completely positive approximations.

Proof. Compose the reconstruction map for an approximation of $\operatorname{id}_A$ with $\theta$. Contractivity of $\theta$ does not increase the error. $\square$

Proposition 5.2. If $D$ has the completely positive approximation property and there are cpc maps $s:A\to D$, $r:D\to A$ with $rs=\operatorname{id}_A$, then $A$ has the same property.

Proof. Compose approximations of $\operatorname{id}_D$ on the finite set $s(F)$ with $s$ and $r$. $\square$

Proposition 5.3. If $A$ has the completely positive approximation property, so do every closed ideal $I\subset A$ and every corner $pAp$, where $p\in A$ is a projection.

Proof. For an ideal, let $F\subset I$ be finite. Choose a positive contraction $e\in I$ with $\|eae-a\|<\varepsilon/2$ for $a\in F$, using an approximate identity of $I$. Choose a cpc factorization $\beta\alpha$ on $A$ with error less than $\varepsilon/2$ on $F$. Restrict $\alpha$ to $I$, and replace $\beta(x)$ by $e\beta(x)e$. The new reconstruction map has range in $I$, is cpc, and has total error less than $\varepsilon$.

For a corner, use the inclusion $pAp\to A$ and the cpc retraction $a\mapsto pap$ in Proposition 5.2. $\square$

Proposition 5.4. Suppose $A$ is the closure of an upward-directed family of C*-subalgebras $A_\lambda$, each having the completely positive approximation property. Then $A$ has the property.

Proof. Approximate a finite set $F\subset A$ by elements $b_a$ of a common $A_\lambda$. Choose cpc maps $\alpha_0:A_\lambda\to M_n$, $\beta_0:M_n\to A_\lambda$ approximating all $b_a$.

The map $\alpha_0$ extends to a cpc map $\alpha:A\to M_n$. To see this when the algebras are nonunital, take a Stinespring dilation of $\alpha_0$ and extend its representation to the unitization. This gives a completely positive extension to $A_\lambda+\mathbb C1$ with value at $1$ at most $1_n$. Apply Theorem 2.3 in the unitization of $A$, and restrict back to $A$. Its norm is at most one. If $A_\lambda$ already contains the ambient unit, apply Theorem 2.3 directly.

Let $\beta$ be $\beta_0$ followed by inclusion in $A$. Then

Choose both errors small enough. $\square$

6. Exercises with solutions

Exercise 1 (A finite algebra inside one matrix algebra; introductory). Embed $M_2\oplus\mathbb C\oplus M_3$ in $M_6$. Give a cpc retraction and verify that it fixes the embedded algebra.

Solution. Use diagonal blocks of sizes $2,1,3$, with projections $p_1,p_2,p_3$. The retraction is $x\mapsto(p_1xp_1,p_2xp_2,p_3xp_3)$. Each compression is completely positive. The direct sum map is unital, so is contractive. On a block diagonal matrix it returns the original three blocks.

Exercise 2 (Error propagation; introductory). Suppose $\theta_i$ and $\theta$ are contractions on a Banach space and $\|\theta_i(a_j)-\theta(a_j)\|\to0$ on a dense subset. Prove convergence everywhere.

Solution. For any $a$, select $a_j$ with $\|a-a_j\|<\delta$. Then $\|\theta_i(a)-\theta(a)\|\le2\delta+\|\theta_i(a_j)-\theta(a_j)\|$. The limit superior is at most $2\delta$, and $\delta$ is arbitrary.

Exercise 3 (A matrix test for positivity; intermediate). For $\beta:M_n\to D$, prove that $\beta$ is completely positive if and only if $C_\beta=[\beta(E_{ij})]$ is positive in $M_n(D)$.

Solution. The matrix $[E_{ij}]\in M_n(M_n)$ is positive: on $\mathbb C^n\otimes\mathbb C^n$ it is the rank-one operator $|\sum_ie_i\otimes e_i\rangle\langle\sum_ie_i\otimes e_i|$. Complete positivity therefore makes $C_\beta$ positive.

Conversely factor $C_\beta=R^*R$, writing its entries as $\beta(E_{ij})=\sum_k r_{ki}^*r_{kj}$. For a positive block matrix $X=[x_{pq}]\in M_m(M_n)$, each matrix

is positive. Indeed it is obtained from the positive scalar matrix indexed by $(p,i),(q,j)$ by multiplying on the left and right by the corresponding rectangular matrices with entries $r_{ki}$. Summing over $k$ gives $[\beta(x_{pq})]\ge0$.

Exercise 4 (A model with no product rule; intermediate). On $C([0,1])$, let $\alpha(f)=(f(0),f(1))$ and $\beta(a,b)(t)=(1-t)a+tb$. Show that $\alpha,\beta$ are ucp, that $\beta\alpha$ fixes the function $t$, and that it does not fix $t^2$.

Solution. Evaluation is a unital *-homomorphism. The two coefficients $1-t,t$ are positive and sum to one, so $\beta$ is ucp by the pointwise matrix test. Both $t$ and $t^2$ have samples $(0,1)$, so their reconstructions are $t$. At $t=1/2$, the reconstructed square has value $1/2$, while $t^2$ has value $1/4$.

Exercise 5 (Why a net is needed; intermediate). Let $I$ be uncountable and $H=\ell^2(I)$. Prove that no sequence of finite-dimensional subspaces $K_n$ makes $P_{K_n}TP_{K_n}\to T$ in norm for every $T\in\mathcal K(H)$.

Solution. The closed span $K$ of all $K_n$ is separable. Every vector of $\ell^2(I)$ has countable support, so a countable dense subset of $K$ has support in some countable $J\subset I$. Choose $i\in I\setminus J$. Then $e_i\perp K_n$ for every $n$. For the rank-one projection $T$ onto $\mathbb Ce_i$, all compressions vanish, and their distance from $T$ is one. The directed family of all finite-dimensional subspaces does approximate every compact operator, as in Example 3.1.

Exercise 6 (Weak approximation and convexity; advanced). Let $\mathcal F$ consist of cpc maps $A\to D$ factoring through matrix algebras. Prove that $\mathcal F$ is convex. Then show that if $\theta$ lies in its pointwise weak closure, it lies in its pointwise norm closure.

Solution. For factorizations $\beta_j\alpha_j$ and nonnegative $t_j$ summing to one, define $\alpha(a)=\bigoplus_j\alpha_j(a)$ and $\beta((x_j)_j)=\sum_jt_j\beta_j(x_j)$. Both are cpc: at every matrix level positivity is preserved, and the norm of the second is at most $\sum_jt_j=1$. Replace the direct sum by one matrix algebra using Section 1.

For a finite $F=\{a_1,\ldots,a_r\}$, evaluate $\mathcal F$ in the Banach space $D^r$, with maximum norm. Its image is convex. The weak and norm closures of a convex subset of a Banach space agree by Hahn–Banach separation. The pointwise weak hypothesis says $(\theta(a_1),\ldots,\theta(a_r))$ belongs to its weak closure, hence to its norm closure. This is exactly the required finite-set approximation.

Exercise 7 (The unit can be repaired; advanced). Suppose $A,D$ are unital, $\theta:A\to D$ is ucp, and cpc factorizations approximate $\theta$ in norm. Prove that the approximating maps can be taken ucp in both directions.

Solution. Start with $\alpha:A\to M_n$, $\beta:M_n\to D$, and put $h=\alpha(1)$. Choose a state $\omega$ on $A$ and define $\alpha_1(a)=\alpha(a)+\omega(a)(1-h)$. This is ucp. Let $d=\beta(1)-\beta(h)$. Then $0\le d\le1-\beta(h)$, so $\|d\|\le\|1-\beta\alpha(1)\|$, and $\|\beta\alpha_1(a)-\beta\alpha(a)\|\le\|a\|\|1-\beta\alpha(1)\|$.