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 $\{s_e: e\in E^1\}$ subject to the Cuntz-Krieger relations:
  • (GA1) $s_e^* s_e = p_{r(e)}$, $s_e^* s_f = 0$ for $e\neq f$.
  • (GA2) $s_e s_e^* \leq p_{s(e)}$.
  • (GA3) $p_v = \sum_{s(e)=v} s_e s_e^*$ when $0<|s^{-1}(v)|<\infty$.

The diagonal MASA $\mathcal{D}_E$ is the abelian $*$-subalgebra generated by projections $p_\mu = s_\mu s_\mu^*$ for finite paths $\mu\in E^k$, $k\geq 0$. Under standard hypotheses (no sinks, every loop has an exit), $\mathcal{D}_E$ is maximal abelian. The algebra admits a canonical gauge action with fixed-point subalgebra $\mathcal{F}_E$ (the core AF-algebra).

2. Endomorphisms from Unitaries and the Support-Preserving Condition

Let

$$\mathcal{U}_E := \{u \in M(C^*(E)): u \text{ unitary, } u p_v = p_v u \quad \forall v\in E^0\}.$$

Every $u\in\mathcal{U}_E$ induces a $*$-endomorphism $\varphi_u$ defined by

$\varphi_u(p_v)=p_v,\qquad \varphi_u(s_e)=u s_e.$

Since $u$ commutes with $p_v$ for all $v$, these $\{u s_e, p_v\}$ satisfy the Cuntz-Krieger relations. If $u$ lies in the minimal unitization of the algebraic part of the core $\mathcal{F}_E$, $\varphi_u$ is injective. The semigroup law is $$\varphi_u \circ \varphi_w = \varphi_{\varphi_u(w)u}$$. Invertibility of $\varphi_u$ is equivalent to $u^*$ belonging to the image of the strict extension of $\varphi_u$.

Localized endomorphism: If $u\in\mathcal{F}_E\cap\mathcal{U}_E$ and belongs to the finite-dimensional span of $\{s_\mu s_\nu^*: |\mu|=|\nu|\leq k\}$, then $\varphi_u$ is localized at level $k$.

Support-preserving or permutative endomorphisms are those $\varphi_u$ such that $\varphi_u(\mathcal{D}_E)\subseteq\mathcal{D}_E$; typically these arise when $u$ is a permutation unitary acting combinatorially at some level.

3. Weyl Groups and Their Combinatorial Structure

The Weyl group $\mathcal{W}_E$ of a graph $\mathrm{C}^*$-algebra $C^*(E)$ is defined as

$$\mathcal{W}_E := \operatorname{Aut}(C^*(E),\mathcal{D}_E)/\operatorname{Aut}_{\mathcal{D}_E}(C^*(E)),$$

where $\operatorname{Aut}(C^*(E),\mathcal{D}_E)$ are automorphisms preserving $\mathcal{D}_E$ globally, and $\operatorname{Aut}_{\mathcal{D}_E}(C^*(E))$ those acting as the identity on $\mathcal{D}_E$. $\mathcal{W}_E$ is countable and discrete for graphs with no sinks and all loops having exits.

Permutative automorphisms arising from permutation unitaries $u \in \mathcal{U}_E\cap\mathcal{F}_E$ and automorphisms of the underlying graph $E$ generate a significant combinatorial subgroup of $\mathcal{W}_E$.

The restricted Weyl group $\mathcal{R}_E$ further requires automorphisms to preserve the core $\mathcal{F}_E$: $$\mathcal{R}_E := \operatorname{Aut}(C^*(E),\mathcal{F}_E,\mathcal{D}_E) / \operatorname{Aut}_{\mathcal{D}_E}(C^*(E)).$$ Structural results include:

• Every automorphism in $\operatorname{Aut}(C^*(E),\mathcal{D}_E)$ arises as $\varphi_u$ for a unique $u\in\mathcal{U}_E$ lying in $\mathcal{U}(\mathcal{D}_E)$.
• $$\operatorname{Aut}(C^*(E),\mathcal{D}_E)\cong \mathcal{U}(\mathcal{D}_E)$$ as topological groups.
• $$\operatorname{Aut}(C^*(E),\mathcal{F}_E,\mathcal{D}_E)$$ is generated (semidirectly) by permutation unitaries in $\mathcal{U}_E\cap\mathcal{F}_E$ and graph-automorphisms.
• If the relative commutant of $\mathcal{F}_E$ in $C^*(E)$ is trivial, the only inner automorphisms in the restricted group come from the finite subgroup $$\{\mathrm{Ad}(w): w\in\mathcal{F}_E \cap N_{\mathcal{F}_E}(\mathcal{D}_E)\}$$.

4. Invertibility Criteria and Combinatorial Analysis

For $u\in \mathcal{F}_E\cap\mathcal{U}_E$ lying in the finite-dimensional algebra $\mathcal{F}_k$, the following are equivalent ((Conti et al., 2011), Thm 5.1):

1. $\varphi_u$ is an automorphism of $C^*(E)$ and $\varphi_u^{-1}$ is localized.
2. The sequence $$\operatorname{Ad}(u_m)(u^*)\in \mathcal{U}(\mathcal{F}_{k+m})$$ stabilizes as $m\to\infty$.
3. The finite ring $A_u\subset \operatorname{End}(\mathcal{F}_{k-1})$ generated by $a_{e,f}(x) := s_e^* u^* x u s_f$ for $e,f\in E^1$ is nilpotent.
4. The limit space $E_u \subset \mathcal{F}_{k-1}$ obtained by inductive intersections lies in $\mathcal{D}_E$.

For the restriction $\varphi_u|_{\mathcal{D}_E}$, a parallel criterion applies: $\varphi_u|_{\mathcal{D}_E}$ is an automorphism iff the finite-dimensional subring $A_u^D$ generated by compressions $a_{e,e}$ to $D_{k-1}$ is nilpotent—corresponding to a descending sequence of subspaces in $D_{k-1}$ falling eventually into $\mathcal{D}_E$.

5. Diagrams and the Combinatorial Approach to Permutative Endomorphisms

Permutation unitaries at exactly level $k$ correspond combinatorially to disjoint families of permutations $$\sigma_{v,w} \in \mathrm{Perm}(E^k_{w\rightarrow v})$$ indexed by vertices $v, w$. The action of $\varphi_u$ on $C^*(E)$ and the finite rings $A_u, A_u^D$ can be encoded in finite labeled graphs.

Two combinatorial conditions arise:

• Condition (b): $$\varphi_u|_{\mathcal{D}_E}\in\operatorname{Aut}(\mathcal{D}_E)$$ iff the associated directed graph on pairs $(a, a')$ for paths $a, a'\in E^{k-1}_{* \rightarrow r(e)}$ 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): $\varphi_u\in\operatorname{Aut}(C^*(E))$ iff in a digraph on pairs $(a,b)\in E^{k-1}\times E^{k-1}$ with edges labeled by $(e,f)$ when $\sigma(e,a)$ and $\sigma(f,b)$ agree on their terminal letter, all cycles except for trivial loops are excluded.

The composite theorem ((Conti et al., 2011), Thm 6.4): Let $u$ be a permutation unitary at level $k$. Then $\varphi_u$ is an automorphism of $C^*(E)$ if and only if both (b) and (d) hold. Notably, (b) $\Leftrightarrow \varphi_u|_{\mathcal{D}_E}$ invertible; combining (d) yields invertibility of $\varphi_u$.

6. Outer Automorphism Criteria and Shift-Commutation

An automorphism in $\mathcal{R}_E$ is inner if and only if the implementing unitary lies in the finite permutation-unitary subgroup $S_E$. Two mechanisms are used to distinguish outerness:

• Gauge-action rigidity: If the only unitaries normalizing $\mathcal{F}_E$ are those in $\mathcal{F}_E$, then every support-preserving automorphism $\varphi_u$ preserving $\mathcal{F}_E$ must come from $u\in\mathcal{F}_E$.
• Shift-commutation: Given $$\alpha\in\operatorname{Aut}(C^*(E),\mathcal{F}_E,\mathcal{D}_E)$$, the induced homeomorphism $\alpha|_{\mathcal{D}_E}$ of the spectrum $X_E$ satisfies an eventual commutation with the one-sided shift $\sigma$; specifically, for some $m\geq 0$,

$$\rho\circ\sigma^m = \sigma^{m+1}\circ\rho \quad \text{on } \mathcal{D}_E,$$

where $\rho=\alpha|_{\mathcal{D}_E}$.

Corollary: If $\varphi_u$ in the combinatorial subgroup of $\mathcal{R}_E$ has infinite order, it represents an infinite-order element in $\operatorname{Out}(C^*(E))$.

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 $k=3$ there exists a permutation unitary $u$ for which Condition (b) holds but (d) fails. Thus, $\varphi_u|_{\mathcal{D}_E}$ is an automorphism, but $\varphi_u$ is a proper (non-surjective) support-preserving endomorphism.
• Simple Kirchberg-algebra: For a strongly connected 4-vertex graph with $K_*(C^*(E))\cong\mathbb{Z}$, a non-graph-automorphism permutation unitary $u$ at level $k=2$ gives $\varphi_u$ of order $2$ in $\mathcal{R}_E$, with both (b) and (d) satisfied, hence $\varphi_u \in \operatorname{Aut}(C^*(E))$.

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

---

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