Baum–Connes Assembly for Minimal Exact Crossed Products

Definitions and mathematical statement

A $G$-$C^*$-algebra $A$ is a $C^*$-algebra equipped with a continuous action of a topological group $G$ by $*$-automorphisms. In the usual locally compact group setting, a crossed-product functor assigns a completion $A\rtimes_E G$ of the convolution $*$-algebra $C_c(G,A)$, functorially in $A$. Exactness means that every $G$-equivariant short exact sequence of coefficient algebras induces a short exact sequence \[ 0\longrightarrow I\rtimes_E G\longrightarrow A\rtimes_E G\longrightarrow (A/I)\rtimes_E G\longrightarrow 0. \] The source calls $\rtimes_E$ the minimal exact crossed-product functor. Minimality ordinarily refers to the order given by canonical quotient maps between crossed-product completions; the precise class of functors being compared is not specified here.

Write $K_*^{\mathrm{top}}(G;A)$ for equivariant topological $K$-theory with coefficients in $A$, and $K_*(B)$ for the operator $K$-theory of a $C^*$-algebra $B$, with degree $*\in{\mathbb{Z}}/2{\mathbb{Z}}$. The requested investigation concerns the reformulated Baum--Connes assembly map \[ \mu^E_{G,A}\colon K_*^{\mathrm{top}}(G;A)\longrightarrow K_*(A\rtimes_E G). \] Describe its construction and concrete realization, and give concrete realizations and computations of $A\rtimes_E G$ for the intended groups and coefficient algebras.

Unresolved scope. The supplied record specifies neither the admissible class of groups and coefficient algebras nor particular cases to compute. The locally compact setting above explains the standard terminology; it is not an additional hypothesis attributed to the source. The record also does not identify the exact version of the minimal exact functor or the hypotheses ensuring the displayed assembly map is defined. Consequently this is a research objective with an incompletely specified domain, rather than a fully quantified conjecture. In particular, the request to "understand" the map does not explicitly assert that it is an isomorphism.

Short English statement

Investigate the Baum--Connes assembly map with target the $K$-theory of the minimal exact crossed product, including concrete descriptions and computations of that crossed product. The intended groups, coefficient algebras, functor convention, and precise success criterion remain unspecified. The supplied record labels the problem partially solved; that label is dated curation, not an independently verified conclusion.

Sources

• Local UnsolvedMath registry, supplied record for permanent ID $30003896$, source problem code catalogue entry, titled "Baum--Connes Assembly for Minimal Exact Crossed Products." The record gives proposed year 2018 and names no proposer.
• Supplied report citation: Non-commutative Geometry, Index Theory and Mathematical Physics (2019), Oberwolfach Reports, pp. 1911--1981, DOI 10.4171/owr/2018/32. https://doi.org/10.4171/owr/2018/32. The supplied background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature reference, labelled "Baum--Guentner--Willett": https://arxiv.org/abs/1311.2343.
• Supplied literature reference, labelled "Buss--Echterhoff--Willett": https://arxiv.org/abs/1804.02725.
• Supplied literature reference, labelled "OWR report": https://doi.org/10.4171/owr/2018/32.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem "partially solved" and states that the minimal exact crossed product clarified the exact crossed-product reformulation, that it agrees with the classical construction for trivial coefficients, and that the Baum--Connes conjecture with general coefficients remains open. This is supplied dated curation derived from the OWR material, not independently verified here; both the status and the cited claims require independent verification.
• The registry labels the statement corrected_verified and reports a manual semantic, typographic, and LaTeX review claiming that the clean statement is self-contained and preserves the target. These are supplied editorial assertions, not independent mathematical verification of this draft.