2.1. Graded rings and $\ast $ -rings

A ring R (not necessarily unital) is graded by a group $\Gamma $ if $R=\bigoplus _{\gamma \in \Gamma } R_\gamma $ for additive subgroups $R_\gamma $ and $R_\gamma R_\delta \subseteq R_{\gamma \delta }$ for all $\gamma ,\delta \in \Gamma .$ The elements of the set $\bigcup _{\gamma \in \Gamma } R_\gamma $ are said to be homogeneous and $\gamma $ is the degree of any nonzero element of $R_\gamma .$ The grading is trivial if $R_\gamma =0$ for every $\gamma \in \Gamma $ which is not the group identity. We adopt the standard definitions of graded ring homomorphisms, graded algebras and graded ideals as defined in [Reference Hazrat4].

A ring R is an involutive ring, or a $*$ -ring, if there is an antiautomorphism $*:R\to R$ of order two. If R is also a K-algebra for some commutative, involutive ring K, then R is a $*$ -algebra if $(kx)^*=k^*x^*$ for all $k\in K$ and $x\in R.$ If R and S are $*$ -rings, a ring homomorphism $\phi : R\to S$ is a $*$ -homomorphism if $\phi (x^*)=\phi (x)^*$ for every $x\in R.$

2.2. Leavitt path algebras

Let E be a directed graph. Let $E^0$ denote the set of vertices, let $E^1$ denote the set of edges and let $\mathbf {s}$ and $\mathbf {r}$ be the source and range maps of E, respectively. A sink of E is a vertex which emits no edges, and an infinite emitter is a vertex which emits infinitely many edges. A vertex of E is regular if it is neither a sink nor an infinite emitter. A path is a single vertex or a sequence of edges $e_1e_2\ldots e_n$ for some positive integer n such that $\mathbf {r}(e_i)=\mathbf {s}(e_{i+1})$ for $i=1,\ldots , n-1.$

Extend a graph E to the graph which has the same vertices and whose edges are $E^1\cup \{e^*\mid e\in E^1\}$ , where the range and source functions are the same as in E for $e\in E^1$ and $\mathbf {s}(e^*)=\mathbf {r}(e)$ and $\mathbf {r}(e^*)=\mathbf {s}(e)$ for the added edges. If K is any field, the Leavitt path algebra $L_K(E)$ of E over K is a free K-algebra generated by the set ${E^0\cup E^1\cup \{e^\ast \mid e\in E^1\}}$ such that, for all vertices $v,w$ and edges $e,f$ :

1. (V) $vw =0$ if $v\neq w$ and $vv=v$ ;

2. (E1) $\mathbf {s}(e)e=e\mathbf {r}(e)=e$ ;

3. (E2) $\mathbf {r}(e)e^\ast =e^\ast \mathbf {s}(e)=e^\ast $ ;

4. (CK1) $e^\ast f=0$ if $e\neq f$ and $e^\ast e=\mathbf {r}(e)$ ; and

5. (CK2) $v=\sum _{e\in \mathbf {s}^{-1}(v)} ee^\ast $ for each regular vertex v.

By the first four axioms, every element of $L_K(E)$ can be represented as a sum of the form $\sum _{i=1}^n k_ip_iq_i^\ast $ for some n, paths $p_i$ and $q_i$ and elements $k_i\in K,$ for $i=1,\ldots ,n$ , where $v^*=v$ for $v\in E^0$ and $p^*=e_n^*\ldots e_1^*$ for a path $p=e_1\ldots e_n.$ Using this representation, one can make $L_K(E)$ into an involutive ring by defining $(\sum _{i=1}^n k_ip_iq_i^\ast )^*=\sum _{i=1}^n k_i^*q_ip_i^\ast $ , where $k_i\mapsto k_i^*$ is any involution on K. For more details on these basic properties, see [Reference Abrams, Ara and Siles Molina1].

If we consider K to be trivially graded by $\mathbb Z, L_K(E)$ is naturally graded by $\mathbb Z$ so that the n-component $L_K(E)_n$ is the K-linear span of the elements $pq^\ast $ for paths $p, q$ with $|p|-|q|=n$ , where $|p|$ denotes the length of a path $p.$ Although one can grade a Leavitt path algebra by any group (see [Reference Hazrat4, Section 1.6.1]), we consider only the natural grading by $\mathbb Z.$

If a K-algebra R contains elements $a_v, b_e$ and $c_{e^*}$ which satisfy the axioms (V), (E1), (E2), (CK1) and (CK2), where $v\in E^0$ and $e\in E^1$ for some graph E, the Universal Property of $L_K(E)$ ensures that the map $\phi : v\mapsto a_v, e\mapsto b_e, e^*\mapsto c_{e^*}$ has a unique K-algebra homomorphism extension $\phi : L_K(E)\to R$ (see [Reference Abrams, Ara and Siles Molina1, Remark 1.2.5]). If R is $\mathbb Z$ -graded and the elements $\phi (v), \phi (e)$ and $\phi (e^*)$ have degrees 0, 1 and $-1$ , respectively, then such an extension is graded and the Graded Uniqueness Theorem [Reference Abrams, Ara and Siles Molina1, Theorem 2.2.15] states that $\phi $ is injective if and only if $\phi (v)\neq 0$ for every $v\in E^0.$ If R is involutive and if $a_v^*=a_v$ for every $v\in E^0$ and $(b_e)^*=c_{e^*}$ for every $e\in E^1,$ then $\phi $ is a $*$ -homomorphism (see [Reference Vaš7, Lemma 4.7]).

2.3. Graded ideals of a Leavitt path algebra

A subset H of $E^0$ is said to be hereditary if $\mathbf {r}(p)\in H$ for any path p such that $\mathbf {s}(p)\in H.$ The set H is saturated if $v\in H$ for any regular vertex v such that $\mathbf {r}(\mathbf {s}^{-1}(v))\subseteq H.$ We recall a construction from [Reference Tomforde6]. If H is hereditary and saturated, let

$$ \begin{align*}B_H&=\{v\in E^0-H\,|\, v\mbox{ an infinite emitter and } \\ &\qquad\qquad\qquad \ \ \mathbf{s}^{-1}(v)\cap \mathbf{r}^{-1}(E^0-H)\mbox{ is nonempty and finite}\}, \end{align*} $$

$$ \begin{align*}v^H=v-\sum_{e\in \mathbf{s}^{-1}(v)\cap \mathbf{r}^{-1}(E^0-H)}ee^*\quad\mbox{for all }v\in B_H.\end{align*} $$

An admissible pair is an ordered pair $(H, S)$ , where $H\subseteq E^0$ is hereditary and saturated and $S\subseteq B_H.$ For such a pair, let $I(H,S)$ denote the graded ideal generated by homogeneous elements $H\cup \{v^H \,|\, v\in S \}.$ The ideal $I(H,S)$ is the K-linear span of the elements $pq^*$ for paths $p,q$ with $\mathbf {r}(p)=\mathbf {r}(q)\in H$ and the elements $pv^Hq^*$ for paths $p,q$ with $\mathbf {r}(p)=\mathbf {r}(q)=v\in S$ (see [Reference Tomforde6, Lemma 5.6]). The converse holds as well: for every graded ideal I, the vertices in I form a hereditary and saturated set H and the set of infinite emitters such that $v^H\in I$ is a subset of $B_H$ (see [Reference Tomforde6, Theorem 5.7] and also [Reference Abrams, Ara and Siles Molina1, Theorem 2.5.8]). For an admissible pair $(H, S),$ let $E_{(H,S)}$ be the graph defined by

$$ \begin{align*} F_1(H,S) &=\{e_1\ldots e_n\mbox{ is a path of }E\mid \mathbf{r}(e_n)\in H, \mathbf{s}(e_n)\notin H\cup S\}, \\ F_2(H,S) &=\{p\mbox{ is a path of }E\mid \mathbf{r}(p)\in S,\; |p|>0\}, \\ \overline{F_i}(H,S) & \mbox{ is a copy of }F_i(H,S) \mbox{ for } i=1,2,\\ E_{(H,S)}^0 &= H\cup S\cup F_1(H,S)\cup F_2(H,S), \\ E_{(H,S)}^1 &=\{e\in E^1\,|\, \mathbf{s}(e)\in H\}\cup \{e\in E^1\,|\, \mathbf{s}(e)\in S, \mathbf{r}(e)\in H\}\cup \overline{F_1}(H,S)\cup \overline{F_2}(H,S), \end{align*} $$

with $\mathbf {s}$ and $\mathbf {r}$ the same as on $E^1$ for $e\in E^1\cap E^1_{(H,S)}$ and $\mathbf {s}(\overline p)=p, \mathbf {r}(\overline p)=\mathbf {r}(p)$ for $\overline p\in \overline {F_1}(H,S)\cup \overline {F_2}(H,S)$ . By [Reference Ruiz and Tomforde5, Theorem 6.1] (also [Reference Abrams, Ara and Siles Molina1, Theorem 2.5.22]), the ideal $I(H,S)$ is isomorphic to $L_K (E_{(H,S)}).$

In [Reference Abrams, Ara and Siles Molina1], the graph $E_{(H,\emptyset )}$ is referred to as the hedgehog graph and the graph $E_{(H,S)}$ is referred to as the generalised hedgehog graph. The term ‘hedgehog’ is chosen since the added edges ending at $H\cup S$ resemble spines of the hedgehog body $H\cup S.$ The isomorphism $L_K(E_{(H,S)})\to I(H,S)$ is defined so that the edges of $E_{(H,S)}$ are mapped to certain homogeneous elements of $I(H,S)$ of positive degree, not necessarily degree one. Thus, the degrees are not necessarily preserved under this map and so this isomorphism may not be graded.

In [Reference Aranda Pino, Rangaswamy and Siles Molina2], the authors consider a version of $E_{(H,S)}$ , which they denote $_HE_S$ (see [Reference Ruiz and Tomforde5] for details on differences in definitions); [Reference Aranda Pino, Rangaswamy and Siles Molina2, Proposition 3.7] states that $I(H,S)$ and $L_K(_HE_S)$ are graded isomorphic. However, the map $L_K(_HE_S)\to I(H,S)$ in the proof is defined analogously to the map $L_K(E_{(H,S)})\to I(H,S)$ and, as we noted above, this map is not necessarily a graded map. The last section of [Reference Ruiz and Tomforde5] contains some further details on a problem with the definition of $_HE_S$ and [Reference Aranda Pino, Rangaswamy and Siles Molina2, Proposition 3.7].

3.1. Graph ${\textit{C}}^{\boldsymbol{*}}$ -algebras

Theorem 3.3 has its graph $C^*$ -algebra version. If E is a graph, then the graph $C^*$ -algebra of E is the universal $C^*$ -algebra generated by mutually orthogonal projections $\{p_v\mid v\in E^0\}$ and partial isometries with mutually orthogonal ranges $\{s_e\mid e\in E^1\}$ satisfying the analogues of the (CK1) and (CK2) axioms and the axiom (CK3) stating that $s_es_e^*\leq p_{\mathbf {s}(e)}$ for every $e\in E^1.$ The term ‘universal’ in the definition means that the $C^*$ -algebra version of the algebraic Universal Property, mentioned before, holds (see [Reference Abrams, Ara and Siles Molina1, Definition 5.2.5]). By letting $s_{e_1\ldots e_n}$ be $s_{e_1}\ldots s_{e_n}$ and $s_v=p_v$ for $e_1,\ldots ,e_n\in E^1$ and $v\in E^0, s_p$ is defined for every path $p.$

The set $\{p_v, s_e\mid v\in E^0, e\in E^1\}$ is referred to as a Cuntz–Krieger E-family. For such an E-family and an element z of the unit sphere $\mathbb T$ , one defines a map $\gamma ^E_z$ by $\gamma ^E_z(p_v)=p_v$ and $\gamma ^E_z(s_e)=zs_e$ and then uniquely extends this map to a $*$ -automorphism of $C^*(E)$ (we assume a homomorphism of a $C^*$ -algebra to be bounded). The gauge action $\gamma ^E$ on $\mathbb T$ is given by $\gamma ^E(z)=\gamma ^E_z.$ Note that $\gamma ^E_z(s_ps_q^*)=z^{|p|-|q|}s_ps_q^*$ for $z\in \mathbb T$ and paths p and $q.$ The presence of the degree $|p|-|q|$ of $pq^*$ in the previous formula explains the connection between this action and the $\mathbb Z$ -grading of $L_{\mathbb C}(E)$ . If the integral of a continuous function $f:\mathbb T\to \mathbb C$ over $\mathbb T$ is defined by $\int _{\mathbb T}f(z)\,dz=\int _0^1 f(e^{2\pi i t})\,dt,$ then the gauge action on $C^*(E)$ determines a $\mathbb Z$ -grading of $C^*(E)$ (see [Reference Abrams, Ara and Siles Molina1, Proposition 5.2.11]) so that

(*) $$ \begin{align}C^*(E)_n=\bigg\{x\in C^*(E)\,\bigg|\, \int_{\mathbb T}z^{-n}\gamma^E_z(x)\,dz=x\bigg\} \end{align} $$

is the completion of $L_{\mathbb C}(E)_n$ and $C^*(E)$ is the completion of $\bigoplus _{n\,\in\,\mathbb Z} C^*(E)_n.$ We note that this grading is not a grading in the algebraic sense, but in the $C^*$ -algebra sense (see the paragraph following [Reference Abrams, Ara and Siles Molina1, Theorem 5.2.9.]).

A closed ideal J of a graph $C^*$ -algebra $C^*(E)$ is gauge-invariant if $\gamma ^E_z(J)=J$ for every $z\in \mathbb T.$ By [Reference Bates, Hong, Raeburn and Szymański3, Theorem 3.6], every such ideal J is the completion of the linear span of the elements $s_ps_q^*$ for paths $p,q$ with $\mathbf {r}(p)=\mathbf {r}(q)\in H$ and the elements $s_pp_v^Hs_q^*$ for paths $p,q$ with $\mathbf {r}(p)=\mathbf {r}(q)=v\in S$ , where $p_v^H=p_v-\sum _{e\in \mathbf {s}^{-1}(v)\cap \mathbf {r}^{-1}(E^0-H)}s_es_e^*$ for $v\in B_H$ , and where $(H, S)$ is the admissible pair uniquely determined by J and defined analogously as for an ideal of $L_K(E)$ . Also as before, an admissible pair $(H,S)$ uniquely determines a closed gauge-invariant ideal $J(H,S)$ which is graded in the $C^*$ -algebra sense.

If R is a $C^*$ -algebra with an action $\beta : \mathbb T\to $ Aut $(R),$ then we say that a $*$ -homomorphism $\phi : C^*(E)\to R$ is gauge-invariant if $\beta _z\circ \phi =\phi \circ \gamma ^E_z$ for every $z\in \mathbb T.$ Thus, if J is a closed gauge-invariant ideal of $C^*(E), F$ is a graph and $\phi : C^*(F)\to J$ is a $*$ -homomorphism, then $\phi $ is gauge-invariant if $\gamma _z^E|_J\circ \phi =\phi \circ \gamma ^F_z$ for every $z\in \mathbb T.$ It is direct to see that every such gauge-invariant map $\phi $ is graded. Indeed, using the formula (*) and the fact that $\phi $ is bounded, one directly checks that $\phi (C^*(F)_n)\subseteq J_n.$

Corollary 3.4. For an admissible pair $(H,S)$ of a graph $E,$ there is a gauge-invariant (thus graded) $*$ -isomorphism

$$ \begin{align*}\phi: C^*(P_{(H,S)})\cong J(H,S).\end{align*} $$

Thus, for every closed gauge-invariant ideal J of a graph $C^*$ -algebra, there is a gauge-invariant (and thus graded) $*$ -isomorphism mapping the graph $C^*$ -algebra of the porcupine graph corresponding to J onto $J.$