Proof Suppose $A^{\dagger }$ is c.e., so $A^{\dagger }$ is the standard presentation ${\mathrm {C}^{*} \langle {\overline {x}}\operatorname {\mathrm {|}}{\mathcal {R}};{\mathbb {K}}\rangle }$ for some c.e. set of rational relations $\mathcal {R}$ in $\overline {x}$ .

Let T be the continuous first-order theory of C*-algebras over $\mathbb {K}$ . We expand our language to include additional constants for the generators $\overline {x}$ . Note if $x_{i}$ is a member of $\overline {x}$ , then there is a relation of the form $\left \lVert x_{i}\right \rVert \leq M$ in $\mathcal {R}$ , so we can put $x_{i}$ in an appropriate sort. Given a rational point $r(\overline {x})$ of $A^{\dagger }$ , we enumerate through all formal deductions from $T \cup \mathcal {R}$ . For each formal deduction, we output a positive dyadic rational $q_{n}$ if $q_{n} \leq q_{i}$ for all $i < n$ and the formal deduction witnesses that $T \cup \mathcal {R} \vdash \left \lVert r(\overline {x})\right \rVert \leq q_{n}$ . Note this procedure is defined uniformly in the computable enumeration of $\mathcal {R}$ . By Pavelka-style completeness, $\left \lVert r(\overline {x})\right \rVert $ is the infimum of all positive dyadic rationals d such that $T \cup \mathcal {R} \vdash \left \lVert r(\overline {x})\right \rVert \leq d$ . Thus, we have enumerated a decreasing sequence of rationals $(q_{n})_{n \in \mathbb {N}}$ such that $\lim _{n \to \infty } q_{n} = \left \lVert r(\overline {x})\right \rVert $ .

Conversely, suppose $A^{\dagger }$ is right-c.e. Then there is an effective procedure which, when given a rational point r of $A^{\dagger }$ , enumerates a decreasing sequence $(q_{n})_{n \in \mathbb {N}}$ of dyadic rationals such that $\lim _{n \to \infty } q_{n} = \left \lVert r\right \rVert $ . Let $\overline {a}$ be such that $A^{\dagger } = (A,\overline {a})$ , and let $\overline {x}$ be a sequence of generators of the same length. Let $\mathcal {R}$ be the set of relations $\left \lVert r(\overline {x})\right \rVert \leq d$ where $r(\overline {a})$ is a rational point of $A^{\dagger }$ and d is one of the positive dyadic rationals enumerated by the procedure given $r(\overline {a})$ . Then $\mathcal {R}$ is c.e., and $A^{\dagger } = {\mathrm {C}^{*} \langle {\overline {x}}\operatorname {\mathrm {|}}{\mathcal {R}};{\mathbb {K}}\rangle }$ .

Proof We write $\left \lVert \cdot \right \rVert _f$ for the norm on the direct sum of all finite-dimensional representations of A. Since A is RFD, we have $\left \lVert \cdot \right \rVert _{A} = \left \lVert \cdot \right \rVert _f$ .

For any $n \in \mathbb {N}$ , consider the set $X_n = \{\overline {m} \in M_n(\mathbb {K})^{|\overline {x}|} {\colon \,} \mathcal {R}[\overline {m}] \text { holds in } M_{n}(\mathbb {K})\}$ as a subset of $\mathbb {R}^{k|\overline {x}|n^2}$ , where k is the dimension of $\mathbb {K}$ over $\mathbb {R}$ . Note $X_n$ is compact since every relation in $\mathcal {R}$ is a closed condition and we require for every generator $x_i$ that there is a relation of the form $\left \lVert x_i\right \rVert \leq M_i$ in $\mathcal {R}$ . Furthermore, $X_n$ is definable in the language of real closed fields since $\mathcal {R}$ is finite and if $\left \lVert p(\overline {x})\right \rVert \leq d$ is a relation in $\mathcal {R}$ , then $\left \lVert p(\overline {m})\right \rVert \leq d$ holds in $M_{n}(\mathbb {K})$ if and only if

$$\begin{align*}(\forall \lambda \in \mathbb{R}) [\textstyle\det_n\big(p(\overline{m})^*p(\overline{m}) - \lambda I\big) = 0 \to \lambda \leq d^2].\end{align*}$$

Let $D_n = \{\xi \in \mathbb {K}^n : \left \lVert \xi \right \rVert \leq 1\}$ . Fix a rational point $q(\overline {x}) \in A^{\dagger }$ . Define $F_{n,q} : X_n \times D_n \to \mathbb {R}$ by $F_{n,q}(\overline {m},\xi ) = \left \lVert q(\overline {m})\xi \right \rVert ^2$ , and let $\alpha _{n,q}$ be the maximum value of $F_{n,q}$ . Following closely, we see there is actually an effective procedure which, when given inputs n and q, returns a formula that defines $\alpha _{n,q}$ in the language of real closed fields. Applying the effective quantifier elimination given by Tarski–Seidenberg, we see this formula can even be made quantifier-free, so $\alpha _{n,q}$ is computable uniformly in n and q.

Observe $\left \lVert q(\overline {x})\right \rVert _A = \left \lVert q(\overline {x})\right \rVert _f \leq \sup \{\alpha _{n,q}^{1/2} {\colon \,} n \in \mathbb {N}\}$ since every finite-dimensional representation of A can be embedded in $M_n(\mathbb {K})$ for sufficiently large n. By the universality of ${\mathrm {C}^{*} \langle {\overline {x}}\operatorname {\mathrm {|}}{\mathcal {R}};{\mathbb {K}}\rangle }$ , we also have $\left \lVert q(\overline {x})\right \rVert _A \geq \sup \{\alpha _{n,q}^{1/2} {\colon \,} n \in \mathbb {N}\}$ . Hence $\left \lVert q(\overline {x})\right \rVert _A = \sup \{\alpha _{n,q}^{1/2} : n \in \mathbb {N}\}$ , so along with Theorem 3.3 we can conclude that $\left \lVert q(\overline {x})\right \rVert $ is computable uniformly in q. Thus $A^{\dagger }$ is computable. This procedure is uniform in $\mathcal {R}$ since the formula which defines $\alpha _{n,q}$ can be effectively determined uniformly in $\mathcal {R}$ , and the procedure from Theorem 3.3 is uniform in $\mathcal {R}$ .

Fact 3.7. $\mathcal {F}(n;\mathbb {K})$ is RFD for every $n \in \mathbb {N} \cup \{\omega \}$ .

Proof (a) $\implies $ (c) clearly holds.

(c) $\implies $ (a). This follows by the standard argument known as Craig’s trick. Since $A^{\dagger }$ is c.e., $A^{\dagger }$ is the standard presentation ${\mathrm {C}^{*} \langle {\overline {x}}\operatorname {\mathrm {|}}{\mathcal {R}};{\mathbb {K}}\rangle }$ for some c.e. set of rational relations $\mathcal {R}$ . For every relation $\left \lVert p(\overline {x})\right \rVert \leq d$ in $\mathcal {R}$ , include the relation $\left \lVert p(\overline {x}) + k0\right \rVert \leq d$ in $\mathcal {S}$ , where k encodes a Turing computation which witnesses that $\left \lVert p(\overline {x})\right \rVert \leq d$ belongs to $\mathcal {R}$ . Then $\mathcal {S}$ is computable and $A^{\dagger }$ is the standard presentation ${\mathrm {C}^{*} \langle {\overline {x}}\operatorname {\mathrm {|}}{\mathcal {S}};{\mathbb {K}}\rangle }$ .

Let $\overline {a}$ be such that $A^{\dagger } = (A,\overline {a})$ , and let $\overline {c}$ be the corresponding sequence of generators for $\mathcal {F}(|\overline {a}|;\mathbb {K})$ . Let N be the word problem of $A^{\dagger }$ .

(b) $\implies $ (c). Given a rational point $r(\overline {a})$ of $A^{\dagger }$ , we enumerate the set of all open rational balls of $\mathcal {F}(|\overline {a}|;\mathbb {K})$ that intersect N. For each rational ball, we output a positive dyadic rational $q_{n}$ if $q_{n} \leq q_{i}$ for all $i < n$ , and the open rational ball is of the form $B(r(\overline {c}), q_{n})$ . Note $B(r(\overline {c}),q_{n})$ intersects N if and only if $\left \lVert r(\overline {a})\right \rVert < q_{n}$ . Then $(q_{n})_{n \in \mathbb {N}}$ is a decreasing sequence of rationals such that $\lim _{n \to \infty } q_{n} = \left \lVert r(\overline {a})\right \rVert $ . By Theorem 3.3, we conclude the presentation $A^{\dagger }$ is c.e.

(c) $\implies $ (b). We enumerate through all rational points of $\mathcal {F}(|\overline {a}|;\mathbb {K})$ and all positive dyadic rationals. For each rational point $r(\overline {c})$ and positive dyadic rational d, we begin an enumeration of the decreasing sequence $(q_{n})_{n \in \mathbb {N}}$ of rationals determined by Theorem 3.3 on input $r(\overline {a})$ , and output $B(r(\overline {c}),d)$ if ever $q_{n} < d$ for some $n \in \mathbb {N}$ . Since $B(r(\overline {c}),d)$ intersects N if and only if $\left \lVert r(\overline {a})\right \rVert < d$ , we have shown N is c.e. closed.

Proof Let $\overline {a}$ be such that $A^{\dagger } = (A,\overline {a})$ , and let $\overline {c}$ be the corresponding sequence of generators for $\mathcal {F}(|\overline {a}|;\mathbb {K})$ . Let N be the word problem of $A^{\dagger }$ .

Suppose $A^{\dagger }$ is a computable presentation. Then $A^{\dagger }$ is a c.e. presentation, so N is c.e. closed by Theorem 3.11. We show N is strongly co-c.e. closed. We enumerate through all rational points $r(\overline {a})$ of $A^{\dagger }$ , positive dyadic rationals d, and $k \in \mathbb {N}$ . The computable presentation $A^{\dagger }$ determines a rational q such that $|\left \lVert r(\overline {a})\right \rVert - q| < 2^{-k}$ . If $d \leq q - 2^{-k}$ , then we output the closed rational ball $\widehat {B}(r(\overline {c}), d)$ . If d is a positive dyadic rational such that $\widehat {B}(r(\overline {c}), d) \cap N = \emptyset $ , then $\left \lVert r(\overline {a})\right \rVert> d$ by Fact 2.11. So, if $k \in \mathbb {N}$ is such that $2^{-k+1} < \left \lVert r(\overline {a})\right \rVert - d$ , then $d \leq q - 2^{-k}$ for any rational q such that $|\left \lVert r(\overline {a})\right \rVert - q| < 2^{-k}$ . Hence $\widehat {B}(r(\overline {c}), d)$ is eventually output.

Conversely, suppose N is strongly computable closed. If $B(r(\overline {c}),d)$ is an open rational ball, then it intersects N if and only if $\left \lVert r(\overline {a})\right \rVert < d$ . Similarly, if $\widehat {B}(r(\overline {c}),e)$ is a closed rational ball, then $\widehat {B}(r(\overline {c}),e) \cap N = \emptyset $ if and only if $\left \lVert r(\overline {a})\right \rVert> e$ by Fact 2.11. We define an effective procedure which witnesses that $A^{\dagger }$ is computable. Given a rational point $r(\overline {a}) \in A^{\dagger }$ and $k \in \mathbb {N}$ , we begin enumerating all open rational balls $B(r(\overline {c}), d_{i})$ , centered at $r(\overline {c})$ , that intersect N. We also begin enumerating all closed rational balls $\widehat {B}(r(\overline {c}),e_{j})$ , centered at $r(\overline {c})$ , such that $\widehat {B}(r(\overline {c}),e_{j}) \cap N = \emptyset $ . If ever $d_{i} - e_{j} < 2^{-k+1}$ for some $i,j \in \mathbb {N}$ , we return $\frac {1}{2}(d_{i} - e_{j})$ .

Question 3.13. Is there a presentation $A^{\dagger }$ of a C*-algebra A over $\mathbb {K}$ such that $A^{\dagger }$ is not computable but has computable closed word problem?

Proof Let $\mathcal {R}$ be a c.e. set of rational relations such that $A^{\dagger }$ is ${\mathrm {C}^{*} \langle {\overline {x}}\operatorname {\mathrm {|}}{\mathcal {R}};{\mathbb {K}}\rangle }$ .

Let T be the continuous first-order theory of C*-algebras over $\mathbb {K}$ . We expand our language to include additional constants for the generators $\overline {x}$ . Fix a rational point $q(\overline {x})$ of $A^{\dagger }$ and a positive dyadic rational $\ell $ such that $\left \lVert q(\overline {x})\right \rVert> \ell $ . Since A is simple, if $r(\overline {x})$ is a rational point of $A^{\dagger }$ and d is a positive dyadic rational, then ${\mathrm {C}^{*} \langle {\overline {x}}\operatorname {\mathrm {|}}{\mathcal {R} \cup \{\left \lVert r(\overline {x})\right \rVert \leq d\}};{\mathbb {K}}\rangle }$ is just A if $\left \lVert r(\overline {x})\right \rVert \leq d$ and $\{0\}$ if $\left \lVert r(\overline {x})\right \rVert> d$ . Hence by Pavelka-style completeness, $\left \lVert r(\overline {x})\right \rVert> d$ if and only if $T \cup \mathcal {R} \cup \{\left \lVert r(\overline {x})\right \rVert \leq d\} \vdash \left \lVert q(\overline {x})\right \rVert \leq \ell $ .

We define an effective procedure which witnesses that $A^{\dagger }$ is computable. Given a rational point $r(\overline {x}) \in A^{\dagger }$ and $k \in \mathbb {N}$ , we apply Theorem 3.3 and enumerate a decreasing sequence $(q_n)_{n \in \mathbb {N}}$ of positive dyadic rationals such that $\lim _{n \to \infty } q_n = \left \lVert r(\overline {x})\right \rVert $ . Let be $a - b$ if $a \geq b$ and $0$ otherwise for $a,b \in \mathbb {R}$ . For each $q_{n}$ , we begin enumerating through all formal deductions from , and we return $q_{n}$ if the formal deduction witnesses that .

Proof (a) $\implies $ (b). Since $\theta $ is computable, $\cos (2\pi \theta )$ and $\sin (2\pi \theta )$ are also computable. Let $(a_k)_{k \in \mathbb {N}}$ be a computable enumeration of rationals such that $|\cos (2\pi \theta ) - a_k| \leq 2^{-k}$ for $k \in \mathbb {N}$ , and let $(b_k)_{k \in \mathbb {N}}$ be a computable enumeration of rationals such that $|\sin (2\pi \theta ) - b_k| \leq 2^{-k}$ for $k \in \mathbb {N}$ . Let

$$ \begin{align*} \mathcal{S} = \operatorname{\mathrm{Iden}}(1;u,v) &\cup \{u^*u = uu^* = v^*v = vv^* = 1\} \\ &\cup \{\left\lVert uv - (a_k +ib_k)vu\right\rVert \leq 2^{-k+1} : k \in \mathbb{N}\}. \end{align*} $$

Then $\mathcal {S}$ is c.e. and $A_{\theta }$ is the standard presentation ${\mathrm {C}^{*} \langle {u,v,1}\operatorname {\mathrm {|}} {\mathcal {S}};{\mathbb {K}}\rangle }$ .

(b) $\implies $ (c). Since $A_{\theta }$ is simple, by Theorem 3.14, $A_{\theta }$ is computable.

(c) $\implies $ (a). Note that $uvu^*v^* - vuv^*u^* = 2i\sin (2\pi \theta )1$ is a rational point of $A_{\theta }$ , so $\sin (2\pi \theta )$ is computable. Similarly, $uvu^*v^* + vuv^*u^* = 2i\cos (2\pi \theta )1$ is a rational point of $A_{\theta }$ , so $\cos (2\pi \theta )$ is computable. The angle $\theta $ can be calculated from $\sin (2\pi \theta )$ and $\cos (2\pi \theta )$ with use of the arctangent function. Thus $\theta $ is computable.

Proof Let $\overline {g}$ be such that $G^{\dagger } = (G,\overline {g})$ . Let $\overline {c}$ be the corresponding sequence of generators for $\mathcal {F}(|\overline {g}|;\mathbb {K})$ . Let e be the identity in G.

(a) $\implies $ (b). Since $G^{\dagger }$ is c.e., ${\mathrm {C}_{\mathrm {uni}}^{*} ({G^{\dagger }};{\mathbb {K}})}$ is c.e. By Theorem 3.11, $N_{\mathrm {uni}}$ is c.e. closed.

We show $N_{\mathrm {red}}$ is strongly co-c.e. closed. We enumerate through all rational points $r(\overline {g})$ of ${\mathrm {C}^{*}_{\mathrm {red}} ({G^{\dagger }};{\mathbb {K}})}$ and all positive dyadic rationals d. Since $G^{\dagger }$ is computable, for each pair $r(\overline {g})$ and d, we can begin an enumeration of all finite sums $\sum _{i=1}^{n} a_{i}h_{i}$ where $a_{1},\ldots ,a_{n}$ are nonzero rationals of $\mathbb {K}$ , $h_{1},\ldots ,h_{n}$ are words on $\overline {g}$ such that $h_{i} \neq h_{j}$ in G for $1 \leq i < j < n$ , and

$$\begin{align*}\left\lVert\sum_{i=1}^{n} a_{i}h_{i}\right\rVert_{\ell^{2}(G)}^{2} = \sum_{i=1}^{n} a_{i}^{2} \leq 1.\end{align*}$$

Again using that $G^{\dagger }$ is computable, we can effectively rewrite the action of $r(\overline {g})$ on $\sum _{i=1}^{n} a_{i}h_{i}$ into the form $\sum _{i=1}^{m} b_{i}f_{i}$ where $b_{1},\ldots ,b_{m}$ are nonzero rationals of $\mathbb {K}$ , and $f_{1},\ldots ,f_{m}$ are words on $\overline {g}$ such that $f_{i} \neq f_{j}$ in G for $1 \leq i < j \leq m$ . Note

$$\begin{align*}\left\lVert r(\overline{g})\right\rVert_{\mathrm{red}} \geq \left\lVert\sum_{i=1}^{m} b_{i}f_{i}\right\rVert_{\ell^{2}(G)} = \bigg(\sum_{i=1}^{m} b_{i}^{2}\bigg)^{1/2}.\end{align*}$$

If $d < (\sum _{i=1}^{m} b_{i}^{2})^{1/2}$ , we output $\widehat {B}(r(\overline {c}),d)$ . Since $\left \lVert r(\overline {g})\right \rVert _{\mathrm {red}}$ is the supremum of all such $\left \lVert \sum _{i=1}^{m} b_{i}f_{i}\right \rVert _{\ell ^{2}(G)}$ , by Fact 2.11, we have shown $N_{\mathrm {red}}$ is strongly co-c.e. closed.

(b) $\implies $ (c). Since $N_{\mathrm {red}}$ is strongly co-c.e. closed, we can enumerate all rational points r such that $\left \lVert r\right \rVert _{\mathrm {red}}> 0$ . Now, we just observe that $\left \lVert r\right \rVert _{\mathrm {uni}}> 0$ if and only if $\left \lVert r\right \rVert _{\mathrm {red}}> 0$ for rational points r.

(c) $\implies $ (a). Let w be a word on generators $\overline {g}$ . Then $\left \lVert w - e\right \rVert _{\mathrm {uni}} = 0$ or $\left \lVert w - e\right \rVert _{\mathrm {uni}} \geq \sqrt {2}$ . Since $N_{\mathrm {uni}}$ is c.e. closed, we can enumerate all open rational balls $B(r(\overline {c}),d)$ that intersect $N_{\mathrm {uni}}$ . If ever $r(\overline {g}) = w - e$ and $d < \sqrt {2}$ , we have determined that $w = e$ in G. Simultaneously, we enumerate all rational points $q(\overline {c})$ which belong $N_{\mathrm {uni}}^{c}$ . If ever $q(\overline {g}) = w - e$ , then $\left \lVert w - e\right \rVert _{\mathrm {uni}}> 0$ , so $w \neq e$ in G.

(b) $\implies $ (d). Since $N_{\mathrm {uni}}$ is c.e. closed, we can enumerate all open rational balls $B(r(\overline {c}),d)$ such that $\left \lVert r(\overline {g})\right \rVert _{\mathrm {uni}} < d$ . Any such open rational ball intersects $N_{\mathrm {red}}$ since $\left \lVert \cdot \right \rVert _{\mathrm {red}} \leq \left \lVert \cdot \right \rVert _{\mathrm {uni}}$ . If $r(\overline {c})$ belongs to $N_{\mathrm {red}}$ , then $\left \lVert r(\overline {g})\right \rVert _{\mathrm {red}} = 0$ , so $\left \lVert r(\overline {g})\right \rVert _{\mathrm {uni}} = 0$ . Hence $B(r(\overline {c}),d)$ will be enumerated for all positive dyadic rationals d.

(d) $\implies $ (a). Let w be a word on generators $\overline {g}$ . Then $\left \lVert w - e\right \rVert _{\mathrm {red}} = 0$ or $\left \lVert w - e\right \rVert _{\mathrm {red}} \geq \sqrt {2}$ . We express $w - e$ as a rational point $q(\overline {g})$ . Since $N_{\mathrm {red}}$ is co-c.e. closed, we can enumerate a sequence of open rational balls $B(r(\overline {c}), d)$ whose union is $N_{\mathrm {red}}^{c}$ . If ever $\left \lVert q(\overline {c}) - r(\overline {c})\right \rVert < d$ , then $q(\overline {c})$ belongs to $N_{\mathrm {red}}^{c}$ , so we have determined $w \neq e$ in G. Simultaneously, we enumerate a sequence of open rational balls $B(s(\overline {c}),e)$ , each which intersects $N_{\mathrm {red}}$ , such that the sequence contains all balls centered at rational points belonging to $N_{\mathrm {red}}$ . If ever $s(\overline {g}) = w - e$ and $d < \sqrt {2}$ , we have determined that $w = e$ in G.

Question 3.21. Is there a computable presentation $G^{\dagger }$ of a countable discrete group G such that ${\mathrm {C}_{\mathrm {uni}}^{*} ({G^{\dagger }};{\mathbb {K}})}$ or ${\mathrm {C}^{*}_{\mathrm {red}} ({G^{\dagger }};{\mathbb {K}})}$ does not have computable closed word problem?

Proof Let $X^{\dagger } = (X,{(x_n)}_{n \in \mathbb {N}})$ be a computably proper presentation of X. For each $n \in \mathbb {N}$ , let $f_n : X \to \mathbb {K}$ be defined by $f_n(z) = \frac {1}{1 + d(x_n,z)}$ , and observe $f_n$ belongs to ${C_0(X;\mathbb {K})}$ . Since ${(f_n)}_{n \in \mathbb {N}}$ separates points and vanishes nowhere, ${\mathrm {C}_{\mathrm {gen}}^{*} ({{(f_n)}_{n \in \mathbb {N}}};{\mathbb {K}})} = C_0(X; \mathbb {K})$ by Stone–Weierstrass. Let ${C_0(X;\mathbb {K})}^{\dagger } = ({C_0(X;\mathbb {K})}, {(f_{n})}_{n \in \mathbb {N}})$ .

We show ${C_0(X;\mathbb {K})}^{\dagger }$ is computable. We are given a rational point $q(f_{\ell _1},\ldots ,f_{\ell _n})$ of ${C_0(X;\mathbb {K})}^{\dagger }$ and a positive integer $k \in \mathbb {N}$ . We must compute a rational r such that $|\left \lVert q(f_{\ell _1},\ldots ,f_{\ell _n})\right \rVert - r| < 2^{-k}$ . By Proposition 2.17, we can effectively determine $j \in \mathbb {N}$ so that, for all $z,w \in X$ , if $\max _i |f_{\ell _i}(z) - f_{\ell _i}(w)| \leq 2 ^{-j}$ , then $|q(f_{\ell _1}(z),\ldots ,f_{\ell _n}(z)) - q(f_{\ell _1}(w),\ldots ,f_{\ell _n}(w))| < 2^{-(k+1)}$ . Let M a positive integer which bounds the sum of the absolute values of the coefficients of q. Let $K = \bigcup _{i=1}^{n}\{z \in X {\colon \,} d(x_{\ell _i}, z) \leq M2^{k+1}\}$ , so K is a finite union of closed rational balls. As $X^{\dagger }$ is computably proper, we can effectively determine a sequence of open rational balls of radius at most $2^{-j}$ , centered at points $x_{\nu _{1}},\ldots ,x_{\nu _{r}}$ , such that the sequence covers K. Let $N = \max _i |q(f_{\ell _1}(x_{\nu _i}),\ldots ,f_{\ell _n}(x_{\nu _i}))|$ , so N is computable since $X^{\dagger }$ is computable.

Now, for all $z \in X$ , either there exists $x_{\nu _{m}}$ such that $d(x_{\nu _{m}}, z) < 2^{-j}$ , or $z \not \in K$ so $|q(f_{\ell _1}(z),\ldots ,f_{\ell _n}(z))| \leq 2^{-(k+1)}$ . If $d(x_{\nu _m}, z) < 2^{-j}$ , then since

$$\begin{align*}\max_i |f_{\ell_i}(x_{\nu_m}) - f_{\ell_i}(z)| = \max_{i}\bigg|\frac{d(x_{\ell_{i}}, z) - d(x_{\ell_{i}}, x_{\nu_{m}})}{(1 + d(x_{\ell_{i}}, x_{\nu_{m}}))(1 + d(x_{\ell_{i}}, z))}\bigg| \leq d(x_{\nu_m},z),\end{align*}$$

we can conclude $|q(f_{\ell _1}(z),\ldots ,f_{\ell _n}(z))| < N + 2^{-(k+1)}$ . Hence

$$\begin{align*}N \leq \left\lVert q(f_{\ell_1},\ldots,f_{\ell_n})\right\rVert \leq N + 2^{-(k+1)}.\end{align*}$$

Since N is computable, we can effectively determine a rational r such that $|\left \lVert q(f_{\ell _1},\ldots ,f_{\ell _n})\right \rVert - r| < 2^{-k}$ , as required.

If $C_0(X;\mathbb {R})$ admits a computable presentation as a real C*-algebra (real Banach algebra), then the real Banach space presentation formed from a computable sequence of the products of the special points is computable.

Proof If $(B, \tau , (z_n)_{n \in \mathbb {N}})$ is a presentation of $(B,\tau )$ , then we consider the induced presentation $(A, (x_n)_{n \in \mathbb {N}})$ on A given by $x_{2n-1} = \frac {1}{2}(z_n + \tau (z_n))$ and $x_{2n} = \frac {1}{2i}(z_n - \tau (z_n))$ for $n \in \mathbb {N}$ . Let $(B,\tau )^{+}$ and $(B,\tau )^{\dagger }$ be computable presentations of $(B,\tau )$ . Let $A^{+}$ and $A^{\dagger }$ be the presentations of A induced by $(B,\tau )^{+}$ and $(B,\tau )^{\dagger }$ , respectively. Then $A^{+}$ and $A^{\dagger }$ are also computable. As A is computably categorical, there exists a computable isomorphism $\eta : A^{+} \to A^{\dagger }$ . Thus $\varphi : B^{+} \to B^{\dagger }$ , defined by $\varphi (z) = \eta \big (\frac {1}{2}(z + \tau (z))\big ) + i\eta \big (\frac {1}{2i}(z - \tau (z))\big )$ , is a computable isomorphism.

Proof Note A is the real C*-algebra determined by $(B,*)$ . If $(A, (x_n)_{n \in \mathbb {N}})$ is a presentation of A, then we call $(B,(x_n)_{n \in \mathbb {N}})$ the induced presentation of B. Furthermore, if $(A,(x_n)_{n \in \mathbb {N}})$ is computable, then $(B,(x_n)_{n \in \mathbb {N}})$ is computable since

$$\begin{align*}\left\lVert r + is\right\rVert =\sqrt{\left\lVert(r+is)^*(r + is)\right\rVert} = \sqrt{\left\lVert r^2 + s^2\right\rVert}\end{align*}$$

for all rational points r and s of $(A,(x_n)_{n \in \mathbb {N}})$ .

(b) $\iff $ (c). This follows since $* : A \to A$ is simply the identity map.

(b) $\implies $ (a). By Lemma 4.4, if A is computably categorical as a real C*-algebra, then $(B,*)$ is computably categorical as a complex C*-algebra with conjugation. Since the conjugation on B is just the $*$ -operation, B is computably categorical as a complex C*-algebra.

(a) $\implies $ (b) Let $A^{+}$ and $A^{\dagger }$ be computable presentations of A. Let $B^{+}$ and $B^{\dagger }$ be the computable presentations on B induced by $A^{+}$ and $A^{\dagger }$ , respectively. As B is computably categorical, there exists a computable isomorphism $\varphi : B^{+} \to B^{\dagger }$ . Then $\varphi \restriction A : A^{+} \to A^{\dagger }$ is a computable isomorphism.

Proof (a) $\iff $ (b) is the result in [Reference Bazhenov, Harrison-Trainor and Melnikov1].

(c) $\iff $ (d) was established above.

As shown in [Reference Harrison-Trainor, Melnikov and Ng19], every computably metrizable Stone space has a computably compact presentation. By Theorem 4.3, (a) $\implies $ (d).

Finally, (c) $\implies $ (b) follows by taking any real C*-algebra presentation which is computable and considering a computable sequence of the products of the special points.

Proof By Theorem 4.5, ${C({[0,1]};{\mathbb {K}})}$ is computably categorical as a C*-algebra over $\mathbb {K}$ if and only if ${C({[0,1]};{\mathbb {R}})}$ is computably categorical as a real Banach algebra. Melnikov and Ng showed in [Reference Melnikov and Ng35] that ${C({[0,1]};{\mathbb {R}})}$ is not computably categorical in the language of real Banach algebras.

Proof We proceed by induction on the size of $\sigma (x) \setminus \{0\}$ . Clearly, if $\operatorname {\mathrm {card}}(\sigma (x) \setminus \{0\}) = 0$ , then $\sigma (x) = \{0\}$ where $0$ is computable.

Otherwise, let $u \in \sigma (x) \setminus \{0\}$ be such that $|u| = \max _{t \in \sigma (x)} |t| = \left \lVert x\right \rVert $ , so u is computable. Let $p(z) = z(z-u) \in \mathbb {R}[z]$ . Then $p(x) = x^2 - ux$ is a self-adjoint computable point of $A^{\dagger }$ and

$$\begin{align*}\operatorname{\mathrm{card}}(\sigma(p(x)) \setminus \{0\}) = \operatorname{\mathrm{card}}(p[\sigma(x)] \setminus \{0\}) < \operatorname{\mathrm{card}}(\sigma(x) \setminus \{0\}).\end{align*}$$

By the inductive hypothesis, every element of $\sigma (p(x))$ is computable. Since $\sigma (p(x)) = p[\sigma (x)]$ , every element of $\sigma (x)$ is a real root of the computable polynomial $p(z) - w \in \mathbb {R}[z]$ for some $w \in \sigma (p(x))$ . It is well known that the computable reals form a real closed field (see [Reference Pour-El and Richards40] for details). Thus every element of $\sigma (x)$ is computable.

Proof Let A be an abelian finite-dimensional real C*-algebra, and let $A^{\dagger }$ be a computable presentation of A. By Fact 2.5, we can identify A with $\mathbb {R}^k \oplus \mathbb {C}^m$ for some positive integers k and m.

Let X be the set of self-adjoint rational points of $A^{\dagger }$ , and let Y be the set of skew-adjoint rational points of $A^{\dagger }$ , so the set of rational points of $A^{\dagger }$ is $X + Y$ . We know X is dense in $\mathbb {R}^{k}\oplus \mathbb {R}^{m}$ and Y is dense in $\{0\}^k \oplus {(i\mathbb {R})}^m$ . In particular, there must exist $x \in X$ such that x has distinct nonzero entries, and $y \in Y$ such that only the first k entries are zero. We view $\mathbb {R}^{k} \oplus \mathbb {R}^{m}$ as a subspace of ${B({\mathbb {R}^{k} \oplus \mathbb {R}^{m}};{\mathbb {R}})}$ where $\mathbb {R}^{k} \oplus \mathbb {R}^{m}$ acts on itself by multiplication. In this sense, the minimal polynomial of x must be of degree $k+m$ with a nonzero constant term. Then $x,\ldots ,x^{k+m}$ are linearly independent over $\mathbb {R}$ , so ${\mathrm {C}_{\mathrm {gen}}^{*} ({x};{\mathbb {R}})} = \mathbb {R}^{k}\oplus \mathbb {R}^{m}$ . Also, we can find $z \in \mathbb {R}^{k} \oplus \mathbb {R}^{m}$ such that the last m entries of $zy$ are i. Hence ${\mathrm {C}_{\mathrm {gen}}^{*} ({x,y};{\mathbb {R}})} = \mathbb {R}^k \oplus \mathbb {C}^m$ . By Corollary 2.19, $(A,x,y)$ is a computable presentation computably isomorphic to $A^{\dagger }$ via the identity map.

The entries of an element $z \in A$ belong to $\sigma _{A}(z)$ since if $a + ib$ is an entry of z for some $a,b \in \mathbb {R}$ , then $(z - a)^{2} + b^{2}$ is not invertible in A. Every element of $\sigma (x)$ is computable by Lemma 5.1, so in particular, every entry of x is computable. Similarly, $y^2$ is self-adjoint, so by Lemma 5.1, every element of $\sigma (y^2)$ is computable. Then every element of $\sigma (y)$ is computable, since $\sigma (y^2) = \sigma (y)^2$ , and the computable complex numbers form an algebraically closed field (see [Reference Specker, Jäger, Läuchli, Scarpellini and Strassen44] for details). In particular, every entry of y is computable. Thus, by Corollary 2.19, $(A,x,y)$ is computably isomorphic to the standard presentation $\mathbb {R}^k \oplus \mathbb {C}^m$ .

Proof By Fact 2.4, any abelian finite-dimensional C*-algebra can be identified with $\mathbb {C}^n$ for some $n \in \mathbb {N}$ . It can be observed, with the use of Theorem 4.5, that $\mathbb {C}^n$ is computably categorical as a complex C*-algebra if and only if $\mathbb {R}^n$ is computably categorical as real C*-algebra.

Proof Let $\mathbb {H}^{\dagger }$ be a computable presentation of $\mathbb {H}$ .

For any nonzero self-adjoint computable point x of $\mathbb {H}^{\dagger }$ , we know $x \in \mathbb {R}$ , and $\left \lVert x\right \rVert $ is computable. Hence $1 = \operatorname {\mathrm {sgn}}(x)\frac {x}{\left \lVert x\right \rVert }$ is a computable point of $\mathbb {H}^{\dagger }$ .

We apply the Gram–Schmidt process to a pair of $\mathbb {R}$ -linearly independent skew-adjoint computable points of $\mathbb {H}^{\dagger }$ , noting that $\langle a,b\rangle _{\mathbb {R}} = \frac {1}{2}(a^*b + b^*a)$ is a computable operation, to get a pair $p,q$ of $\mathbb {R}$ -orthonormal skew-adjoint computable points of $\mathbb {H}^{\dagger }$ .

Then $p^2 = -p^*p = -1 = -q^*q = q^2$ , and $0 = \langle p, q \rangle _{\mathbb {R}} = -\frac {1}{2}(pq + qp)$ so $pq = -qp$ . Thus there is an automorphism $\varphi $ of $\mathbb {H}$ which sends $1 \mapsto 1$ , $p \mapsto i$ , $q \mapsto j$ , and $pq \mapsto k$ . By Corollary 2.19, $(\mathbb {H},1,p,q)$ is a computable presentation computably isomorphic to $\mathbb {H}^{\dagger }$ . Therefore, $\varphi $ is a computable isomorphism from $\mathbb {H}^{\dagger }$ to the standard presentation $\mathbb {H}$ .

Proof We proceed by induction on the $\mathbb {K}$ -dimension of A. Let X be the set of self-adjoint elements of A. Let Z be the subset of $A^{\dagger }$ -computable points of X, so Z is dense in X.

For $z \in Z$ , ${\mathrm {C}_{\mathrm {gen}}^{*} ({z};{\mathbb {K}})} \cong \mathbb {K}^n$ for some $n \in \mathbb {N}$ . By Theorem 5.2 or Corollary 5.3, $\mathbb {K}^{n}$ is computably categorical as a C*-algebra over $\mathbb {K}$ . In particular, there must be a family F of minimal projections in ${\mathrm {C}_{\mathrm {gen}}^{*} ({z};{\mathbb {K}})}$ , each computable with respect to $({\mathrm {C}_{\mathrm {gen}}^{*} ({z};{\mathbb {K}})},z)$ , such that $z \in {\operatorname {\mathrm {span}}_{{\mathbb {R}}}({F})}$ . Since every projection in F is computable with respect to $({\mathrm {C}_{\mathrm {gen}}^{*} ({z};{\mathbb {K}})},z)$ , they must be also computable with respect to $A^{\dagger }$ by Proposition 2.18.

If $1$ is a minimal projection in A, then $X = \mathbb {R} = {\operatorname {\mathrm {span}}_{{\mathbb {R}}}({1})}$ and $1$ is $A^{\dagger }$ -computable. In that case, we just let $M = \{1\}$ .

Suppose $1$ is not a minimal projection in A. Then there exists $z \in Z$ such that ${\mathrm {C}_{\mathrm {gen}}^{*} ({z};{\mathbb {K}})}$ is not isomorphic to $\mathbb {K}$ , so must contain a copy of $\mathbb {K} \oplus \mathbb {K}$ . Hence, there is a nontrivial computable projection q in $A^{\dagger }$ . Let N be the set of all nontrivial projections computable with respect to $A^{\dagger }$ . If $1$ is $A^{\dagger }$ -computable, then $1 - q \in N$ , so ${\operatorname {\mathrm {span}}_{{\mathbb {R}}}({N})} = {\operatorname {\mathrm {span}}_{{\mathbb {R}}}({N \cup \{1\}})} = {\operatorname {\mathrm {span}}_{{\mathbb {R}}}({Z})} = X$ . If $1$ is not $A^{\dagger }$ -computable, we directly see ${\operatorname {\mathrm {span}}_{{\mathbb {R}}}({N})} = {\operatorname {\mathrm {span}}_{{\mathbb {R}}}({Z})} = X$ .

Let K be a finite subset of N such that ${\operatorname {\mathrm {span}}_{{\mathbb {R}}}({K})} {\kern-1.5pt}={\kern-1.5pt} {\operatorname {\mathrm {span}}_{{\mathbb {R}}}({N})} {\kern-1.5pt}={\kern-1.5pt} X$ . For any $p {\kern-1.5pt}\in{\kern-1.5pt} K$ , $pAp$ is a C*-subalgebra of A over $\mathbb {K}$ of strictly less dimension. By the inductive hypothesis, there is a finite set $M_p$ of computable minimal projections in $pA^{\dagger } p$ such that ${\operatorname {\mathrm {span}}_{{\mathbb {R}}}({M_p})}$ is the set of self-adjoint elements of $pAp$ . In particular, $p \in {\operatorname {\mathrm {span}}_{{\mathbb {R}}}({M_p})}$ . Every projection r which is minimal in $pAp$ is also minimal in A since for $t \in A$ if $t \leq r$ , then $t \leq p$ , so $t \in pAp$ . Also, each $r \in M_p$ is computable with respect to $A^{\dagger }$ . If we let $M = \bigcup _{p \in K} M_p$ , then M satisfies the required conditions.

Proof Let ${(A \oplus B)}^+$ be a computable presentation of $A \oplus B$ . By Theorem 5.5, there is a finite set M of computable minimal projections in ${(A \oplus B)}^+$ such that ${\operatorname {\mathrm {span}}_{{\mathbb {R}}}({M})}$ is the set of self-adjoint elements of $A \oplus B$ . Any minimal projection in $A \oplus B$ must belong to either A or B by minimality. For $X = A$ or $X = B$ , let $M_{X}$ be the subset of M which belongs to X. Also, let $Z_{X}$ be the set of products $qz_{1} \cdots z_{n}p$ where $z_{1},\ldots ,z_{n}$ are special points of $(A \oplus B)^{+}$ and $p,q \in M_{X}$ . Note $Z_A \subseteq A$ and $Z_B \subseteq B$ . Since $1 \in {\operatorname {\mathrm {span}}_{{\mathbb {R}}}({M})}$ , we have ${\mathrm {C}_{\mathrm {gen}}^{*} ({Z_A \cup Z_B};{\mathbb {K}})} = A \oplus B$ . Hence ${\mathrm {C}_{\mathrm {gen}}^{*} ({Z_A};{\mathbb {K}})} = A$ and ${\mathrm {C}_{\mathrm {gen}}^{*} ({Z_B};{\mathbb {K}})} = B$ . Then $(A, Z_A)$ and $(B, Z_B)$ are computable presentations of A and $B,$ respectively, by Proposition 2.18, and $(A,Z_{A}) \oplus (B,Z_{B})$ is a computable presentation computably isomorphic to $(A \oplus B)^{+}$ by Corollary 2.19.

Let $A^{\dagger }$ and $B^{\dagger }$ be computable presentations of A and $B,$ respectively. Since A is computably categorical, there exists a computable isomorphism $\varphi $ from $A^{\dagger }$ to $(A, Z_A)$ . Similarly, since B is computably categorical, there exists a computable isomorphism $\psi $ from $B^{\dagger }$ to $(B, Z_B)$ . Then $\varphi \oplus \psi : A^{\dagger } \oplus B^{\dagger } \to {(A \oplus B)}^+$ is a computable isomorphism.

Proof We proceed by induction on the length of the sequence. We denote ${B({\mathcal {H}};{\mathbb {D}})} $ by A for clarity.

It can be observed that ${\mathrm {C}_{\mathrm {gen}}^{*} ({pAp};{\mathbb {K}})} = pAp = {B({p\mathcal {H}};{\mathbb {D}})} \cong \mathbb {D}$ for any minimal projection p in A.

Now, let ${(p_i)}_{i=1}^{n+1}$ be as stated such that ${\mathrm {C}_{\mathrm {gen}}^{*} ({\bigcup _{i=1}^n p_i A p_i};{\mathbb {K}})} \cong M_n(\mathbb {D})$ . Since $M_n(\mathbb {D}) \cong {\mathrm {C}_{\mathrm {gen}}^{*} ({\bigcup _{i=1}^n p_i A p_i};{\mathbb {K}})} \subseteq {B({q_n\mathcal {H}};{\mathbb {D}})}$ , by dimensionality we must have $\mathrm {C}_{\mathrm {gen}}^{*} (\bigcup _{i=1}^n p_i A p_i;{\mathbb {K}}) = {B({q_n\mathcal {H}};{\mathbb {D}})}$ . Then $q_n$ must be of $\mathbb {D}$ -rank n, so $q_{n+1}$ is of $\mathbb {D}$ -rank $n+1$ as $p_{n+1}$ does not commute with $q_n$ . Hence

$$\begin{align*}{\mathrm{C}_{\mathrm{gen}}^{*} \bigg(\kern2pt{\bigcup_{i=1}^{n+1} p_i A p_i};{\mathbb{K}} \bigg)} \subseteq {B({q_{n+1}\mathcal{H}};{\mathbb{D}})} \cong M_{n+1}(\mathbb{D}). \end{align*}$$

For each $a \in \mathbb {D}$ , we let $R_{a} : \mathcal {H} \to \mathcal {H}$ be multiplication on the right by a. Let $w \in {B({q_{n+1}\mathcal {H}};{\mathbb {K}})}$ commute with ${\mathrm {C}_{\mathrm {gen}}^{*} ({\bigcup _{i=1}^{n+1} p_i A p_i};{\mathbb {K}})}$ . In particular, $wq_n$ belongs to the commutant of ${B({q_n\mathcal {H}};{\mathbb {D}})}$ in ${B({q_n\mathcal {H}};{\mathbb {K}})}$ , so is of the form $R_x q_n$ for some $x \in \mathbb {D}$ . Also, $wp_{n+1}$ belongs to the commutant of ${B({p_{n+1}\mathcal {H}};{\mathbb {D}})} = p_{n+1}Ap_{n+1}$ in ${B({p_{n+1}\mathcal {H}};{\mathbb {K}})}$ , so is of the form $R_y p_{n+1}$ for some $y \in \mathbb {D}$ . As w commutes with $p_{n+1}$ , we have that

$$\begin{align*}R_x p_{n+1}q_n = p_{n+1}R_x q_n = p_{n+1}wq_n = wp_{n+1}q_n = R_y p_{n+1}q_n.\end{align*}$$

Since $p_{n+1}$ does not commute with $q_n$ , $p_{n+1}q_n$ is nonzero, so $x = y$ . Then $w = R_{x}q_{n+1}$ since $q_{n+1}\mathcal {H} = p_{n+1}\mathcal {H} + q_n\mathcal {H}$ . Hence w commutes with ${B({q_{n+1}\mathcal {H}};{\mathbb {D}})}$ . Thus, by Definition 2.13,

$$\begin{align*}{\mathrm{C}_{\mathrm{gen}}^{*} \bigg({\bigcup_{i=1}^{n+1} p_i A p_i}; {\mathbb{K}}\bigg)} = {\mathrm{C}_{\mathrm{gen}}^{*} \bigg({\bigcup_{i=1}^{n+1}p_i A p_i};{\mathbb{K}} \bigg)}" = {B({q_{n+1}\mathcal{H}};{\mathbb{D}})}" = {B({q_{n+1}\mathcal{H}};{\mathbb{D}})} .\end{align*}$$

Proof We proceed by induction on n.

Observe $\mathbb {D}$ is computably categorical as a C*-algebra over $\mathbb {K}$ by Theorem 5.2, Corollary 5.3, or Lemma 5.4.

Now, assume $M_{n}(\mathbb {D})$ is computably categorical. For ease of reading, we denote $M_{n+1}(\mathbb {D})$ by A. Let $A^{\dagger }$ be a computable presentation of A.

To make use of the inductive hypothesis, we determine a subalgebra B isomorphic to $M_{n}(\mathbb {D})$ which is computably included in $A^{\dagger }$ . By Theorem 5.5, there is a finite set M of computable minimal projections in $A^{\dagger }$ such that ${\operatorname {\mathrm {span}}_{{\mathbb {R}}}({M})}$ is the set of self-adjoint elements of A. We construct a sequence $(p_i)_{i=1}^{n+1}$ of minimal projections from M so that $p_{k+1}$ does not commute with $q_k$ for $1 \leq k < n+1$ , where $q_k = \bigvee _{i=1}^k p_i$ . Choose $p_1 \in M$ . Suppose we have constructed $(p_i)_{i=1}^k$ for some $k \leq n$ . If $q_k$ commutes with every projection in M, then $q_k$ commutes with every self-adjoint element of A, so $q_{k} = dI$ for some nonzero $d \in \mathbb {D}$ . However, $q_k$ has rank at most k, while $dI$ has rank $n+1$ . Thus, we may choose $p_{k+1} \in M$ which does not commute with $q_k$ . Let $B = {\mathrm {C}_{\mathrm {gen}}^{*} ({\bigcup _{i=1}^n p_iAp_i};{\mathbb {K}})}$ . Let $B^{\dagger }$ be the presentation of B formed from the special points of $p_1A^{\dagger }p_{1},\ldots ,p_nA^{\dagger }p_{n}$ . Then $B^{\dagger }$ is computable and the inclusion from $B^{\dagger }$ into $A^{\dagger }$ is computable by Proposition 2.18. By Lemma 5.7, we have ${\mathrm {C}_{\mathrm {gen}}^{*} ({\bigcup _{i=1}^{n+1} p_iAp_i};{\mathbb {K}})} \cong M_{n+1}(\mathbb {D})$ and $B \cong M_n(\mathbb {D})$ . Hence ${\mathrm {C}_{\mathrm {gen}}^{*} ({\bigcup _{i=1}^{n+1} p_iAp_i};{\mathbb {K}})} = A$ and $B = {B({q_n\mathbb {D}^{n+1}};{\mathbb {D}})}$ .

Let Z be the standard orthonormal basis of $\mathbb {D}$ over $\mathbb {K}$ . By the inductive hypothesis, $M_n(\mathbb {D})$ is computably categorical over $\mathbb {K}$ . In particular, there is a system of matrix units $(f^z_{kj} {\colon \,} 1 \leq k,j \leq n, z \in Z)$ , computable with respect to $B^{\dagger }$ so computable with respect to $A^{\dagger }$ , such that $B = {\mathrm {C}_{\mathrm {gen}}^{*} ({\{f^z_{kj} {\colon \,} 1 \leq k,j \leq n, z \in Z\}};{\mathbb {K}})}$ .

Let $(e^z_{kj} {\colon \,} 1 \leq k,j \leq n+1, z \in Z)$ be the standard system of matrix units for $M_{n+1}(\mathbb {D})$ . We find a unitary U in $M_{n+1}(\mathbb {D})$ such that $U^{*}f_{kj}^{z}U = e^{z}_{kj}$ for $1 \leq k,j \leq n$ and $z \in Z$ . To that end, observe we can extend a $\mathbb {D}$ -orthonormal basis on $(\sum _{k=1}^n f_{kk})(\mathbb {D}^{n+1})$ to one for $\mathbb {D}^{n+1}$ , so there exists a unitary $V \in M_{n+1}(\mathbb {D})$ such that $V^*BV = M_n(\mathbb {D})$ . Then $(V^*f^z_{kj}V {\colon \,} 1 \leq k,j \leq n, z\in Z)$ forms a system of matrix units for $M_n(\mathbb {D})$ . Before we find U, we show there exists a unitary u in $M_{n}(\mathbb {D})$ such that $u^{*}V^{*}f^{z}_{kj}Vu = e^{z}_{kj}$ for $1 \leq k,j \leq n$ and $z \in Z$ . There are two cases. If the center of $\mathbb {D}$ is $\mathbb {K}$ , then u exists by Corollary 2.21. If the center of $\mathbb {D}$ is not $\mathbb {K}$ , it must be that $\mathbb {D} = \mathbb {C}$ , $\mathbb {K} = \mathbb {R}$ , and $Z = \{1,i\}$ . Then $\sum _{k=1}^{n} V^*f_{kk}^i V$ belongs to the center of $M_{n}(\mathbb {C})$ , and ${(\sum _{k=1}^{n} V^*f_{kk}^i V)}^2 = -I$ , so $\sum _{k=1}^{n} V^*f_{kk}^i V = \pm i$ . By possibly replacing $f_{kj}^i$ by their negations for $1 \leq k,j \leq n$ , we may assume $\sum _{k=1}^{n} f_{kk}^i = i$ . Then there exists a unitary matrix $u \in M_n(\mathbb {C})$ such that $u^*V^*f_{kj}Vu = e_{kj}$ for $1 \leq k,j \leq n$ , hence $u^*V^*f_{kj}^z Vu = e^z_{kj}$ for $1 \leq k,j \leq n$ and $z \in Z$ . For both cases, we let $U = V(u + e_{(n+1)(n+1)})$ .

Using U, we show we can construct a system of matrix units for A that is computable with respect to $A^{\dagger }$ . As $U^*p_{n+1}U$ is a projection which does not commute with the identity in $M_n(\mathbb {D})$ , there must exist $1 \leq \ell \leq n$ such that the $(n+1,\ell )$ th entry of $U^*p_{n+1}U$ is nonzero. For $z \in Z$ , let $w_z = p_{n+1}f_{\ell \ell }^z - \sum _{k=1}^n f_{kk}p_{n+1}f_{\ell \ell }^z$ . Then there exists a nonzero $d \in \mathbb {D}$ such that $U^*w_z U = d e^z_{(n+1)\ell }$ for $z \in Z$ . Note $\left \lVert d\right \rVert = \left \lVert w_1\right \rVert $ is computable. Now, we are ready to extend the system of matrix units. For $z \in Z$ , let

$$ \begin{align*} f^z_{(n+1)k} &= \frac{1}{\left\lVert d\right\rVert} w_z f_{\ell k} \text{ for } 1 \leq k \leq n,\\ f^z_{k(n+1)} &= \frac{1}{\left\lVert d\right\rVert} f^z_{k\ell}w_1^* \text{ for } 1 \leq k \leq n,\\ \text{and } f^z_{(n+1)(n+1)} &= \frac{1}{\left\lVert d\right\rVert^2}w_z w_1^*. \end{align*} $$

Then for $z \in Z$ , we observe

$$ \begin{align*} U^*f^z_{ij}U &= e^z_{kj} \text{ for } 1 \leq k,j \leq n,\\ U^*f^z_{(n+1)k}U &= \frac{d}{\left\lVert d\right\rVert}e^z_{(n+1)k} \text{ for } 1 \leq k \leq n,\\ U^*f^z_{k(n+1)}U &= e^z_{k(n+1)}\frac{d^*}{\left\lVert d\right\rVert} \text{ for } 1 \leq k \leq n,\\ \text{and } U^*f^z_{(n+1)(n+1)}U &= \frac{d}{\left\lVert d\right\rVert}e^z_{(n+1)(n+1)}\frac{d^*}{\left\lVert d\right\rVert}. \end{align*} $$

We only need to tweak our unitary conjugation. Let $x_{1},\ldots ,x_{n+1}$ be the standard $\mathbb {D}$ -orthonormal basis for $\mathbb {D}^{n+1}$ . If $W \in M_{n+1}(\mathbb {D})$ is the unitary matrix which sends $x_{i}$ to itself for $1 \leq i \leq n$ and sends $x_{n+1}$ to $x_{n+1}\frac {d}{\left \lVert d\right \rVert }$ , then ${(UW)}^*f^z_{kj}UW = e^z_{kj}$ for $1 \leq k,j \leq n+1$ and $z \in Z$ . Note each member of $(f^z_{kj} {\colon \,} 1 \leq k,j \leq n+1, z \in Z)$ is computable with respect to $A^{\dagger }$ , so by Corollary 2.19, $(A, (f^{z}_{kj} {\colon \,} 1 \leq k,j \leq n+1, z \in Z))$ is computably isomorphic to $A^{\dagger }$ via the identity map. Thus conjugation by $UW$ gives a computable isomorphism from $A^{\dagger }$ to the standard presentation $M_{n+1}(\mathbb {D})$ .