Cartan Subalgebras in Classifiable $C^*$-Algebras

Definitions and mathematical statement

For a $C^*$-algebra $A$, a Cartan subalgebra is a commutative $C^*$-subalgebra $B\subseteq A$ satisfying the following conditions: $B$ contains an approximate identity for $A$; $B$ is maximal abelian in $A$; the normalizer \[ N_A(B)=\{a\in A:aBa^*\subseteq B,\ a^*Ba\subseteq B\} \] generates $A$ as a $C^*$-algebra; and there is a faithful conditional expectation $E:A\to B$. Here a conditional expectation is a completely positive contractive projection onto $B$ satisfying $E(b_1ab_2)=b_1E(a)b_2$, and faithfulness means that $E(a^*a)=0$ implies $a=0$. A maximal abelian subalgebra is also called a MASA. Write $\widehat B$ for the character space of $B$, so that $B\cong C_0(\widehat B)$.

Let $\mathcal W$ denote the Razak--Jacelon algebra, let $\mathcal Z$ denote the Jiang--Su algebra, and let \[ A_{\mathrm{CAR}}=M_{2^\infty}=\bigotimes_{n=1}^{\infty}M_2({\mathbb{C}}), \] with the unital connecting embeddings $x\mapsto x\otimes 1_2$. Let $\mathcal O_2$ be the universal unital $C^*$-algebra generated by isometries $s_1,s_2$ satisfying $s_1s_1^*+s_2s_2^*=1$.

The supplied record collects the following questions, rather than specifying a single conjecture.

1. Does there exist a Cartan subalgebra $B\subseteq\mathcal W$?
2. Does $\mathcal Z$ have a distinguished one-dimensional Cartan subalgebra? If "one-dimensional" refers to the covering dimension of the spectrum, the existence component would ask for a Cartan subalgebra $B\subseteq\mathcal Z$ with $\dim(\widehat B)=1$. This interpretation and the meaning of "distinguished" are not specified in the supplied record; the latter could impose an additional selection or uniqueness requirement that cannot be reconstructed here.
3. Do strongly self-absorbing Cartan pairs admit useful existence and uniqueness theorems? The record does not define strong self-absorption for a pair, specify the class of pairs, or identify the equivalence relation and hypotheses for uniqueness. This remains a research-program question without a determinate quantified assertion.
4. For a Cartan inclusion $\iota:B\hookrightarrow A$, characterize the induced homomorphisms \[ K_i(\iota):K_i(B)\longrightarrow K_i(A),\qquad i\in\{0,1\}. \] These are the maps supplied by functoriality of operator $K$-theory. The record does not specify the inclusions under consideration or the additional characterization sought.
5. What equivalence relation models Popa intertwining for Cartan subalgebras? The ambient algebra, the version of intertwining, and the objects on which the requested equivalence relation should be defined are not supplied.
6. How do Cartan subalgebras interact with completely positive approximation? The approximation property, topology of approximation, and compatibility required with the Cartan inclusion are unspecified.
7. Does there exist a Cartan subalgebra $B\subseteq A_{\mathrm{CAR}}$ such that $\widehat B$ is connected? Since $A_{\mathrm{CAR}}$ is unital, such a Cartan subalgebra contains its unit and has compact spectrum. Connectedness is the requested condition; a spectrum that is merely not totally disconnected does not by itself satisfy it.
8. When does an equivariant embedding into $M_{2^\infty}$ have MASA image? For a specified domain algebra $D$, group $G$, and actions $\beta$ on $D$ and $\gamma$ on $M_{2^\infty}$, equivariance of an injective $*$-homomorphism $\varphi:D\to M_{2^\infty}$ would mean \[ \varphi\circ\beta_g=\gamma_g\circ\varphi\quad(g\in G). \] The requested conclusion would be that $\varphi(D)$ is maximal abelian. This is only a template: the domain, group, actions, unitality requirements, and hypotheses intended by the source are missing.
9. Is every automorphism $\alpha\in\operatorname{Aut}(\mathcal O_2)$ of order exactly two strongly approximately inner? Using the usual fixed-point implementation convention, this asks whether every $\alpha$ satisfying $\alpha^2=\operatorname{id}$ and $\alpha\ne\operatorname{id}$ admits a sequence of unitaries $u_n\in\mathcal O_2^\alpha$, where \[ \mathcal O_2^\alpha=\{a\in\mathcal O_2:\alpha(a)=a\}, \] such that \[ \lim_{n\to\infty}\|\alpha(a)-u_nau_n^*\|=0 \quad\text{for every }a\in\mathcal O_2. \] The supplied record does not explicitly define strong approximate innerness; this convention requires confirmation against the cited report.

The supplied status is partially solved. Its dated assessment attributes Cartan constructions for $\mathcal W$ and $\mathcal Z$, in the classifiable simple setting, to Li, and CAR diagonals with non-totally-disconnected spectrum to Blackadar. Those statements do not by themselves settle the additional "distinguished one-dimensional" requirement or the connected-spectrum question. The supplied assessment also says that the remaining classification, equivariant, and automorphism questions are not settled in general. These are supplied curation claims, not independently verified conclusions.

Short English statement

This record collects nine questions about Cartan subalgebras, their invariants and approximation properties, equivariant embeddings, and order-two automorphisms of $\mathcal O_2$. Several questions lack hypotheses or definitions needed for a unique precise formulation. The supplied partially solved label and literature assessment require independent review; in particular, non-totally-disconnected spectrum does not establish connected spectrum.

Sources

• Local UnsolvedMath registry: supplied complete record with permanent ID $30003588$, exact source problem code catalogue entry, title "Cartan Subalgebras in Classifiable $C^*$-Algebras," in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied.
• Supplied report citation: Mini-Workshop: MASAs and Automorphisms of C*-Algebras (2018), Oberwolfach Reports, pp. 2601--2629, DOI 10.4171/owr/2017/42. https://doi.org/10.4171/owr/2017/42. The registry gives proposed year 2017 and the tag owr-year:2017; the citation's parenthetical year 2018 is retained without resolving this discrepancy. The supplied background describes Oberwolfach Reports as documenting workshops and their open-problem discussions.
• Li, Cartan subalgebras in classifiable simple C*-algebras. https://arxiv.org/abs/1802.01190. The supplied assessment attributes Cartan constructions for the Razak--Jacelon and Jiang--Su algebras in the classifiable simple setting to this reference.
• Blackadar, Symmetries of the CAR algebra. https://doi.org/10.2307/1971472. The supplied assessment attributes CAR diagonals with non-totally-disconnected spectrum to this reference; it does not state that these spectra are connected.
• Literature assessment date, copied exactly: 2026-08-21. The supplied assessment labels the compound list partially solved and states that the remaining classification, equivariant, and automorphism questions are not settled in general. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment. The cited report and papers have not been independently consulted for this draft.