Strong Approximate Innerness of Quasi-Free Actions

Definitions and mathematical statement

Use the following standard finite-matrix interpretation of the terminology. Let $d\geq 1$, let $M=(M_{ij})\in\{0,1\}^{d\times d}$, and let $B=\mathcal O_M$ be a unital Cuntz--Krieger algebra with canonical generating partial isometries $s_1,\ldots,s_d$. Their range projections are mutually orthogonal and satisfy \[ \sum_{i=1}^{d}s_i s_i^*=1, \qquad s_i^*s_i=\sum_{j=1}^{d}M_{ij}s_j s_j^* \quad(1\leq i\leq d). \] Assume that $B$ is simple and purely infinite. Here simplicity means that $B$ has no nonzero proper closed two-sided ideals; for a simple $C^*$-algebra, pure infiniteness means that every nonzero hereditary $C^*$-subalgebra contains an infinite projection. A projection $p$ is infinite if there are a projection $q<p$ and a partial isometry $v$ with $v^*v=p$ and $vv^*=q$.

Let $G$ be a finite abelian group and let $\alpha:G\to\operatorname{Aut}(B)$ be an action. Interpret "by diagonal quasi-free automorphisms" to mean that there are characters $\chi_i:G\to\mathbb T$, where $\mathbb T=\{z\in{\mathbb{C}}:|z|=1\}$, such that \[ \alpha_g(s_i)=\chi_i(g)s_i \quad(g\in G,\ 1\leq i\leq d). \] Write \[ B^\alpha=\{b\in B:\alpha_h(b)=b\text{ for every }h\in G\}. \] Use the fixed-point unitary formulation of strong approximate innerness: for every $g\in G$, there is a sequence $(u_{g,n})_{n\geq1}$ of unitaries in $B^\alpha$ such that \[ \lim_{n\to\infty}\|\alpha_g(b)-u_{g,n}b u_{g,n}^*\|=0 \quad\text{for every }b\in B. \] This condition imposes no additional representation law on the implementing unitaries.

Question. For every simple purely infinite Cuntz--Krieger algebra $B$ as above, every finite abelian group $G$, and every action $\alpha$ by diagonal quasi-free automorphisms, is $\alpha$ strongly approximately inner in this sense?

Unresolved qualifications. The supplied statement does not assume that the action is outer or faithful, and it does not explicitly impose aperiodicity on the defining matrix. Those hypotheses must not be added to the question. The supplied dated assessment reports a result under outerness and aperiodicity hypotheses, while leaving the broader extracted formulation unsettled. The source record does not reproduce the original definitions or detailed matrix conventions; the finite-matrix, diagonal-action, and strong-approximate-innerness conventions above require comparison with the cited source and its intended reference to Izumi.

Short English statement

Must every finite abelian group action on a simple purely infinite Cuntz--Krieger algebra that multiplies each canonical generator by a scalar be approximated, for each group element and in pointwise norm, by conjugation with unitaries fixed by the whole action? The supplied question assumes neither outerness nor explicit matrix aperiodicity. Its supplied partially solved status is dated curation, not an independently verified conclusion.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003587$, source problem code catalogue entry, title "Strong Approximate Innerness of Quasi-Free Actions," in the Oberwolfach Reports Open Problems collection. No individual proposer is identified. The proposed year is recorded as 2017.
• Supplied source 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 supplied background describes this report as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source labelled "Barlak--Szabo": https://doi.org/10.7146/math.scand.a-114823. No article title or theorem number is supplied.
• Supplied literature source labelled "OWR report": https://doi.org/10.4171/owr/2017/42.
• Literature assessment date: 2026-08-21. This is supplied dated curation, not independently verified. It labels the problem partially solved and reports strong approximate innerness for outer finite abelian diagonal quasi-free actions on simple Cuntz--Krieger algebras with aperiodic matrices, while describing the broader extracted formulation as not fully settled. The supplied background requests independent verification of time-sensitive status and cited claims.
• The record attributes the terminology "strongly approximately inner" to Izumi's sense, but supplies no separate bibliographic reference to Izumi. The record's corrected_verified label and reported manual semantic, typographic, and LaTeX review are supplied metadata, not verification performed for this draft.