Support-Preserving Endomorphisms

• Support-preserving endomorphisms are *-endomorphisms of graph C*-algebras that preserve the diagonal MASA via permutation unitaries and combinatorial structures.
• They implement the Cuntz-Krieger relations by leveraging localized unitaries and finite-dimensional subalgebras, with invertibility characterized by nilpotency in associated finite rings.
• These endomorphisms connect the combinatorial aspects of graph theory with Weyl group structures, offering insights into inner versus outer automorphisms and shift-commutation dynamics.

A support-preserving endomorphism of a graph $\mathrm{C}^*$-algebra $C^*(E)$ is a $*$-endomorphism that globally preserves the diagonal maximal abelian subalgebra (MASA) $\mathcal{D}_E$, often implemented by unitaries subject to compatibility conditions with the graph structure. These endomorphisms, also called "permutative" when realized by permutation unitaries, have deep connections to the structure theory of $C^*$-algebras, Weyl groups, and the interplay of combinatorial and operator-algebraic methods (Conti et al., 2011).

1. The Structure of Graph $\mathrm{C}^*$-Algebras and the Diagonal MASA

Given a countable directed graph $E=(E^0,E^1,r,s)$, the graph $\mathrm{C}^*$-algebra $C^*(E)$ is generated by

• Mutually orthogonal projections $\{p_v: v\in E^0\}$.
• Partial isometries $C^*(E)$0 subject to the Cuntz-Krieger relations:
  • (GA1) $C^*(E)$1, $C^*(E)$2 for $C^*(E)$3.
  • (GA2) $C^*(E)$4.
  • (GA3) $C^*(E)$5 when $C^*(E)$6.

The diagonal MASA $C^*(E)$7 is the abelian $C^*(E)$8-subalgebra generated by projections $C^*(E)$9 for finite paths $*$0, $*$1. Under standard hypotheses (no sinks, every loop has an exit), $*$2 is maximal abelian. The algebra admits a canonical gauge action with fixed-point subalgebra $*$3 (the core AF-algebra).

2. Endomorphisms from Unitaries and the Support-Preserving Condition

Let

$*$4

Every $*$5 induces a $*$6-endomorphism $*$7 defined by

$*$8

Since $*$9 commutes with $\mathcal{D}_E$0 for all $\mathcal{D}_E$1, these $\mathcal{D}_E$2 satisfy the Cuntz-Krieger relations. If $\mathcal{D}_E$3 lies in the minimal unitization of the algebraic part of the core $\mathcal{D}_E$4, $\mathcal{D}_E$5 is injective. The semigroup law is $\mathcal{D}_E$6. Invertibility of $\mathcal{D}_E$7 is equivalent to $\mathcal{D}_E$8 belonging to the image of the strict extension of $\mathcal{D}_E$9.

Localized endomorphism: If $C^*$0 and belongs to the finite-dimensional span of $C^*$1, then $C^*$2 is localized at level $C^*$3.

Support-preserving or permutative endomorphisms are those $C^*$4 such that $C^*$5; typically these arise when $C^*$6 is a permutation unitary acting combinatorially at some level.

3. Weyl Groups and Their Combinatorial Structure

The Weyl group $C^*$7 of a graph $C^*$8-algebra $C^*$9 is defined as

$\mathrm{C}^*$0

where $\mathrm{C}^*$1 are automorphisms preserving $\mathrm{C}^*$2 globally, and $\mathrm{C}^*$3 those acting as the identity on $\mathrm{C}^*$4. $\mathrm{C}^*$5 is countable and discrete for graphs with no sinks and all loops having exits.

Permutative automorphisms arising from permutation unitaries $\mathrm{C}^*$6 and automorphisms of the underlying graph $\mathrm{C}^*$7 generate a significant combinatorial subgroup of $\mathrm{C}^*$8.

The restricted Weyl group $\mathrm{C}^*$9 further requires automorphisms to preserve the core $E=(E^0,E^1,r,s)$0: $E=(E^0,E^1,r,s)$1 Structural results include:

• Every automorphism in $E=(E^0,E^1,r,s)$2 arises as $E=(E^0,E^1,r,s)$3 for a unique $E=(E^0,E^1,r,s)$4 lying in $E=(E^0,E^1,r,s)$5.
• $E=(E^0,E^1,r,s)$6 as topological groups.
• $E=(E^0,E^1,r,s)$7 is generated (semidirectly) by permutation unitaries in $E=(E^0,E^1,r,s)$8 and graph-automorphisms.
• If the relative commutant of $E=(E^0,E^1,r,s)$9 in $\mathrm{C}^*$0 is trivial, the only inner automorphisms in the restricted group come from the finite subgroup $\mathrm{C}^*$1.

4. Invertibility Criteria and Combinatorial Analysis

For $\mathrm{C}^*$2 lying in the finite-dimensional algebra $\mathrm{C}^*$3, the following are equivalent ((Conti et al., 2011), Thm 5.1):

1. $\mathrm{C}^*$4 is an automorphism of $\mathrm{C}^*$5 and $\mathrm{C}^*$6 is localized.
2. The sequence $\mathrm{C}^*$7 stabilizes as $\mathrm{C}^*$8.
3. The finite ring $\mathrm{C}^*$9 generated by $C^*(E)$0 for $C^*(E)$1 is nilpotent.
4. The limit space $C^*(E)$2 obtained by inductive intersections lies in $C^*(E)$3.

For the restriction $C^*(E)$4, a parallel criterion applies: $C^*(E)$5 is an automorphism iff the finite-dimensional subring $C^*(E)$6 generated by compressions $C^*(E)$7 to $C^*(E)$8 is nilpotent—corresponding to a descending sequence of subspaces in $C^*(E)$9 falling eventually into $\{p_v: v\in E^0\}$0.

5. Diagrams and the Combinatorial Approach to Permutative Endomorphisms

Permutation unitaries at exactly level $\{p_v: v\in E^0\}$1 correspond combinatorially to disjoint families of permutations $\{p_v: v\in E^0\}$2 indexed by vertices $\{p_v: v\in E^0\}$3. The action of $\{p_v: v\in E^0\}$4 on $\{p_v: v\in E^0\}$5 and the finite rings $\{p_v: v\in E^0\}$6 can be encoded in finite labeled graphs.

Two combinatorial conditions arise:

• Condition (b): $\{p_v: v\in E^0\}$7 iff the associated directed graph on pairs $\{p_v: v\in E^0\}$8 for paths $\{p_v: v\in E^0\}$9 has the property that no two distinct fixed points exist under sufficiently long colored paths. Equivalently, a partial order forces upward flow of each "off-diagonal" vertex.
• Condition (d): $C^*(E)$00 iff in a digraph on pairs $C^*(E)$01 with edges labeled by $C^*(E)$02 when $C^*(E)$03 and $C^*(E)$04 agree on their terminal letter, all cycles except for trivial loops are excluded.

The composite theorem ((Conti et al., 2011), Thm 6.4): Let $C^*(E)$05 be a permutation unitary at level $C^*(E)$06. Then $C^*(E)$07 is an automorphism of $C^*(E)$08 if and only if both (b) and (d) hold. Notably, (b) $C^*(E)$09 invertible; combining (d) yields invertibility of $C^*(E)$10.

6. Outer Automorphism Criteria and Shift-Commutation

An automorphism in $C^*(E)$11 is inner if and only if the implementing unitary lies in the finite permutation-unitary subgroup $C^*(E)$12. Two mechanisms are used to distinguish outerness:

• Gauge-action rigidity: If the only unitaries normalizing $C^*(E)$13 are those in $C^*(E)$14, then every support-preserving automorphism $C^*(E)$15 preserving $C^*(E)$16 must come from $C^*(E)$17.
• Shift-commutation: Given $C^*(E)$18, the induced homeomorphism $C^*(E)$19 of the spectrum $C^*(E)$20 satisfies an eventual commutation with the one-sided shift $C^*(E)$21; specifically, for some $C^*(E)$22,

$C^*(E)$23

where $C^*(E)$24.

Corollary: If $C^*(E)$25 in the combinatorial subgroup of $C^*(E)$26 has infinite order, it represents an infinite-order element in $C^*(E)$27.

7. Illustrative Examples

Two canonical examples illustrate the criteria for support-preserving endomorphisms:

• Fibonacci-graph: For a graph with two vertices and three edges in a two-cycle plus tail configuration, at level $C^*(E)$28 there exists a permutation unitary $C^*(E)$29 for which Condition (b) holds but (d) fails. Thus, $C^*(E)$30 is an automorphism, but $C^*(E)$31 is a proper (non-surjective) support-preserving endomorphism.
• Simple Kirchberg-algebra: For a strongly connected 4-vertex graph with $C^*(E)$32, a non-graph-automorphism permutation unitary $C^*(E)$33 at level $C^*(E)$34 gives $C^*(E)$35 of order $C^*(E)$36 in $C^*(E)$37, with both (b) and (d) satisfied, hence $C^*(E)$38.

These cases explicitly employ the combinatorial digraph constructions to verify invertibility and restriction to the diagonal.

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For full proofs, detailed constructions, and comprehensive combinatorial models, see Sections 3–6 of (Conti et al., 2011).