An intrinsic characterization of Condition (K) for inverse semigroups and their tight C*-algebras

For an inverse semigroup $S$ with zero, LaLonde, Milan and Scott defined Condition (K) by requiring that $S/I$ satisfy Condition (L) (that is, $(S/I)/\leftrightarrow$ is fundamental) for every saturated ideal $I$ of $S$, generalizing the classical Condition (K) for directed graphs. Under Hausdorffness of $S$, Condition (K) is equivalent to strong effectivity of the tight groupoid $\mathcal{G}_{\mathrm{tight}}(S)$ whenever the semilattice $E(S)$ of idempotents satisfies the trapping condition, and, with amenability plus the trapping condition and existence of finite covers, it yields lattice isomorphisms between saturated ideals of $S$ and the ideals of the associated Steinberg and tight C*-algebras. The definition quantifies over all saturated ideals, which makes it hard to verify in practice, and the authors explicitly state that they have been unable to give an intrinsic formulation: they conjecture that Condition (K) is equivalent to $S$ being fundamental together with some additional property of $E(S)$. The open problem is to prove or refute this conjecture and, in the affirmative, to identify the additional property and prove the equivalence with the quotient-based definition.

Background

An inverse semigroup is a semigroup in which every element $s$ has a unique "inverse" $s^*$ with $s=ss^*s$ and $s^*=s^*ss^*$; inverse semigroups model partial symmetries, and their idempotents $e=s^*s$ form a commutative semilattice $E(S)$ under multiplication, ordered by the natural partial order $t\leq s$ when $t=se$ for some idempotent $e$. A nonzero subset $I$ with $SIS\subseteq I$ is an ideal, and an ideal $I$ is saturated if $e\in I$ whenever $e\to\{e_1,\dots,e_n\}$ (every nonzero $x\leq e$ satisfies $xe_i\neq 0$ for some $i$) with all $e_i\in I$. A congruence on $S$ is an equivalence relation compatible with multiplication; the Rees congruence of an ideal $I$ identifies exactly the elements of $I$ to zero. $S$ is fundamental if any element commuting with every idempotent is itself idempotent, and $E(S)$ is 0-disjunctive if for $0<e<f$ there is $0<e'<f$ with $ee'=0$. Exel's program assigns to $S$ a tight groupoid $\mathcal{G}_{\mathrm{tight}}(S)$, an étale locally compact groupoid whose unit space is the space of tight filters (the closure, in the space of nonzero filters of $E(S)$, of the set of ultrafilters) and whose C*-algebra is the tight C*-algebra $C^*(\mathcal{G}_{\mathrm{tight}}(S))$; a groupoid is effective if its isotropy has trivial interior, and strongly effective if every reduction to a closed invariant subset of the unit space is effective. In graph algebras, Condition (L) (every cycle has an exit) and Condition (K) (no vertex is the base of exactly one simple loop) govern the Cuntz–Krieger uniqueness theorem and the correspondence between ideals and saturated hereditary sets of vertices. LaLonde–Milan–Scott extended these to inverse semigroups using the congruence $\leftrightarrow$ introduced by Lenz, where $a\to b$ if every nonzero $x\leq a$ meets $b$'s downset — this is exactly the case $a\to\{b\}$ of the covering relation used in the definition of saturated ideal above — and $a\leftrightarrow b$ if $a\to b$ and $b\to a$: $S$ satisfies Condition (L) when $S/\leftrightarrow$ is fundamental, and Condition (K) when every saturated-ideal quotient $S/I$ satisfies Condition (L).

They proved that for the inverse semigroup $S_\Lambda$ of a row-finite graph this recovers the graph conditions, that for Hausdorff $S$ whose semilattice satisfies the trapping condition Condition (K) is equivalent to strong effectivity of $\mathcal{G}_{\mathrm{tight}}(S)$ (Theorem 5.2), and that under the trapping condition plus existence of finite covers, saturated ideals of $S$ correspond bijectively to closed invariant subsets of the tight spectrum (Theorem 4.17), giving lattice isomorphisms onto the basic ideals of the Steinberg algebra and the ideals of the C*-algebra (Theorems 5.4 and 5.6; the C*-algebra statement additionally assumes that $S$ is amenable). Here the trapping condition means that for all $0<f<e$ in $E(S)$ there exist $e_1,\dots,e_m\leq e$ with $e\to\{e_1,\dots,e_m,f\}$ and $e_if=0$ for each $i$ (Definition 4.11 of the source), and $E(S)$ admits finite covers if every invariant order ideal $X$ of $E(S)$ has a finite cover $C\subseteq X$, that is, every nonzero $x\in X$ satisfies $xc\neq 0$ for some $c\in C$ (Definition 4.12 of the source). Hausdorffness of $S$ means that $a^{\downarrow}\cap b^{\downarrow}$ is finitely generated for every $a,b\in S$ (the standard convention used in the source, which does not restate it). As an easily checkable sufficient condition they characterized, in Theorem 5.9, the property that all congruences on $S$ are Rees congruences: it holds exactly when every quotient $S/I$ is fundamental with 0-disjunctive semilattice of idempotents. However, verifying Condition (K) directly requires checking every quotient by a saturated ideal, which the authors note makes the definition difficult to apply; they conjecture that an equivalent definition should be phrased intrinsically, and specifically that it should take the form "$S$ is fundamental and $E(S)$ satisfies some additional property", but state they were unable to give such a definition.

Problem Statement

For a countable inverse semigroup $S$ with zero, give an intrinsic characterization of Condition (K) — that is, a property phrased directly in terms of the algebraic structure of $S$ and its semilattice $E(S)$ of idempotents, without quantifying over the set of all saturated ideals or their quotients — that is provably equivalent to the definition of LaLonde–Milan–Scott: $S/I$ satisfies Condition (L) (equivalently, $(S/I)/\leftrightarrow$ is fundamental) for every saturated ideal $I$ of $S$. In particular, prove or refute their conjecture that such a characterization exists in the form "$S$ is fundamental and $E(S)$ satisfies an additional property": in the affirmative case, state the additional property precisely and prove both implications of the equivalence; in the negative case, exhibit a witness refuting the conjectured form itself — for instance an inverse semigroup satisfying Condition (K) without being fundamental, or two fundamental inverse semigroups with isomorphic semilattices of idempotents on which Condition (K) differs — or otherwise prove that no such intrinsic characterization of the conjectured form exists. As in the source, Definition 5.1 and the conjecture are posed for an arbitrary inverse semigroup with zero — countable, per the source's standing convention — with no Hausdorffness, trapping, finite-cover, or amenability hypothesis: those hypotheses appear only in the separate Theorems 5.2, 5.4 and 5.6, so an equivalence established only on such a restricted class would constitute partial progress rather than a resolution of the problem.

References

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