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Non-computability of K-theory for computably presented $\mathrm {C}^*$-algebras

Part of: Computability and recursion theory Selfadjoint operator algebras

Published online by Cambridge University Press:  29 May 2026

Christopher J. Eagle Open the ORCID record for Christopher J. Eagle [Opens in a new window] ,
Isaac Goldbring Open the ORCID record for Isaac Goldbring [Opens in a new window] ,
Timothy H. McNicholl Open the ORCID record for Timothy H. McNicholl [Opens in a new window]  and
Russell Miller Open the ORCID record for Russell Miller [Opens in a new window]
Christopher J. Eagle*
Affiliation:
University of Victoria , Canada
Isaac Goldbring
Affiliation:
University of California , Irvine, USA e-mail: igoldbri@uci.edu
Timothy H. McNicholl
Affiliation:
Iowa State University , USA e-mail: mcnichol@iastate.edu
Russell Miller
Affiliation:
Queens College & CUNY Graduate Center , USA e-mail: russell.miller@qc.cuny.edu
*
e-mail: eaglec@uvic.ca
Rights & Permissions [Opens in a new window]

Abstract

We give an example of a unital $\mathrm {C}^*$-algebra $\mathbf {A}$ with a computable presentation and for which neither $K_0(\mathbf {A})$ nor $K_1(\mathbf {A})$ has a computable presentation.

Keywords

Computable analysisC*-algebrasK-theory

MSC classification

Primary: 03D45: Theory of numerations, effectively presented structures 03D78: Computation over the reals 46L35: Classifications of $C^*$-algebras 46L80: $K$-theory and operator algebras (including cyclic theory)

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Canadian Mathematical Bulletin , First View , pp. 1 - 6
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This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0), which permits unrestricted re-use, distribution and reproduction, provided the original article is properly cited.
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© The Author(s), 2026. Published by Cambridge University Press on behalf of Canadian Mathematical Society

1 Introduction

In [Reference Eagle, Goldbring, McNicholl and MillerEGMM], we initiated the study of the effective content of the construction of the K-theory of a $\mathrm {C}^*$ normal upper C Superscript asterisk -algebra. In particular, we constructed a computable functor that takes as input a c.e. presentation of a $\mathrm {C}^*$ normal upper C Superscript asterisk -algebra $\mathbf {A}$ bold upper A and outputs a c.e. presentation of the group $K_0(\mathbf {A})$ upper K 0 left parenthesis bold upper A right parenthesis and another such functor outputting a c.e. presentation of $K_1(\mathbf {A})$ upper K 1 left parenthesis bold upper A right parenthesis . In some cases, such as when $\mathbf {A}$ bold upper A is an AF algebra, our functor maps computable presentations of $\mathbf {A}$ bold upper A to computable presentations of $K_0(\mathbf {A})$ upper K 0 left parenthesis bold upper A right parenthesis (see [Reference Eagle, Goldbring and McNichollEGM, Main Theorem 3]). This naturally raises the question of whether there is a $\mathrm {C}^*$ normal upper C Superscript asterisk -algebra $\mathbf {A}$ bold upper A such that $\mathbf {A}$ bold upper A has a computable presentation but $K_0(\mathbf {A})$ upper K 0 left parenthesis bold upper A right parenthesis does not (see [Reference Eagle, Goldbring, McNicholl and MillerEGMM, Question 2.17]). In this note, we give an example of a $\mathrm {C}^*$ normal upper C Superscript asterisk -algebra $\mathbf {A}$ bold upper A such that $\mathbf {A}$ bold upper A has a computable presentation but neither $K_0(\mathbf {A})$ upper K 0 left parenthesis bold upper A right parenthesis nor $K_1(\mathbf {A})$ upper K 1 left parenthesis bold upper A right parenthesis has a computable presentation. We refer the reader to [Reference Eagle, Goldbring, McNicholl and MillerEGMM] for background on computability theory for $\mathrm {C}^*$ normal upper C Superscript asterisk -algebras and the computable $K_0$ upper K 0 and $K_1$ upper K 1 functors.

2 A group with no computable presentation

Fix, for the remainder of the note, a set $R \subseteq \mathbb {N}$ upper R subset of or equal to double struck upper N that is c.e. but not computable (e.g., we can take $R = 0'$ upper R equals 0 prime , the halting problem). Enumerate the prime numbers in order as $(p_n)_{n \in \mathbb {N}}$ left parenthesis p Subscript n Baseline right parenthesis Subscript n element of double struck upper N .

For each $n \in \mathbb {N}$ n element of double struck upper N , set

$$\begin{align*}G_n = \begin{cases}\mathbb{Z} & n \in R \\ \mathbb{Z}/p_n\mathbb{Z} & n \not\in R.\end{cases}\end{align*}$$ upper G Subscript n Baseline equals StartLayout Enlarged left brace 1st Row 1st Column double struck upper Z 2nd Column n element of upper R 2nd Row 1st Column double struck upper Z divided by p Subscript n Baseline double struck upper Z 2nd Column n not an element of upper R period EndLayout

For the rest of the note, set $G = \bigoplus _{n \in \mathbb {N}}G_n$ upper G equals circled plus Underscript n element of double struck upper N Endscripts upper G Subscript n .

Proposition 2.1 There is no computable presentation of the group G.

Proof Suppose, toward a contradiction, that $G^\#$ upper G Superscript number sign is a computable presentation of G. Let $P = \{n \in \mathbb {N} : G \text { contains an element of order }p_n\}$ upper P equals StartSet n element of double struck upper N colon upper G contains an element of order p Subscript n Baseline EndSet . Note that G has an element of order $p_n$ p Subscript n if and only if $G_n = \mathbb {Z}/p_n\mathbb {Z}$ upper G Subscript n Baseline equals double struck upper Z divided by p Subscript n Baseline double struck upper Z if and only if $n \not \in R$ n not an element of upper R , that is, $P = \mathbb {N} \setminus R$ upper P equals double struck upper N minus upper R .

Since $G^\#$ upper G Superscript number sign is computable, we can use it to successively compute the orders of each element of G, whence P is a c.e. set. Since R was chosen to be a c.e. set, this implies that R is a computable set, contradicting our choice of R.

Remark 2.2 It is not too difficult to show that the above group G admits a c.e. presentation. However, using [Reference Eagle, Goldbring, McNicholl and MillerEGMM, Corollary 2.16], this will be a consequence of the fact that G is isomorphic to $K_0(\mathbf {A})$ upper K 0 left parenthesis bold upper A right parenthesis for a computably presentable $\mathrm {C}^*$ normal upper C Superscript asterisk -algebra $\mathbf {A}$ bold upper A .

3 Non-computable $K_0$ upper K 0

For each $n \in \mathbb {N}$ n element of double struck upper N , set

$$\begin{align*}\mathbf{B}_n = \begin{cases}\mathbb{C} & n \in R \\ \mathcal{O}_{p_n+1} & n \not\in R.\end{cases}\end{align*}$$ bold upper B Subscript n Baseline equals StartLayout Enlarged left brace 1st Row 1st Column double struck upper C 2nd Column n element of upper R 2nd Row 1st Column script upper O Subscript p Sub Subscript n Subscript plus 1 Baseline 2nd Column n not an element of upper R period EndLayout

Here, $\mathcal {O}_k$ script upper O Subscript k is the Cuntz algebra on k generators, that is, $\mathcal {O}_k$ script upper O Subscript k is the universal $\mathrm {C}^*$ normal upper C Superscript asterisk -algebra generated by partial isometries $s_1, \ldots , s_k$ s 1 comma ellipsis comma s Subscript k Baseline such that each $s_i^*s_i=1$ s Subscript i Superscript asterisk Baseline s Subscript i Baseline equals 1 and $\sum _{i=1}^ks_is_i^* = 1$ sigma summation Underscript i equals 1 Overscript k Endscripts s Subscript i Baseline s Subscript i Superscript asterisk Baseline equals 1 . It is well-known that each $\mathcal {O}_k$ script upper O Subscript k is simple [Reference CuntzCun77, Remarks after Theorem 1.12] and has $K_0(\mathcal {O}_k) \cong \mathbb {Z}/(k-1)\mathbb {Z}$ upper K 0 left parenthesis script upper O Subscript k Baseline right parenthesis approximately equals double struck upper Z divided by left parenthesis k minus 1 right parenthesis double struck upper Z [Reference CuntzCun81, Theorem 3.7]. We will also need the fact that the standard presentations of the Cuntz algebras are computable, uniformly in k (see [Reference FoxFox24, Corollary 3.16]).

Lemma 3.1 The algebras $\mathbf {B}_n$ bold upper B Subscript n have computable presentations, uniformly in n.

Proof We begin by noting that it suffices to show that each $\mathbf {B}_n$ bold upper B Subscript n has a c.e. presentation, uniformly in n. Indeed, by a result of Fox [Reference FoxFox24, Theorem 3.14], any c.e. presentation of a simple $\mathrm {C}^*$ normal upper C Superscript asterisk -algebra is computable (and the proof shows that the computability of the presentation is indeed uniform in the c.e. presentation).

Toward this end, fix n and fix a computable enumeration $(R_s)_{s\in \mathbb {N}}$ left parenthesis upper R Subscript s Baseline right parenthesis Subscript s element of double struck upper N of R, that is, $R_s$ upper R Subscript s consists of those natural numbers that have been determined to belong to R by stage s. Let $q_k$ q Subscript k denote the kth special point of the standard presentation of $\mathcal {O}_{p_n+1}$ script upper O Subscript p Sub Subscript n Subscript plus 1 . For all $k,s\in \mathbb {N}$ k comma s element of double struck upper N with $s\geq k$ s greater than or equals k , we define $p_{k,s}\in \mathbf {B}_n$ p Subscript k comma s Baseline element of bold upper B Subscript n as follows:

  • • If $n\notin R_s$ n not an element of upper R Subscript s , set $p_{k,s}:=q_k$ p Subscript k comma s Baseline colon equals q Subscript k .

  • • If $n\in R_s\setminus R_k$ n element of upper R Subscript s minus upper R Subscript k , set $p_{k,s}:=0$ p Subscript k comma s Baseline colon equals 0 .

  • • If $n\in R_k$ n element of upper R Subscript k , set $p_{k,s}:=1$ p Subscript k comma s Baseline colon equals 1 .

If $s<k$ s less than k , we declare that $p_{k,s}$ p Subscript k comma s is undefined. Given $k\in \mathbb {N}$ k element of double struck upper N , note that $p_{k,s}$ p Subscript k comma s is eventually defined and constant; we denote this eventual constant value by $p_k$ p Subscript k and we set ${Q:=\{p_k \ : \ k\in \mathbb {N}\}}$ upper Q colon equals StartSet p Subscript k Baseline colon k element of double struck upper N EndSet .

Next note that in the case that $n\notin R$ n not an element of upper R , we have $p_k=q_k$ p Subscript k Baseline equals q Subscript k for all k. On the other hand, if $n\in R$ n element of upper R and $s_0\in \mathbb {N}$ s 0 element of double struck upper N is the least s for which $n\in R_s$ n element of upper R Subscript s , then we have $p_k=0$ p Subscript k Baseline equals 0 for $k<s_0$ k less than s 0 and $p_k=1$ p Subscript k Baseline equals 1 for $k\geq s_0$ k greater than or equals s 0 . As a result, in either case, Q generates a dense $*$ asterisk -subalgebra of $\mathbf {B}_n$ bold upper B Subscript n , whence we may consider $p_k$ p Subscript k as the kth special point of a presentation $\mathbf {B}_n^\#$ bold upper B Subscript n Superscript number sign of $\mathbf {B}_n$ bold upper B Subscript n .

It remains to observe that $\mathbf {B}_n^\#$ bold upper B Subscript n Superscript number sign is a c.e. presentation of $\mathbf {B}_n$ bold upper B Subscript n , uniformly in n. To see this, consider a rational $*$ asterisk -polynomial $\rho (x_0,\ldots ,x_m)$ rho left parenthesis x 0 comma ellipsis comma x Subscript m Baseline right parenthesis . Let $(r_s)_{s\in \mathbb {N}}$ left parenthesis r Subscript s Baseline right parenthesis Subscript s element of double struck upper N be a computable nonincreasing sequence converging to $\|\rho (q_0,\ldots ,q_m)\|$ StartMetric rho left parenthesis q 0 comma ellipsis comma q Subscript m Baseline right parenthesis EndMetric in $\mathcal {O}_{p_{n+1}}$ script upper O Subscript p Sub Subscript n plus 1 . We define a sequence $(t_s)_{s\geq m}$ left parenthesis t Subscript s Baseline right parenthesis Subscript s greater than or equals m as follows. If $n\notin A_s$ n not an element of upper A Subscript s , set $t_s:=r_s$ t Subscript s Baseline colon equals r Subscript s . If $n\in R_s\setminus R_k$ n element of upper R Subscript s minus upper R Subscript k , set $t_s:=\rho (\vec 0)$ t Subscript s Baseline colon equals rho left parenthesis ModifyingAbove 0 With right arrow right parenthesis . If $n\in R_k$ n element of upper R Subscript k , set $t_s:=\rho (\vec 1)$ t Subscript s Baseline colon equals rho left parenthesis ModifyingAbove 1 With right arrow right parenthesis . It follows that $(t_s)_{s\in \mathbb {N}}$ left parenthesis t Subscript s Baseline right parenthesis Subscript s element of double struck upper N is a computable nonincreasing sequence, uniformly in n, which converges to $\rho (p_0,\ldots ,p_m)$ rho left parenthesis p 0 comma ellipsis comma p Subscript m Baseline right parenthesis , completing the proof.

Set $\mathbf {B} = \bigoplus _{n \in \mathbb {N}}\mathbf {B}_n$ bold upper B equals circled plus Underscript n element of double struck upper N Endscripts bold upper B Subscript n .

Proposition 3.2 $\mathbf {B}$ bold upper B has a computable presentation.

Proof Using Lemma 3.1, fix presentations $\mathbf {B}_n^\#$ bold upper B Subscript n Superscript number sign for the algebras $\mathbf {B}_n$ bold upper B Subscript n such that these presentations are computable uniformly in n. For each N, set $\mathbf {B}_N = \bigoplus _{n=1}^N\mathbf {B}_n$ bold upper B Subscript upper N Baseline equals circled plus Underscript n equals 1 Overscript upper N Endscripts bold upper B Subscript n . Then each $\mathbf {B}_N$ bold upper B Subscript upper N has a computable presentation $\mathbf {B}_N^\#$ bold upper B Subscript upper N Superscript number sign , uniformly in N, obtained by taking the direct sum of the presentations $\mathbf {B}_1^\#, \ldots , \mathbf {B}_N^\#$ bold upper B 1 Superscript number sign Baseline comma ellipsis comma bold upper B Subscript upper N Superscript number sign . The inductive limit presentation constructed from the presentations $\mathbf {B}_N^\#$ bold upper B Subscript upper N Superscript number sign as described in [Reference GoldbringGol24, Section 2] is then a computable presentation of $\lim _{N}\mathbf {B}_N=\mathbf {B}$ limit Underscript upper N Endscripts bold upper B Subscript upper N Baseline equals bold upper B , by [Reference GoldbringGol24, Lemma 2.7(3)].

Theorem 3.3 $K_0(\mathbf {B}) \cong G$ upper K 0 left parenthesis bold upper B right parenthesis approximately equals upper G and $K_1(\mathbf {B}) = 0$ upper K 1 left parenthesis bold upper B right parenthesis equals 0 .

Proof If $n \in R,$ n element of upper R comma then $\mathbf {B}_n = \mathbb {C}$ bold upper B Subscript n Baseline equals double struck upper C , so $K_0(\mathbf {B}_n) = \mathbb{Z}$ upper K 0 left parenthesis bold upper B Subscript n Baseline right parenthesis equals double struck upper Z , while if $n \not \in R,$ n not an element of upper R comma then $\mathbf {B}_n = \mathcal {O}_{p_n+1}$ bold upper B Subscript n Baseline equals script upper O Subscript p Sub Subscript n Subscript plus 1 , so $K_0(\mathbf {B}_n) = \mathbb {Z}/p_n\mathbb {Z}$ upper K 0 left parenthesis bold upper B Subscript n Baseline right parenthesis equals double struck upper Z divided by p Subscript n Baseline double struck upper Z . That is, $K_0(\mathbf {B}_n) = G_n$ upper K 0 left parenthesis bold upper B Subscript n Baseline right parenthesis equals upper G Subscript n for all n.

Applying the fact that $K_0$ upper K 0 commutes with direct sums and direct limits [Reference Rørdam, Larsen and LaustsenRLL00, Proposition 4.3.4 and Theorem 6.3.2], we have

$$ \begin{align*} K_0(\mathbf{B}) &= K_0\left(\bigoplus_{n \in \mathbb{N}}\mathbf{B}_n\right) \\ &\cong \bigoplus_{n\in\mathbb{N}}K_0(\mathbf{B}_n) \\ &\cong \bigoplus_{n \in \mathbb{N}}G_n \\ &= G.\\[-37pt] \end{align*} $$ StartLayout 1st Row 1st Column upper K 0 left parenthesis bold upper B right parenthesis 2nd Column equals upper K 0 left parenthesis circled plus Underscript n element of double struck upper N Endscripts bold upper B Subscript n Baseline right parenthesis 2nd Row 1st Column Blank 2nd Column approximately equals circled plus Underscript n element of double struck upper N Endscripts upper K 0 left parenthesis bold upper B Subscript n Baseline right parenthesis 3rd Row 1st Column Blank 2nd Column approximately equals circled plus Underscript n element of double struck upper N Endscripts upper G Subscript n Baseline 4th Row 1st Column Blank 2nd Column equals upper G period EndLayout

Putting together the results so far, we have the following corollary.

Corollary 3.4 $\mathbf {B}$ bold upper B is a computably presentable $\mathrm {C}^*$ normal upper C Superscript asterisk -algebra and $K_0(\mathbf {B})$ upper K 0 left parenthesis bold upper B right parenthesis has no computable presentation.

4 Non-computable $K_1$ upper K 1

Using the algebra $\mathbf {B}$ bold upper B from the previous section, let $\mathbf {C}$ bold upper C be the suspension of $\mathbf {B}$ bold upper B , that is, $\mathbf {C} = S\mathbf {B} := C_0(0, 1) \otimes \mathbf {B}$ bold upper C equals upper S bold upper B colon equals upper C 0 left parenthesis 0 comma 1 right parenthesis circled times bold upper B .

Proposition 4.1 $\mathbf {C}$ bold upper C has a computable presentation.

Proof This follows immediately from Proposition 3.2 and [Reference Eagle, Goldbring, McNicholl and MillerEGMM, Lemma 3.1].

Proposition 4.2 $K_0(\mathbf {C}) \cong G$ upper K 0 left parenthesis bold upper C right parenthesis approximately equals upper G and $K_1(\mathbf {C}) = 0$ upper K 1 left parenthesis bold upper C right parenthesis equals 0 .

Proof Since $\mathbf {C} = S\mathbf {B}$ bold upper C equals upper S bold upper B , we have $K_0(\mathbf {C}) \cong K_1(\mathbf {B})$ upper K 0 left parenthesis bold upper C right parenthesis approximately equals upper K 1 left parenthesis bold upper B right parenthesis and $K_1(\mathbf {C}) \cong K_0(\mathbf {B})$ upper K 1 left parenthesis bold upper C right parenthesis approximately equals upper K 0 left parenthesis bold upper B right parenthesis (see [Reference Rørdam, Larsen and LaustsenRLL00, Theorem 10.1.3 and Corollary 11.3.1]).

We therefore have the following corollary.

Corollary 4.3 $\mathbf {C}$ bold upper C is a computably presentable $\mathrm {C}^*$ normal upper C Superscript asterisk -algebra and $K_1(\mathbf {C})$ upper K 1 left parenthesis bold upper C right parenthesis has no computable presentation.

5 The main examples

Putting together the examples from the previous two sections, we get a single computably presentable $\mathrm {C}^*$ normal upper C Superscript asterisk -algebra where neither $K_0$ upper K 0 nor $K_1$ upper K 1 is computably presented.

Theorem 5.1 There is a computably presentable $\mathrm {C}^*$ normal upper C Superscript asterisk -algebra $\mathbf {A}$ bold upper A such that neither $K_0(\mathbf {A})$ upper K 0 left parenthesis bold upper A right parenthesis nor $K_1(\mathbf {A})$ upper K 1 left parenthesis bold upper A right parenthesis are computably presentable.

Proof Using the algebras $\mathbf {B}$ bold upper B and $\mathbf {C}$ bold upper C defined earlier, let $\mathbf {A} = \mathbf {B} \oplus \mathbf {C}$ bold upper A equals bold upper B circled plus bold upper C . The algebra $\mathbf {A}$ bold upper A has a computable presentation because $\mathbf {B}$ bold upper B and $\mathbf {C}$ bold upper C have computable presentations.

As mentioned above, $K_0$ upper K 0 commutes with direct sums, so Propositions 3.3 and 4.2 yield that

$$\begin{align*}K_0(\mathbf{A}) \cong K_0(\mathbf{B}) \oplus K_0(\mathbf{C}) \cong G \oplus 0 \cong G.\end{align*}$$ StartLayout 1st Row upper K 0 left parenthesis bold upper A right parenthesis approximately equals upper K 0 left parenthesis bold upper B right parenthesis circled plus upper K 0 left parenthesis bold upper C right parenthesis approximately equals upper G circled plus 0 approximately equals upper G period EndLayout

Likewise, $K_1$ upper K 1 commutes with direct sums [Reference Rørdam, Larsen and LaustsenRLL00, Proposition 8.2.6], so we have

$$\begin{align*}K_1(\mathbf{A}) \cong K_1(\mathbf{B}) \oplus K_1(\mathbf{C}) \cong 0 \oplus G \cong G.\end{align*}$$ StartLayout 1st Row upper K 1 left parenthesis bold upper A right parenthesis approximately equals upper K 1 left parenthesis bold upper B right parenthesis circled plus upper K 1 left parenthesis bold upper C right parenthesis approximately equals 0 circled plus upper G approximately equals upper G period EndLayout

By Proposition 2.1, neither $K_0(\mathbf {A})$ upper K 0 left parenthesis bold upper A right parenthesis nor $K_1(\mathbf {A})$ upper K 1 left parenthesis bold upper A right parenthesis has a computable presentation.

Finally, we reach our main result.

Theorem 5.2 There is a computably presentable, unital, nuclear $\mathrm {C}^*$ normal upper C Superscript asterisk -algebra $\mathbf {D}$ bold upper D such that neither $K_0(\mathbf {D})$ upper K 0 left parenthesis bold upper D right parenthesis nor $K_1(\mathbf {D})$ upper K 1 left parenthesis bold upper D right parenthesis is computably presentable.

Proof Let $\mathbf {D}$ bold upper D be the unitization of the algebra $\mathbf {A}$ bold upper A used in the proof of Theorem 5.1. Note that since each $\mathbf {B}_n$ bold upper B Subscript n is nuclear, $\mathbf {D}$ bold upper D is nuclear as well. Since $\mathbf {A}$ bold upper A has a computable presentation, $\mathbf {D}$ bold upper D does as well by [Reference Eagle, Goldbring, McNicholl and MillerEGMM, Proposition 2.21]. We have $K_1(\mathbf {D}) \cong K_1(\mathbf {A})$ upper K 1 left parenthesis bold upper D right parenthesis approximately equals upper K 1 left parenthesis bold upper A right parenthesis [Reference Rørdam, Larsen and LaustsenRLL00, Equation (8.4)], so $K_1(\mathbf {D})$ upper K 1 left parenthesis bold upper D right parenthesis has no computable presentation.

For $K_0$ upper K 0 , we have $K_0(\mathbf {D}) \cong K_0(\mathbf {A}) \oplus \mathbb {Z}$ upper K 0 left parenthesis bold upper D right parenthesis approximately equals upper K 0 left parenthesis bold upper A right parenthesis circled plus double struck upper Z [Reference Rørdam, Larsen and LaustsenRLL00, Example 4.3.5]. Since $G \oplus \mathbb{Z} \cong G$ upper G circled plus double struck upper Z approximately equals upper G , $K_0(\mathbf{D}))$ upper K 0 left parenthesis bold upper D right parenthesis right parenthesis has no computable presentation.

6 A stably finite example

The algebra $\mathbf {D}$ bold upper D in the previous section is infinite in the sense that it has a proper isometry. As mentioned in the introduction, AF algebras have the property that any computable presentation of them yields a computable induced presentation on the $K_0$ upper K 0 group. Since AF algebras are finite, one might suspect that the distinction at play here is the finite versus infinite one. In this section, we show that one can in fact find stably finite $\mathrm {C}^*$ normal upper C Superscript asterisk -algebras satisfying the conclusion of Theorem 5.2.

For each $n\geq 2$ n greater than or equals 2 , let $\Gamma _n:=\langle a,b \ | \ bab^{-1}=a^n\rangle $ normal upper Gamma Subscript n Baseline colon equals left angle bracket a comma b vertical bar b a b Superscript negative 1 Baseline equals a Superscript n Baseline right angle bracket , the so-called Baumslag–Solitar group $BS(1,n)$ upper B upper S left parenthesis 1 comma n right parenthesis . The following lemma captures the key facts we need about the group $\mathrm {C}^*$ normal upper C Superscript asterisk -algebras of the Baumslag–Solitar groups.

Lemma 6.1 For every $n \geq 2$ n greater than or equals 2 , $C^*(\Gamma _n)$ upper C Superscript asterisk Baseline left parenthesis normal upper Gamma Subscript n Baseline right parenthesis is a stably finite residually finite-dimensional $\mathrm {C}^*$ normal upper C Superscript asterisk -algebra and has $K_0(C^*(\Gamma _n)) \cong \mathbb {Z}$ upper K 0 left parenthesis upper C Superscript asterisk Baseline left parenthesis normal upper Gamma Subscript n Baseline right parenthesis right parenthesis approximately equals double struck upper Z and $K_1(C^*(\Gamma _n)) \cong \mathbb {Z} \oplus \mathbb {Z}/(n-1)\mathbb {Z}$ upper K 1 left parenthesis upper C Superscript asterisk Baseline left parenthesis normal upper Gamma Subscript n Baseline right parenthesis right parenthesis approximately equals double struck upper Z circled plus double struck upper Z divided by left parenthesis n minus 1 right parenthesis double struck upper Z .

Proof We may view $\Gamma _n = \langle a, b \ | \ bab^{-1} = a^n\rangle $ normal upper Gamma Subscript n Baseline equals left angle bracket a comma b vertical bar b a b Superscript negative 1 Baseline equals a Superscript n Baseline right angle bracket as a semi-direct product $\mathbb {Z}[1/n]\rtimes \mathbb {Z}$ double struck upper Z left bracket 1 divided by n right bracket right normal factor semidirect product double struck upper Z , where the $\mathbb {Z}$ double struck upper Z factor corresponds to the subgroup $\langle b \rangle ,$ left angle bracket b right angle bracket comma which acts by taking nth powers. Thus, $\Gamma _n$ normal upper Gamma Subscript n is an extension of $\mathbb {Z}[1/n]$ double struck upper Z left bracket 1 divided by n right bracket by $\mathbb {Z}$ double struck upper Z , and hence is an (elementary) amenable group. Therefore, $C^*(\Gamma _n) \cong C^*_r(\Gamma _n)$ upper C Superscript asterisk Baseline left parenthesis normal upper Gamma Subscript n Baseline right parenthesis approximately equals upper C Subscript r Superscript asterisk Baseline left parenthesis normal upper Gamma Subscript n Baseline right parenthesis (see [Reference Brown and OzawaBO08, Theorem 2.6.8]). It thus suffices to show that $C^*_r(\Gamma _n)$ upper C Subscript r Superscript asterisk Baseline left parenthesis normal upper Gamma Subscript n Baseline right parenthesis is residually finite dimensional and stably finite. The latter property is shared by all reduced group $\mathrm {C}^*$ normal upper C Superscript asterisk -algebras, because they have faithful tracial states (see [Reference Brown and OzawaBO08, Proposition 2.5.3]), so we only need to show that $C^*_r(\Gamma _n)$ upper C Subscript r Superscript asterisk Baseline left parenthesis normal upper Gamma Subscript n Baseline right parenthesis is residually finite dimensional.

Each $\Gamma _n$ normal upper Gamma Subscript n is a linear group, as it is isomorphic to the subgroup

$$ \begin{align*}\left\{\begin{bmatrix}n^k & s \\ 0 & 1\end{bmatrix} : k \in \mathbb{Z}, s \in \mathbb{Z}[1/n]\right\}\end{align*} $$ Start 1 By 1 Matrix 1st Row Start 2 By 2 Matrix 1st Row 1st Column n Superscript k Baseline 2nd Column s 2nd Row 1st Column 0 2nd Column 1 EndMatrix colon k element of double struck upper Z comma s element of double struck upper Z left bracket 1 divided by n right bracket EndMatrix

of $\operatorname {GL}_2(\mathbb {C})$ upper G upper L 2 left parenthesis double struck upper C right parenthesis . Thus, by [Reference Bekka and LouvetBL00, Theorem 4.3], each $C^*_r(\Gamma _n)$ upper C Subscript r Superscript asterisk Baseline left parenthesis normal upper Gamma Subscript n Baseline right parenthesis is residually finite dimensional. Again, we have $C^*(\Gamma _n) \cong C^*_r(\Gamma _n)$ upper C Superscript asterisk Baseline left parenthesis normal upper Gamma Subscript n Baseline right parenthesis approximately equals upper C Subscript r Superscript asterisk Baseline left parenthesis normal upper Gamma Subscript n Baseline right parenthesis , so the former is also residually finite dimensional. The statements about the K-theory of $C^*(\Gamma _n)$ upper C Superscript asterisk Baseline left parenthesis normal upper Gamma Subscript n Baseline right parenthesis are proved in [Reference Pooya and ValettePV18, Theorem 1].

Now define $\mathbf {B}_n$ bold upper B Subscript n as in Section 3, except replace $\mathcal {O}_{p_n+1}$ script upper O Subscript p Sub Subscript n Subscript plus 1 by $C^*(\Gamma _{p_n+1})$ upper C Superscript asterisk Baseline left parenthesis normal upper Gamma Subscript p Sub Subscript n Subscript plus 1 Baseline right parenthesis when $n\notin R$ n not an element of upper R . Equip $C^*(\Gamma _{p_n+1})$ upper C Superscript asterisk Baseline left parenthesis normal upper Gamma Subscript p Sub Subscript n Subscript plus 1 Baseline right parenthesis with its standard presentation. Now one observes that Lemma 3.1 continues to hold in this setting. Indeed, the standard presentation of $C^*(\Gamma _{p_n+1})$ upper C Superscript asterisk Baseline left parenthesis normal upper Gamma Subscript p Sub Subscript n Subscript plus 1 Baseline right parenthesis is clearly c.e. uniformly in n. Moreover, this presentation is in fact computable uniformly in n by [Reference FoxFox24, Theorem 3.5], since $C^*(\Gamma _{p_n+1})$ upper C Superscript asterisk Baseline left parenthesis normal upper Gamma Subscript p Sub Subscript n Subscript plus 1 Baseline right parenthesis is residually finite dimensional, thereby establishing Lemma 3.1 with the redefined $\mathbf {B}_n$ bold upper B Subscript n . As a result, setting $\mathbf {B}:=\bigoplus _n \mathbf {B}_n$ bold upper B colon equals circled plus Underscript n Endscripts bold upper B Subscript n , we have that $K_1(\mathbf {B})\cong G$ upper K 1 left parenthesis bold upper B right parenthesis approximately equals upper G , where G is the group admitting no computable presentation defined in Section 2, and $K_0(\mathbf {B}) = \bigoplus _n \mathbb {Z}$ upper K 0 left parenthesis bold upper B right parenthesis equals circled plus Underscript n Endscripts double struck upper Z . The algebra $\mathbf {B}$ bold upper B is stably finite because each $\mathbf {B}_n$ bold upper B Subscript n is stably finite. We also note that since each $\Gamma _n$ normal upper Gamma Subscript n is amenable each $\mathbf {B}_n$ bold upper B Subscript n is nuclear, and hence $\mathbf {B}$ bold upper B is nuclear as well.

We have thus found a stably finite $\mathrm {C}^*$ normal upper C Superscript asterisk -algebra $\mathbf {B}$ bold upper B that is computably presentable (the validity of Proposition 3.2 depended only on the conclusion of Lemma 3.1) for which $K_1(\mathbf {B})$ upper K 1 left parenthesis bold upper B right parenthesis is not computably presentable. Setting $\mathbf {A}:=\widetilde {S\mathbf {B}}$ bold upper A colon equals ModifyingAbove upper S bold upper B With tilde , the unitization of the suspension of $\mathbf {B}$ bold upper B , we have that $K_0(\mathbf {A})\cong K_1(\mathbf {B})\oplus \mathbb Z\cong G\oplus \mathbb {Z}\cong G$ upper K 0 left parenthesis bold upper A right parenthesis approximately equals upper K 1 left parenthesis bold upper B right parenthesis circled plus double struck upper Z approximately equals upper G circled plus double struck upper Z approximately equals upper G . Note also that $\mathbf {A}$ bold upper A is computably presentable, stably finite and that $K_1(\mathbf {A})\cong K_0(\mathbf {B})=\bigoplus _n \mathbb {Z}$ upper K 1 left parenthesis bold upper A right parenthesis approximately equals upper K 0 left parenthesis bold upper B right parenthesis equals circled plus Underscript n Endscripts double struck upper Z . Finally, setting $\mathbf {E}:=\mathbf {A}\oplus \mathbf {B}$ bold upper E colon equals bold upper A circled plus bold upper B yields a computably presentable, stably finite $\mathrm {C}^*$ normal upper C Superscript asterisk -algebra with $K_0(\mathbf {E})\cong K_0(\mathbf {A})\oplus K_0(\mathbf {B})\cong G$ upper K 0 left parenthesis bold upper E right parenthesis approximately equals upper K 0 left parenthesis bold upper A right parenthesis circled plus upper K 0 left parenthesis bold upper B right parenthesis approximately equals upper G and $K_1(\mathbf {E})\cong K_1(\mathbf {A}) \oplus K_1(\mathbf {B})\cong G$ upper K 1 left parenthesis bold upper E right parenthesis approximately equals upper K 1 left parenthesis bold upper A right parenthesis circled plus upper K 1 left parenthesis bold upper B right parenthesis approximately equals upper G , neither of which are computably presentable. In summary, we have shown the following theorem.

Theorem 6.2 There is a computably presentable, stably finite, unital, nuclear $\mathrm {C}^*$ normal upper C Superscript asterisk -algebra $\mathbf {E}$ bold upper E such that neither $K_0(\mathbf {E})$ upper K 0 left parenthesis bold upper E right parenthesis nor $K_1(\mathbf {E})$ upper K 1 left parenthesis bold upper E right parenthesis has a computable presentation.

Acknowledgements

The results of this note were obtained during the authors’ visit to the American Institute of Mathematics (AIM) in January 2026 as part of their SQuaREs program. The authors thank AIM for their hospitality and for creating a wonderful working environment during the writing of this article.

Footnotes

C.E. was supported by NSERC Discovery Grant RGPIN-2021-02459. I.G. was supported by NSF Grant DMS-2054477. R.M. was supported by NSF Grant DMS-2348891, by Simons Foundation Award No. MP-TSM-00007933, and by several grants from the City University of New York PSC-CUNY Research Award Program.

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