Relative Ultragraph Algebras

• Relative ultragraph algebras are C*-algebras associated with a pair (𝒢, X) where Cuntz–Krieger relations are imposed only on vertices in X.
• They interpolate between full ultragraph algebras and Toeplitz-type algebras, capturing dynamics that include both return phenomena and escape behavior.
• A key feature is the reduction to a modified ultragraph 𝒢_X under finite-range conditions, enabling the application of classical uniqueness theorems and branching system frameworks.

Relative ultragraph algebras are $C^*$-algebras attached to a pair $(\mathcal G,X)$, where $\mathcal G$ is an ultragraph and $X\subseteq \mathrm{Reg}(\mathcal G)$ is a chosen subset of regular vertices. They are defined by imposing the ultragraph Cuntz–Krieger relation only at vertices in $X$, so they interpolate between the full ultragraph algebra $C^*(\mathcal G)$ and a Toeplitz-type algebra in which some regular-vertex relations are omitted. In the current literature, this notion is introduced explicitly as a distinct class of operator algebras motivated by branching-system representations and by infinite Markov interval maps with escape behavior; at the same time, it sits inside a broader ultragraph program involving quotient constructions, uniqueness theorems, partial crossed products, groupoids, and Leavitt path algebras (Eidt et al., 27 Aug 2025).

1. Ultragraphic background and the full ultragraph relations

An ultragraph is a quadruple

$\mathcal G=(G^0,\mathcal G^1,r,s),$

where $G^0$ and $\mathcal G^1$ are countable sets, $s:\mathcal G^1\to G^0$ is the source map, and $(\mathcal G,X)$0 is the range map. The essential feature distinguishing ultragraphs from directed graphs is that an edge may have a set-valued range. In the relative-ultragraph framework, one works with the Boolean algebra $(\mathcal G,X)$1 generated by the singletons $(\mathcal G,X)$2 for $(\mathcal G,X)$3 and the sets $(\mathcal G,X)$4 for $(\mathcal G,X)$5, closed under finite unions and intersections (Eidt et al., 27 Aug 2025).

The full ultragraph $(\mathcal G,X)$6-algebra $(\mathcal G,X)$7 is the universal $(\mathcal G,X)$8-algebra generated by partial isometries $(\mathcal G,X)$9 with orthogonal ranges and projections $\mathcal G$0 satisfying

$\mathcal G$1

$\mathcal G$2

and

$\mathcal G$3

The last relation is the Cuntz–Krieger relation at regular vertices (Eidt et al., 27 Aug 2025).

The algebraic side of ultragraph theory uses closely parallel generators and relations. For a unital commutative ring $\mathcal G$4, the ultragraph Leavitt path algebra $\mathcal G$5 is generated by $\mathcal G$6 and $\mathcal G$7 subject to the analogous Boolean, source-range, orthogonality, and finite-emitter relations. This algebraic formalism is important historically because quotient ultragraphs, admissible pairs, uniqueness theorems, and ideal classification were first organized there in a way closely aligned with later relative constructions (Imanfar et al., 2017).

2. Definition of relative ultragraph algebras

A relative ultragraph is a pair $\mathcal G$8, where $\mathcal G$9 and

$X\subseteq \mathrm{Reg}(\mathcal G)$0

The relative ultragraph $X\subseteq \mathrm{Reg}(\mathcal G)$1-algebra $X\subseteq \mathrm{Reg}(\mathcal G)$2 is the universal $X\subseteq \mathrm{Reg}(\mathcal G)$3-algebra generated by partial isometries $X\subseteq \mathrm{Reg}(\mathcal G)$4 with orthogonal ranges and projections $X\subseteq \mathrm{Reg}(\mathcal G)$5 satisfying the full ultragraph relations except that the Cuntz–Krieger relation is imposed only for vertices in $X\subseteq \mathrm{Reg}(\mathcal G)$6: $X\subseteq \mathrm{Reg}(\mathcal G)$7 Thus the defining distinction is that regular vertices in $X\subseteq \mathrm{Reg}(\mathcal G)$8 retain only the inequality $X\subseteq \mathrm{Reg}(\mathcal G)$9, not the equality. The construction is explicitly designed to provide structural flexibility and to encode dynamical situations in which some vertices satisfy a return relation and others correspond to escape behavior (Eidt et al., 27 Aug 2025).

A central structural theorem reduces the relative theory to the ordinary ultragraph theory. Setting

$X$0

the associated ultragraph $X$1 is obtained by adjoining a new vertex $X$2 for each $X$3, and a new edge $X$4 for each $X$5 with $X$6. Its vertex and edge sets are

$X$7

with

$X$8

The paper proves a decomposition lemma for the Boolean algebra $X$9: $C^*(\mathcal G)$0

There is then a canonical $C^*(\mathcal G)$1-homomorphism

$C^*(\mathcal G)$2

given by

$C^*(\mathcal G)$3

for $C^*(\mathcal G)$4, and

$C^*(\mathcal G)$5

If

$C^*(\mathcal G)$6

then $C^*(\mathcal G)$7 is a $C^*(\mathcal G)$8-isomorphism, with inverse $C^*(\mathcal G)$9. Under this finiteness hypothesis, relative ultragraph algebras are therefore not merely analogous to ordinary ultragraph algebras; they are concretely representable as ordinary ultragraph algebras for a canonically modified ultragraph $\mathcal G=(G^0,\mathcal G^1,r,s),$0 (Eidt et al., 27 Aug 2025).

3. Relative Condition (L) and injectivity criteria

The uniqueness theory for relative ultragraph algebras is controlled by a modified exit condition. Writing again $\mathcal G=(G^0,\mathcal G^1,r,s),$1, the ultragraph $\mathcal G=(G^0,\mathcal G^1,r,s),$2 satisfies Relative Condition (L) if, for every cycle without exits in $\mathcal G=(G^0,\mathcal G^1,r,s),$3, there exists an edge $\mathcal G=(G^0,\mathcal G^1,r,s),$4 in the cycle such that

$\mathcal G=(G^0,\mathcal G^1,r,s),$5

The paper proves that

$\mathcal G=(G^0,\mathcal G^1,r,s),$6

This transfers the ordinary ultragraph no-exit criterion to the relative setting (Eidt et al., 27 Aug 2025).

A second basic object is the defect projection

$\mathcal G=(G^0,\mathcal G^1,r,s),$7

defined for $\mathcal G=(G^0,\mathcal G^1,r,s),$8. These projections measure the failure of the full Cuntz–Krieger equality at vertices outside $\mathcal G=(G^0,\mathcal G^1,r,s),$9.

The injectivity theorem for a $G^0$0-homomorphism $G^0$1 assumes $G^0$2 is finite for every $G^0$3 and states that $G^0$4 is injective if and only if the following four conditions hold:

• $G^0$5 for all $G^0$6;
• for each $G^0$7 with $G^0$8, $G^0$9;
• for all $\mathcal G^1$0, $\mathcal G^1$1;
• for any simple cycle $\mathcal G^1$2 with no exit in $\mathcal G^1$3 and $\mathcal G^1$4 for all $\mathcal G^1$5, the spectrum of $\mathcal G^1$6 contains the unit circle.

A preceding result shows that under Relative Condition (L), the spectral condition is unnecessary: the first three nonvanishing conditions already imply injectivity (Eidt et al., 27 Aug 2025).

These criteria are the relative analogue of the general Cuntz–Krieger uniqueness theorem. They preserve the familiar ultragraph pattern in which injectivity is governed simultaneously by nonzero diagonal data and by cycle behavior, but they refine the diagonal test by separating ordinary vertex projections $\mathcal G^1$7, range projections $\mathcal G^1$8 for omitted vertices, and defect projections $\mathcal G^1$9.

4. Relative branching systems and infinite Markov interval maps

A relative $s:\mathcal G^1\to G^0$0-branching system on a measure space $s:\mathcal G^1\to G^0$1 is a family

$s:\mathcal G^1\to G^0$2

such that the sets $s:\mathcal G^1\to G^0$3 are pairwise disjoint modulo null sets, the sets $s:\mathcal G^1\to G^0$4 reproduce the Boolean operations in $s:\mathcal G^1\to G^0$5, one has $s:\mathcal G^1\to G^0$6 modulo null sets, and the covering relation

$s:\mathcal G^1\to G^0$7

is imposed only for $s:\mathcal G^1\to G^0$8. For each edge $s:\mathcal G^1\to G^0$9, there are measurable maps

$(\mathcal G,X)$00

satisfying the expected inverse and absolute-continuity conditions (Eidt et al., 27 Aug 2025).

Every such branching system induces a representation

$(\mathcal G,X)$01

with

$(\mathcal G,X)$02

and

$(\mathcal G,X)$03

If $(\mathcal G,X)$04 satisfies Relative Condition (L), the induced representation is injective provided

$(\mathcal G,X)$05

Without Relative Condition (L), the cycle obstruction reappears in dynamical form: for every simple no-exit cycle $(\mathcal G,X)$06 with $(\mathcal G,X)$07, one requires positive-measure sets $(\mathcal G,X)$08 that avoid returning to themselves under finitely many iterates of $(\mathcal G,X)$09 (Eidt et al., 27 Aug 2025).

The motivating dynamical model is the class $(\mathcal G,X)$10 of infinite partition Markov interval maps on an interval $(\mathcal G,X)$11. Here $(\mathcal G,X)$12 is partitioned into countably many closed intervals $(\mathcal G,X)$13, arranged consecutively, such that $(\mathcal G,X)$14 is injective on each interior $(\mathcal G,X)$15, $(\mathcal G,X)$16 for the boundary set $(\mathcal G,X)$17, and $(\mathcal G,X)$18 is a union of partition intervals. The intervals between consecutive $(\mathcal G,X)$19 are the escape sets

$(\mathcal G,X)$20

From the associated $(\mathcal G,X)$21-$(\mathcal G,X)$22 matrix one constructs an ultragraph $(\mathcal G,X)$23 with vertices $(\mathcal G,X)$24 and edges $(\mathcal G,X)$25,

$(\mathcal G,X)$26

For

$(\mathcal G,X)$27

and $(\mathcal G,X)$28, the set $(\mathcal G,X)$29 consists of points that eventually map to the same escape point as $(\mathcal G,X)$30. On $(\mathcal G,X)$31, the operators

$(\mathcal G,X)$32

$(\mathcal G,X)$33

define a representation of $(\mathcal G,X)$34; if $(\mathcal G,X)$35 is chosen so that

$(\mathcal G,X)$36

the same formulas define a representation

$(\mathcal G,X)$37

The paper proves that $(\mathcal G,X)$38 is exactly the representation induced by a naturally associated relative branching system, so the interval-map construction is not external to the relative theory but one of its canonical realization mechanisms (Eidt et al., 27 Aug 2025).

5. Quotient constructions, ideal theory, and dynamical precursors

Before the explicit introduction of relative ultragraph $(\mathcal G,X)$39-algebras, the algebraic theory already contained a closely related quotient apparatus. For an ultragraph $(\mathcal G,X)$40, hereditary and saturated subcollections $(\mathcal G,X)$41, together with subsets $(\mathcal G,X)$42 of breaking vertices, form admissible pairs $(\mathcal G,X)$43. The quotient ultragraph $(\mathcal G,X)$44 is built using equivalence classes

$(\mathcal G,X)$45

and its Leavitt path algebra $(\mathcal G,X)$46 is universal for the corresponding quotient relations. The fundamental theorem is

$(\mathcal G,X)$47

and the map $(\mathcal G,X)$48 is a bijection from admissible pairs onto the graded basic ideals of $(\mathcal G,X)$49. Moreover,

$(\mathcal G,X)$50

These quotient ultragraph algebras are not the same object as the later relative ultragraph $(\mathcal G,X)$51-algebras, but they provide the clearest algebraic precursor to a relative viewpoint based on selectively imposed relations and controlled quotients (Imanfar et al., 2017).

The operator-algebraic dynamical infrastructure was developed independently. For ultragraphs with sinks satisfying the finiteness hypothesis (RFUM2), the boundary ultrapath space becomes locally compact, Hausdorff, and metrizable, and the full ultragraph algebra can be realized both as a partial crossed product and as a groupoid $(\mathcal G,X)$52-algebra. That paper is not primarily about relative ultragraph algebras, but it states that the partial crossed-product description generalizes corresponding results for relative graph $(\mathcal G,X)$53-algebras from Carlsen–Larsen, with the same mechanism applying to ultragraphs (Tasca et al., 2020).

A related algebraic-dynamical development studies partial skew groupoid rings. It does not introduce a new notion of relative ultragraph algebra, but it identifies quotient partial actions and the residual intersection property as the correct tools for controlling ideals after passing to invariant quotients. In the ultragraph application, the final equivalence

$(\mathcal G,X)$54

is obtained through the chain

$(\mathcal G,X)$55

which is precisely the quotient-sensitive mechanism later described as relevant to relative variants (Bagio et al., 2022).

6. Position within the broader ultragraph literature

Relative ultragraph algebras emerged within a mature ultragraph research program rather than in isolation. Branching systems for ultragraph $(\mathcal G,X)$56-algebras, concrete Hilbert-space representations, Perron–Frobenius operators, and a generalized Cuntz–Krieger uniqueness theorem were established earlier; in particular, every permutative representation is unitarily equivalent to one arising from a branching system, and the faithfulness problem is tied to cycles without exits and projection kernels (Gonçalves et al., 2016). This branching-system paradigm is directly continued in the relative setting of (Eidt et al., 27 Aug 2025).

Thermodynamic analysis was also already available for full ultragraph algebras. For ultragraphs satisfying (RFUM), KMS$(\mathcal G,X)$57-states and ground states were characterized via the partial crossed-product model, with equivalent descriptions in terms of states on $(\mathcal G,X)$58, regular Borel probability measures, finitely additive functions on generalized vertices, and states on the core algebra. That paper explicitly states that it does not develop relative ultragraph algebras in detail, but it places its results in the broader context of relative graph algebras, quotient-type constructions, and the partial crossed-product model (Castro et al., 2017).

On the algebraic side, ultragraph Leavitt path algebras have been realized as partial skew group rings and as Steinberg algebras. The partial skew group ring model yields simplicity criteria in terms of Condition (L) and the absence of nontrivial hereditary and saturated subcollections, as well as an artinianity criterion equivalent to finiteness and acyclicity (Gonçalves et al., 2017). The Steinberg-algebra model realizes $(\mathcal G,X)$59 as the convolution algebra of an ample ultragraph groupoid, from which semiprimitivity, strong grading criteria, irreducible representations, graded von Neumann regularity, and simplicity criteria are derived (Hazrat et al., 2020). Every ultragraph Leavitt path algebra is also Morita equivalent, as a ring, to a graph Leavitt path algebra; that thesis explicitly connects this result to the algebraic Exel–Laca construction and to the broader relative-ultragraph viewpoint (Firrisa, 2020).

More recently, finite-dimensional branching-system methods have been used to characterize residual finite-dimensionality for ultragraph algebras. For ultragraphs satisfying (RFUM2), residual finite-dimensionality of $(\mathcal G,X)$60, residual finite-dimensionality of $(\mathcal G,X)$61, and a graph-theoretic condition involving terminal boundary sets, no-exit cycles, absence of infinite receivers, and absence of infinite backward chains are equivalent (Gonçalves et al., 1 Jul 2026). Although this result is not formulated for relative ultragraph algebras, it reinforces the same methodological pattern that the relative theory adopts: boundary-path combinatorics, orbit decompositions, and branching-system representations are the decisive structural tools.

Within the present literature, the dedicated theory of relative ultragraph algebras is therefore centered on four ingredients: a universal presentation with Cuntz–Krieger relations imposed only on $(\mathcal G,X)$62, reduction to an ordinary ultragraph algebra $(\mathcal G,X)$63 under a finite-range hypothesis, injectivity theorems formulated through projections, defect projections, and cycle spectra, and a branching-system/Markov-map realization that makes the relative nature of the relations dynamically visible (Eidt et al., 27 Aug 2025).