1 Introduction
Roughly speaking, if parts of an object are similar to the whole, repeating the structure of the object at all scales, then we call the object self-similar. If a group or a groupoid acts self-similarly on a space, then we simply call it a self-similar group or a self-similar groupoid. Self-similar groups were introduced by Grigorchuk in [Reference Gupta and Sidki2] and Gupta and Sidki in [Reference Grigorchuk3] to answer the question of existence of groups with intermediate growth. Recently, operator algebraists have made use of self-similar groups to study
$C^*$
-algebras (for example, [Reference Exel and Pardo1, Reference Laca, Raeburn, Rammage and Whittaker4]). Since a groupoid is a generalisation of a group, it is then natural to think of this notion of self-similarity on a groupoid, as introduced in [Reference Laca, Raeburn, Rammage and Whittaker5]. Self-similar groups act on the path-spaces of graphs with a single vertex. To study self-similar actions on more general directed graphs and the associated Cuntz–Krieger algebras, Laca et al. in [Reference Laca, Raeburn, Rammage and Whittaker5] introduced the notion of a self-similar groupoid. In [Reference Laca, Raeburn, Rammage and Whittaker5], the authors are primarily interested in computing KMS states, so, informed by results about graph
$C^*$
-algebras, they restricted their attention to finite graphs. They also built their self-similar groupoids by generalising the process whereby automata are used to build self-similar groups, so by definition their self-similar actions satisfy a faithfulness condition that simplifies their analysis and, in particular, guarantees that
$C^*(\mathcal {G})$
embeds in
$C^*(\mathcal {G}, E)$
.
Another approach to self-similar actions on graphs with multiple vertices was developed by Exel and Pardo [Reference Exel and Pardo1] and does not require a faithfulness condition. We combine and generalise the constructions in [Reference Exel and Pardo1, Reference Laca, Raeburn, Rammage and Whittaker5]. We consider self-similar actions of groupoids
$\mathcal {G}$
on the path spaces
$E^*$
of row-finite directed graphs E that are not necessarily faithful in the sense of [Reference Laca, Raeburn, Rammage and Whittaker5]. We develop a new faithfulness condition that is weaker than both faithfulness as in [Reference Laca, Raeburn, Rammage and Whittaker5] and pseudo-faithfulness as in [Reference Exel and Pardo1], but still guarantees that
$C^*(\mathcal {G})$
embeds in
$C^*(\mathcal {G}, E)$
, and we prove a gauge-invariant uniqueness theorem. In particular, our theorems apply to conventional actions of groups on graphs (see Example 3.7). We also depart from [Reference Laca, Raeburn, Rammage and Whittaker5] in that we work solely with generators and relations, without employing the machinery of Hilbert modules and Cuntz–Pimsner algebras.
The paper is organised as follows. We define our notion of a self-similar groupoid
$(\mathcal {G}, E)$
in Definition 2.1 and construct the associated
$C^*$
-algebras
$C^*(\mathcal {G}, E)$
in Section 3 following the approach of [Reference Raeburn6]. We introduce our injectivity condition in our key technical result Proposition 3.6. We analyse the fixed-point algebra
$C^*(\mathcal {G},E)^{\gamma }$
for the gauge action
$\gamma $
in Section 4. By applying all the results in the previous sections, we prove the gauge-invariant uniqueness theorem in Theorem 5.1.
2 Self-similar groupoids
Recall that a groupoid
$\mathcal {G}$
is a small category with inverses. We write
$\mathcal {G}^{(0)}$
for the set of identity morphisms and
$r, s : \mathcal {G} \rightarrow \mathcal {G}^{(0)}$
for the maps induced by the codomain and domain range maps. Throughout this paper,
$\mathcal {G}$
will denote a countable discrete groupoid. We will assume that
$\mathcal {G}$
is amenable in the sense of [Reference Sims, Whitehead and Whittaker7]. Since
$\mathcal {G}$
is discrete, this is equivalent to requiring that its full and reduced
$C^*$
-algebras coincide, and is also equivalent to requiring that each of its isotropy groups is amenable.
As in [Reference Raeburn6], a (directed) graph is a quadruple
$E=(E^0, E^1, r, s)$
consisting of countable sets
$E^0$
,
$E^1$
and maps
$r, s: E^1 \rightarrow E^0$
. Elements of
$E^1$
are called edges and elements of
$E^0$
are called vertices. We will assume that all our graphs are row-finite and have no sources in the sense that
$0 < \lvert r^{-1}(v) \rvert < \infty $
for all
$v \in E^0$
. Let
$e, f \in E^1$
with
$s(e)=r(f)$
. Then,
$ef$
is a path of length 2 and we write
$\lvert ef \rvert =2$
. In general, a path
$\mu $
of length n in E is a sequence
$\mu _1\mu _2 \cdots \mu _n$
such that
$s(\,\mu _i)=r(\,\mu _{i+1})$
for
$1\leq i \leq n-1$
. The vertices are viewed as paths of length 0. The paths of length n are collected in a set denoted by
$E^n$
. We let
$E^*:=\bigcup _{k\geq 0} E^k$
. It is natural to extend the maps
$r,s$
to
$E^*$
by putting
$r(\,\mu )=r(\,\mu _1)$
and
$s(\,\mu )=s(\,\mu _{\lvert \,\mu \rvert })$
where
$\lvert \,\mu \rvert>1$
, and
$r(v)=v=s(v)$
for
$v \in E^0$
.
Definition 2.1. Let E be a row-finite graph with no sources and let
$\mathcal {G}$
be a groupoid with
$\mathcal {G}^{(0)}=E^0$
. Write
$$ \begin{align*} \mathcal{G}\ast E^*:= \{(g, \mu) \in \mathcal{G}\times E^* \mid s(g)=r(\,\mu)\} \end{align*} $$
and
$$ \begin{align*} E^* \ast \mathcal{G} := \{(\,\mu, g) \mid s(\,\mu)=r(g)\}. \end{align*} $$
We will often denote the element
$(\,\mu , g)\in E^* \ast \mathcal {G}$
by the shorthand
$\mu g$
. A self-similar action of
$\mathcal {G}$
on
$E^*$
consists of two maps: (1) an action
$(g,\mu )\mapsto g\cdot \mu $
of
$\mathcal {G}$
on the set
$E^*$
and (2) a map
$\varphi : \mathcal {G}\ast E^* \rightarrow \mathcal {G}$
such that:
-
(i)
$g\cdot (\kern1pt\mu \beta )=(g\cdot \mu )(\varphi (g, \mu )\cdot \beta )$
; -
(ii)
$r(g\cdot \mu )=g\cdot r(\,\mu )$
and
$s(g\cdot \mu )=\varphi (g, \mu ) \cdot s(\,\mu )$
; -
(iii)
$\lvert g\cdot \mu \rvert =\lvert \,\mu \rvert $
; -
(iv)
$\varphi (g, v)=g$
; -
(v)
$\varphi (gh, \mu )=\varphi (g, h\cdot \mu )\varphi (h, \mu )$
; -
(vi)
$\varphi (g, \mu \beta )=\varphi (\varphi (g,\mu ),\beta )$
; and -
(vii)
$\varphi (g^{-1}, \mu )=(\varphi (g, g^{-1}\cdot \mu ))^{-1}$
.
We write this self-similar action of the groupoid
$\mathcal {G}$
on
$E^*$
as a pair
$(\mathcal {G},E)$
and call it a self-similar groupoid
$(\mathcal {G},E)$
.
3 The universal
$C^*$
-algebra
$C^*(\mathcal {G}, E)$
Recall that a Toeplitz–Cuntz–Krieger family for a row-finite directed graph E with no sources consists of partial isometries
$\{T_e \mid e \in E^1\}$
and mutually orthogonal projections
$\{W_v \mid v \in E^0\}$
satisfying
$T_e^*T_e=W_{s(e)}$
and
$W_v \geq \sum _{e \in vE^1}T_eT^*_e$
for all
$v \in E^0$
. It is a Cuntz–Krieger E-family if
$W_v=\sum _{e\in vE^1}T_eT^*_e$
for all
$v \in E^0$
. A unitary representation in a unital
$C^*$
-algebra A of a discrete groupoid
$\mathcal {G}$
is a family
$\{U_g \mid g \in \mathcal {G}\}$
of partial isometries such that
$U_gU_h=\delta _{s(g),r(h)}U_{gh}$
and
$U_{g^{-1}}=U_g^*$
for all
$g, h \in \mathcal {G}$
, and such that
$\sum _{v\in \mathcal {G}^{(0)}}U_v=1_A$
.
Definition 3.1. Let
$(\mathcal {G},E)$
be a self-similar groupoid. A Toeplitz
$(\mathcal {G}, E)$
-family consists of partial isometries
$\{T_e \mid e \in E^1\}$
and a unitary representation
$\{W_g \mid g \in \mathcal {G}\}$
of
$\mathcal {G}$
such that
$\{T_e \mid e \in E^1\} \cup \{W_v \mid v \in E^0\} $
is a Toeplitz–Cuntz–Krieger E-family. It is a Cuntz–Krieger
$(\mathcal {G}, E)$
-family if
$\{T_e, W_v\}$
is a Cuntz–Krieger E-family.
Example 3.2. Suppose that
$\mathcal {G}$
acts self-similarly on E. Let
$\mathcal {H}:=l^2(E^* \ast \mathcal {G})$
with orthonormal basis
$\{e_{\mu g} \mid \mu \in E^*, g \in \mathcal {G} \}$
. For
$e \in E^1$
and
$h \in \mathcal {G}$
, let
$T_e, W_h \in B(\mathcal {H})$
be the operators such that
$$ \begin{align*} T_e e_{\mu g} &= \begin{cases} e_{e \mu g} & \mathrm{if} \ s(e)=r(\,\mu), \\ 0 & \mathrm{otherwise}, \end{cases} \\[6pt] W_h e_{\mu g} &= \begin{cases} e_{(h \cdot \mu) (\varphi (h, \mu)\cdot g)} & \mathrm{if} \ s(h)= r(\,\mu), \\ 0 & \mathrm{otherwise}. \end{cases} \end{align*} $$
For
$v \in E^0$
,
$W_v$
is the projection onto
$l^2(\{\,\mu g\mid r(\,\mu )=v\}) \subset \mathcal {H}$
, and a routine calculation shows that for
$e \in E^1$
,
$$ \begin{align*} T_e^* e_{\mu g} = \begin{cases} e_{\mu' g} & \mathrm{if} \ \mu=e\mu', \\ 0 & \mathrm{otherwise}. \end{cases} \end{align*} $$
It is routine to check that the family
$\{T_e \mid e \in E^1\} \cup \{ W_h \mid h \in \mathcal {G}\}$
is a Toeplitz
$(\mathcal {G}, E)$
-family in
$B(\mathcal {H})$
.
The proofs of the following two lemmas are more or less identical to those of the cited results in [Reference Laca, Raeburn, Rammage and Whittaker4–Reference Raeburn6].
Lemma 3.3 (See [Reference Laca, Raeburn, Rammage and Whittaker4, Lemma 3.4] and [Reference Laca, Raeburn, Rammage and Whittaker5, Lemma 4.6])
Let
$(\mathcal {G},E)$
be a self-similar groupoid. Suppose that
$\{T_e,W_g\}$
is a Toeplitz
$(\mathcal {G}, E)$
-family in a
$C^*$
-algebra B. Then for all
$\mu , \beta , \alpha , \rho \in E^*$
,
$g, h \in \mathcal {G}$
,
$$ \begin{align*}(T_{\mu} W_g T^*_{\beta})(T_{\alpha} W_h T^*_{\rho})=\begin{cases} T_{\mu(g \cdot \alpha')} W_{\varphi(g,\alpha')h} T^{*}_{\rho} & \mathrm{if} \ \alpha=\beta \alpha', \\ T_{\mu} W_{g\varphi(h, h^{-1}\cdot \beta')} T^{*}_{\rho(h^{-1}\cdot \beta')} & \mathrm{if} \ \beta=\alpha\beta', \\ 0 & \mathrm{otherwise}. \end{cases} \end{align*} $$
Lemma 3.4 (See [Reference Raeburn6, Corollary 1.16])
Let
$(\mathcal {G},E)$
be a self-similar groupoid. Suppose that
$\{T_e,W_g\}$
is a Toeplitz
$(\mathcal {G}, E)$
-family. Then
$$ \begin{align*}C^*(T,W)=\overline{\mathrm{span}}\{T_{\mu} W_g T^*_{\beta} \mid \mu, \beta \in E^*, g \in \mathcal{G}^{s(\,\mu)}_{s(\,\beta)}, s(\,\mu)=g\cdot s(\,\beta)\}.\end{align*} $$
A standard argument along the lines of Propositions 1.20 and 1.21 of [Reference Raeburn6] shows that there exists a
$C^*$
-algebra
$\mathcal {T}C^*(\mathcal {G}, E)$
generated by a Toeplitz
$(\mathcal {G},E)$
-family
$\{t_e, w_g\}$
that is universal in the sense that for any Toeplitz
$(\mathcal {G},E)$
-family
$\{T_e, W_g\}$
, there is a homomorphism
$\pi _{T,W}: \mathcal {T}C^*(\mathcal {G}, E) \rightarrow C^*(T,W)$
such that
$\pi _{T,W}(t_e)=T_e$
for all
$e \in E^1$
and
$\pi _{T,W}(w_g)=W_g$
for all
$g \in \mathcal {G}$
.
Let I be the ideal of
$\mathcal {T}C^*(\mathcal {G}, E)$
generated by
$\{w_v-\sum _{r(e)=v}t_et^*_e \mid v \in E^0\}$
. Then
$s_e:=t_e+I$
for all
$e \in E^1$
and
$u_g:=w_g+I$
for all
$g \in \mathcal {G}$
defines a Cuntz–Krieger
$(\mathcal {G}, E)$
-family and
$C^*(\mathcal {G}, E):=\mathcal {T}C^*(\mathcal {G}, E)/I$
is universal for Cuntz–Krieger
$(\mathcal {G}, E)$
- families. We will need to know that the generators of
$C^*(\mathcal {G}, E)$
are nonzero. For this, we construct a concrete Cuntz–Krieger
$(\mathcal {G}, E)$
-family (see Proposition 3.6).
Lemma 3.5. Let
$(\mathcal {G}, E)$
be a self-similar groupoid. Let
$\pi : C^*(T,W) \rightarrow B(l^2(E^* \ast \mathcal {G}))$
be the representation induced by the Toeplitz
$(\mathcal {G}, E)$
-family
$\{T_e, W_g\}$
of Example 3.2. For every
$a \in I$
and every
$\varepsilon>0$
, there exists
$N \in \mathbb {N}$
such that for all
$n \geq N$
,
$$ \begin{align*} \lVert\pi(a)|_{\overline{\mathrm{span}}\{e_{\lambda g} \mid \lambda \in E^n, g \in \mathcal{G}^{s(\lambda)}\}}\rVert <\varepsilon. \end{align*} $$
Proof. First, note that for
$v \in E^0, \lambda \in E^*$
and
$k \in \mathcal {G}^{s(\lambda )}$
,
$$ \begin{align} \bigg(W_v-\sum_{e\in vE^1}T_eT_e^*\bigg ) e_{\lambda k} & = \begin{cases} 0 & \mathrm{if} \ \lambda \neq v, \\ e_{\lambda k} & \mathrm{otherwise}. \end{cases} \end{align} $$
Now fix
$v \in E^0, \mu , \beta \in E^*$
and
$g \in \mathcal {G}^{s(\,\mu )}_v, h \in \mathcal {G}^{s(\,\beta )}_v$
. Then,
$$ \begin{align*} T_{\mu} W_g & \bigg(W_v-\sum_{e\in vE^1} T_eT^*_e \bigg)W^*_h T^*_{\beta} e_{\lambda k} \\ &= \begin{cases} T_{\mu} W_g (W_v-\sum_{e\in vE^1} T_eT^*_e) e_{(h^{-1} \cdot \lambda^{'}) \varphi(h^{-1}, \lambda^{'})k} & \mathrm{if} \ \lambda=\beta \lambda^{'}, \\ 0 & \mathrm{otherwise}. \end{cases} \end{align*} $$
By (3.1), this equals
$0$
if
$\lvert \lambda ^{'}\rvert>0$
. Hence,
$$ \begin{align} \bigg\lVert T_{\mu} W_g \bigg (W_v-\sum_{e\in vE^1} T_eT^*_e\bigg )W^*_h T^*_{\beta} e_{\lambda k}\bigg \rVert = 0 \quad \textrm{whenever} \ \lvert \lambda\rvert> \lvert \,\beta\rvert. \end{align} $$
Fix a finite linear combination
$a_0=\sum _{a_{\mu , g, h, \beta }}T_{\mu } W_g(W_v-\sum _{e\in vE^1}T_eT^*_e)W_h^*T_{\beta }^*$
. Let
$N=\mathrm { \max }\{\lvert \,\beta \rvert \mid a_{\mu ,g,h,\beta }\neq 0\}$
. Then (3.2) implies that
$\lVert a_0 e_{\lambda k}\rVert = 0$
whenever
$\lvert \lambda \rvert> N$
.
Finally, fix
$a \in I$
, and
$\varepsilon>0$
. A routine argument gives
$$ \begin{align*} I=\overline{\mathrm{span}}\bigg \{t_{\mu} w_g \bigg (w_v-\sum_{e\in vE^1}t_et_e^*\bigg )w_h^*t^*_{\beta} \mid \mu, \beta \in E^*, g,h \in \mathcal{G}, v \in E^0\bigg \}. \end{align*} $$
So there exists
$$ \begin{align*} a_0\in \mathrm{span}\bigg \{T_{\mu} W_g \bigg (W_v-\sum_{e\in vE^1}T_eT^*_e\bigg ) W^*_hT^*_{\beta} \mid \mu, \beta \in E^*, g, h \in \mathcal{G}, v \in E^0 \bigg \}, \end{align*} $$
such that
$\lVert \pi (a)-a_0\rVert <\varepsilon .$
Take N as above and fix
$n \geq N$
. Then,
$$ \begin{align*} \lVert \pi(a)|_{\overline{\mathrm{span}}\{e_{\lambda k} \mid \lambda\in E^n, k\in \mathcal{G}^{s(\lambda)}\}}\lVert \leq \rVert \pi(a)-a_0\lVert + \rVert a_0|_{\overline{\mathrm{span}}\{e_{\lambda k} \mid \lambda\in E^n, k\in \mathcal{G}^{s(\lambda)}\}}\rVert <\varepsilon.\\[-34pt] \end{align*} $$
The following proposition will be used in describing our fixed-point algebra in the next section. Let G be a discrete group and let
$\mathcal {H}=l^2(G)=\overline {\mathrm {span}}\{\delta _g \mid g \in G\}$
. For
$g \in G$
, define
$\lambda _g \in \mathcal {U}(l^2(G))$
by
$\lambda _g(\delta _h)=\delta _{gh}$
for all
$h \in G$
. We get a representation
$\lambda : C^*(G)\rightarrow B(\mathcal {H})$
such that
$\lambda (u_g)=\lambda _g$
for all
$g\in G$
; we call this the regular representation. If G is amenable, then the representation
$\lambda $
is faithful. Since our groupoid is an amenable (discrete) groupoid, its (discrete) isotropy groups are also amenable.
Proposition 3.6. Let
$(\mathcal {G},E)$
be a self-similar groupoid. Let
$\{s_e, u_g\}$
be the universal Cuntz–Krieger
$(\mathcal {G},E)$
-family in
$C^*(\mathcal {G},E)$
. Then each
$s_e$
and each
$u_g$
is nonzero. Fix
$v \in E^0$
. The universal property of
$C^*(\mathcal {G}^v_v)$
gives a homomorphism
$\pi _u: C^*(\mathcal {G}^v_v) \rightarrow C^*(\mathcal {G},E)$
such that
$\pi _u(\delta _h)=u_h$
for all
$h \in \mathcal {G}^v_v$
. Suppose that for each
$k \in \mathbb {N}$
, there exists
$\lambda \in vE^k$
such that the map
$g\mapsto (g\cdot \lambda )\varphi (g,\lambda )$
is injective. Then
$\pi _u$
is injective.
Proof. By the universal property, it suffices to construct a Cuntz–Krieger
$(\mathcal {G},E)$
-family
$\{S_e, U_g\}$
consisting of nonzero partial isometries. If
$\{S_e, U_g\}$
is a Cuntz–Krieger
$(\mathcal {G}, E)$
-family and each
$U_v\neq 0$
, then
$S_e \neq 0$
for all
$ e\in E^1$
and
$U_g \neq 0$
for all
$g \in \mathcal {G}$
, because
$U_{s(e)}=S_e^*S_e$
and
$U_{s(g)}=U_g^*U_g$
. So, it suffices to construct a
$(\mathcal {G},E)$
-family with
$U_v \neq 0$
for all
$v \in E^0$
.
Let
$\{T_e, W_g\}$
be the Toeplitz
$(\mathcal {G},E)$
-family of Example 3.2. For
$v \in E^0$
, we have
$W_v \cdot e_{\mu s(\,\mu )}=e_{\mu s(\,\mu )}$
for all
$\mu \in vE^*$
. So,
$$ \begin{align*} \lVert W_v|_{\overline{\mathrm{span}}\{e_{\lambda g} \mid \lambda\in vE^n, g\in \mathcal{G}^{s(\lambda)}\}}\rVert = 1. \end{align*} $$
Thus, Lemma 3.5 gives
$W_v\not \in I$
. Therefore,
$S_e:=T_e + I \ \mathrm {and} \ U_g:=W_g + I$
is a
$(\mathcal {G}, E)$
-family with each
$U_v \neq 0$
.
Now fix
$v \in E^0$
. Let
$\pi _W: C^*(\mathcal {G}^v_v) \rightarrow B(l^2(E^* \ast \mathcal {G}))$
be the homomorphism such that
$\pi _W(\delta _h)=W_h$
. Fix
$k\in \mathbb {N}$
. Choose
$\lambda \in vE^*$
such that the map
$h\mapsto (h\cdot \lambda )\varphi (h,\lambda )$
is injective. Let
$\mathcal {H}_{\lambda }:=\overline {\mathrm {span}} \{e_{(h\cdot \lambda ) \varphi (h,\lambda )} \mid h \in \mathcal {G}^v_v\} \subseteq l^2(E^*\ast \mathcal {G})$
. By construction,
$\mathcal {H}_{\lambda }$
is invariant for
$\pi _W$
.
Since the map
$g\mapsto (g\cdot \lambda )\varphi (g,\lambda )$
is injective, there is an inner-product preserving map
$\phi _{\lambda } : l^2(\mathcal {G}^v_v) \rightarrow \mathcal {H}_{\lambda }$
that maps the element
$e_g$
of the orthonormal basis of
$l^2(\mathcal {G}^v_v)$
to the element
$e_{(g \cdot \lambda ) \varphi (g, \lambda )}$
of the orthonormal basis of
$\mathcal {H}_{\lambda }$
. For
$h \in \mathcal {G}^v_v$
, define
$V^{\lambda }_h \in \mathcal {U}(l^2(\mathcal {G}^v_v))$
by
$V^{\lambda }_h =\phi ^*_{\lambda } W_h \phi _{\lambda } $
. We get
$$ \begin{align*} V^{\lambda}_h e_g = \phi^*_{\lambda} W_h \phi_{\lambda} e_g = \phi^*_{\lambda} W_h e_{(g \cdot \lambda)(\varphi(g, \lambda))} &=\phi^*_{\lambda} e_{(h \cdot (g \cdot \lambda))(\varphi(h, g \cdot \lambda)\varphi(g, \lambda))} \nonumber \\ &=\phi^*_{\lambda} e_{((hg) \cdot \lambda)(\varphi(hg, \lambda))}= e_{hg}. \nonumber \end{align*} $$
Hence,
$\{V^{\lambda }_h \mid h \in \mathcal {G}^v_v\}\subseteq B(l^2(\mathcal {G}^v_v))$
is the regular representation of
$\mathcal {G}^v_v$
and induces a faithful representation of
$C^*(\mathcal {G}^v_v)$
. Hence, the reduction of
$\pi _W$
to
$\mathcal {H}_{\lambda }$
is injective, so its reduction to
$l^2(E^k \ast \mathcal {G})$
is injective. Since k was arbitrary, the reduction of
$\pi _W$
to
$l^2(E^k \ast \mathcal {G})$
is injective, and hence isometric for all k.
Now, fix
$a \in C^*(\mathcal {G}^v_v)\setminus \{0\}$
. Then for all k,
$$ \begin{align*} \lVert \pi_W(a)|_{\overline{\mathrm{span}}\{e_{\mu g} \mid \mu \in E^k, g \in \mathcal{G}^{s(\,\mu)}\}}\rVert = \lVert a\rVert \neq 0. \end{align*} $$
Thus, Lemma 3.5 implies
$a \not \in I$
. We have
$\pi _u(a)=a+I\neq 0$
. Therefore, the homomorphism
$\pi _u$
is injective.
To see that our faithfulness condition is strictly weaker than that of [Reference Laca, Raeburn, Rammage and Whittaker5], we provide the following example.
Example 3.7. Let E be the graph with one vertex and n edges
$e_0, \ldots , e_{n-1}$
and let
$\mathcal {G}=\mathbb {Z}$
. Define an action of
$\mathcal {G}$
on E by
$m \cdot e_i=e_{i+m}$
where addition is mod n, and define
$\varphi (m, e_i)=m$
for all m. Then
$\mathcal {G}$
does not act faithfully in the sense of [Reference Laca, Raeburn, Rammage and Whittaker5], because
$n \cdot e_i=e_i$
for all i. However, the map
$(m, \lambda ) \mapsto (m \cdot \lambda , \varphi (m, \lambda ))$
is injective for each
$\lambda $
because
$\varphi (m, \lambda )=m$
and then
$\lambda =\varphi (m,\lambda )^{-1} \cdot (m \cdot \lambda )$
. It is routine to see using universal properties that
$C^*(\mathcal {G}, E)= \mathcal {O}_n \rtimes \mathbb {Z}$
.
4 The gauge action and the core
Let
$\{s_e, u_g\}$
be the universal Cuntz–Krieger
$(\mathcal {G},E)$
-family in
$C^*(\mathcal {G}, E)$
. Then for
$z \in \mathbb {T}$
, the family
$\{zs_e, u_g\}$
is also a Cuntz–Krieger
$(\mathcal {G},E)$
-family. So, the universal property gives a homomorphism
$\gamma _z:C^*(\mathcal {G}, E) \rightarrow C^*(\mathcal {G}, E)$
such that
$\gamma _z(s_e)=zs_e$
and
$\gamma _z(u_g)=u_g$
for all
$e,g$
. Since
$\gamma _1$
agrees with the identity and
$\gamma _z \circ \gamma _w$
agrees with
$\gamma _{zw}$
on generators,
$z \mapsto \gamma _z$
is an action. A standard
$\varepsilon /3$
argument shows that it is a strongly continuous action, which we call the gauge action on
$C^*(\mathcal {G}, E)$
. The fixed-point algebra of
$\gamma $
is the
$*$
-subalgebra
$$ \begin{align*}C^*(\mathcal{G}, E)^{\gamma}:=\{a\in C^*(\mathcal{G},E) \mid \gamma_z(a)=a \ \mathrm{for \ all} \ z\in \mathbb{T}\}\end{align*} $$
of
$C^*(\mathcal {G}, E)$
. The following corollary describes
$C^*(\mathcal {G}, E)^{\gamma }$
concretely.
Corollary 4.1. Let
$(\mathcal {G},E)$
be a self-similar groupoid and let
$\Phi : C^*(\mathcal {G}, E) \rightarrow C^*(\mathcal {G},E)^{\gamma }$
be the conditional expectation,
$\Phi (a)=\int _{\mathbb {T}} \gamma _z(a)\,dz$
. Then,
$$ \begin{align*} \Phi (s_{\mu} u_g s^*_{\beta})=\delta_{\lvert \,\mu \rvert, \lvert \,\beta \rvert}s_{\mu} u_g s_{\beta}^* \quad\mbox{for } \mu, \beta \in E^* \mbox{ and } g\in \mathcal{G}_{s(\,\beta)}^{s(\,\mu)}. \end{align*} $$
Further,
$C^*(\mathcal {G}, E)^{\gamma }=\overline {\mathrm {span}}\{s_{\mu } u_g s^*_{\beta } \mid s(\,\mu )=g\cdot s(\,\beta ) \ \mathrm {and} \ \lvert \,\mu \rvert =\lvert \,\beta \rvert \}. $
Proof. We have
$\gamma _z(s_{\mu } u_g s^*_{\beta })=z^{ \lvert \,\mu \rvert -\lvert \,\beta \rvert }s_{\mu } u_g s^*_{\beta }$
, so
$\Phi (s_{\mu } u_g s^*_{\beta }) = \delta _{\lvert \,\mu \rvert , \lvert \,\beta \rvert } s_{\mu } u_g s^*_{\beta }$
. Moreover,
$\Phi (C^*(\mathcal {G},E))=\overline {\mathrm {span}}\{s_{\mu } u_g s^*_{\beta } \mid s(\,\mu )=g\cdot s(\,\beta ) \ \mathrm {and} \ \lvert \,\mu \rvert =\lvert \,\beta \rvert \}.$
Proposition 3.2 of [Reference Raeburn6] shows that
$\Phi (C^*(\mathcal {G},E))=C^*(\mathcal {G},E)^{\gamma }$
.
Let
$(\mathcal {G},E)$
be a self-similar groupoid and let
$\{S_e, U_g\}$
be a Cuntz–Krieger
$(\mathcal {G}, E)$
-family. For
$k\in \mathbb {N}$
, we define
$$ \begin{align*} \mathcal{F}_k(S, U):=\overline{\mathrm{span}}\{S_{\mu} U_g S^*_{\beta} \mid \mu, \beta \in E^k, g \in \mathcal{G}^{s(\,\mu)}_{s(\,\beta)}, s(\,\mu)=g\cdot s(\,\beta) \}. \end{align*} $$
We define a relation
$\sim $
on
$E^0$
by
$v\sim w$
if and only if
$\mathcal {G}^v_w\neq \emptyset $
. Then
$\sim $
is an equivalence relation. For
$\xi \in E^0/{\sim }$
, define
$$ \begin{align*} \mathcal{F}_k (S, U, \xi):=\overline{\mathrm{span}}\{S_{\mu} U_g S^*_{\beta} \mid \mu, \beta \in E^k,g \in \mathcal{G}^{s(\,\mu)}_{s(\,\beta)}, s(\,\mu)=g\cdot s(\,\beta)\in \xi\}. \end{align*} $$
When
$\{S_e, U_g\}$
is the universal family
$\{s_e, u_g\}$
in
$C^*(\mathcal {G}, E)$
, we write
$\mathcal {F}_k:=\mathcal {F}_k(s,u)$
and
$\mathcal {F}_k(\xi ):=\mathcal {F}_k(s, u, \xi )$
.
Notation 4.2. For the next few results, fix a self-similar groupoid
$(\mathcal {G}, E)$
, an element
$\xi \in E^0/\sim $
, a vertex
$v \in \xi $
and for each
$u \in \xi $
, an element
$g_u \in \mathcal {G}^u_v$
(take
$g_v =v$
). We call
$\{g_u\mid u\in \xi \}$
a spanning tree for
$\mathcal {G}|_{\xi }$
. We denote
$E^k \xi :=\{\,\mu \in E^k \mid \ s(\,\mu ) \in \xi \}$
.
Proposition 4.3. With Notation 4.2, let
$\{S_e, U_g\}$
be a Cuntz–Krieger
$(\mathcal {G}, E)$
-family. For
$h \in \mathcal {G}^v_v$
and
$\mu \in E^*$
, define
$V_{h,\mu }:=S_{\mu } U_{g_{s(\,\mu )}}U_h(S_{\mu } U_{g_{s(\,\mu )}})^*$
. For each
$k \in \mathbb {N}$
, the series
$\sum _{\mu \in E^k \xi } V_{h,\mu }$
converges strictly to a partial unitary
$\overline {V}_h$
in
$\mathcal {M}C^*(S, U)$
and
$\overline {V}_h \mathcal {F}_k(S, U, \xi )\subseteq \mathcal {F}_k(S, U, \xi )$
.
Proof. Fix
$h \in \mathcal {G}^v_v$
. For
$\mu \in E^k\xi $
,
$$ \begin{align} V_{h,\mu}V_{h,\mu}^* = S_{\mu} S^*_{\mu}= V_{h,\mu}^*V_{h,\mu}. \end{align} $$
For
$\mu \neq \beta \in E^k\xi $
,
$S_{\mu }S^*_{\mu }S_{\beta }S^*_{\beta }=0$
. So,
$S^*_{\mu }S_{\beta }=0$
. Therefore, for
$F\subseteq E^k\xi $
finite,
$$ \begin{align*} \bigg(\sum_{\mu \in F} V_{h,\mu}\bigg)\bigg(\sum_{\beta \in F} V_{h,\beta}\bigg)^*=\sum_{\mu \in F} V_{h,\mu} V^*_{h,\mu} =\sum_{\mu \in F}S_{\mu}S^*_{\mu}. \end{align*} $$
Now, fix
$a\in C^*(S, U)$
. Then
$a= \lim _K \sum _{v\in K}P_v a$
, where K ranges over all finite subsets of
$E^0$
. Let
$P_K =\sum _{v \in K}P_v$
. Fix
$\varepsilon>0$
. There exists a finite set
$K' \subseteq E^0$
such that
$\lVert P_Ka-a \rVert <\varepsilon /2$
for all finite
$K \supseteq K'$
.
Let
$F \subseteq E^k \xi $
be the finite set
$F=KE^k\xi $
. For
$F', F" \supseteq F$
,
$$ \begin{align*} \bigg \lVert \sum_{\mu \in F'}S_{\mu} S^{*}_{\mu} a- \sum_{\beta \in F"}S_{\beta} S^{*}_{\beta} a\bigg \rVert & \leq \bigg\lVert \sum_{\mu \in F' \setminus F"}S_{\mu} S^*_{\mu} a \bigg\rVert + \bigg\lVert\sum_{\beta \in F"\setminus F'}S_{\beta} S^*_{\beta} a \bigg\rVert \\ \nonumber & \leq \lVert (1-P_K)a\rVert + \lVert(1-P_K)a\rVert < \varepsilon. \nonumber \end{align*} $$
So,
$(\sum _{\mu \in F}S_{\mu } S^*_{\mu } a )_{F \subseteq E^k \xi }$
is Cauchy and hence converges. Thus,
$\sum _{\mu \in E^k \xi } S_{\mu } S^*_{\mu }$
converges strictly to a projection
$P_{\xi } \in \mathcal {M}C^*(S, U)$
. Equation (4.1) shows that
$\sum _{\mu \in F} V^*_{h,\mu }V_{h,\mu }$
also converges strictly to
$P_{\xi }$
. Therefore,
$\sum _{\mu \in E^k\xi } V_{h,\mu }$
converges strictly to a unitary
$\overline {V}_h \in P_{\xi }\mathcal {M}C^*(S, U)P_{\xi }$
.
Now fix a spanning element
$S_{\alpha } U_l S^*_{\beta }$
of
$\mathcal {F}_k(S, U, \xi )$
. For each
$\mu \in E^k\xi $
, we obtain
$$ \begin{align*} V_{h, \mu}S_{\alpha} U_l S^*_{\beta} &=\delta_{\mu, \alpha}S_{\mu}U_{g'l} S^*_{\beta} \quad\text{for some} \ g'\in\mathcal{G}^{s(\,\mu)}_{s(\,\mu)}, \end{align*} $$
which implies that
$$ \begin{align*} \overline{V}_h S_{\alpha} U_l S^*_{\beta} =\sum_{\mu \in E^k \xi} \delta_{\mu, \alpha} S_{\mu} U_{g'l} S^*_{\beta}& = S_{\alpha} U_{g'l} S^*_{\beta} \quad \text{for some} \ g'\in\mathcal{G}^{s(\alpha)}_{s(\alpha)},\\ & \in \mathcal{F}_k(S, U, \xi). \end{align*} $$
Hence,
$\overline {V}_h \mathcal {F}_k(S, U, \xi )\subseteq \mathcal {F}_k(S, U, \xi )$
.
Proposition 4.4. Fix
$\xi \in E^0/{\sim } \ \text {and} \ v \in \xi $
. Let
$\{S_e, U_g\}$
be a Cuntz–Krieger
$(\mathcal {G}, E)$
-family. For
$h \in \mathcal {G}^v_v$
, let
$\overline {V}_h$
be as in Proposition 4.3. Then there is a homomorphism
$\pi _{\overline {V}}: C^*(\mathcal {G}^v_v)\rightarrow \mathcal {M}C^*(S, U)$
that maps
$\delta _h$
to
$\overline {V}_h$
.
Proof. Let
$h, k \in \mathcal {G}^v_v$
. Routine calculations show that for
$k\geq 1$
and
$\mu \in E^k\xi $
, we have
$V_{h, \mu }V_{k, \mu }=V_{hk, \mu } $
and
$V^*_{h, \mu } = S_{\mu } U_{g_{s(\,\mu )}}U_{h^{-1}}(S_{\mu } U_{g_{s(\,\mu )}})^* = V_{h^{-1}, \mu }. $
This implies that for any finite
$F \subseteq E^k\xi $
,
$$ \begin{align*} \sum_{\mu \in F} V_{h,\mu} \sum_{\mu \in F} V_{k, \mu}=\sum_{\mu \in F} V_{hk,\mu}. \end{align*} $$
Thus,
$\overline {V}_h \overline {V}_k = \sum _{\mu \in E^k \xi }V_{h,\mu }\sum _{\mu \in E^k \xi }V_{k,\mu }=\overline {V}_{hk}$
and
$$ \begin{align*} \overline{V}_h^*=\sum_{\mu \in E^k \xi} V^*_{h,\mu}=\sum_{\mu \in E^k \xi}V_{h^{-1},\mu}=\overline{V}_{h^{-1}}. \end{align*} $$
So, the universal property of
$C^*(\mathcal {G}^v_v)$
gives a homomorphism
$$ \begin{align*} \pi_{\overline{V}}: C^*(\mathcal{G}^v_v)\rightarrow \mathcal{M}(C^*(S, U)\quad \mathrm{such \ that} \ \pi_{\overline{V}}(\delta_h)=\overline{V}_h.\\[-34pt] \end{align*} $$
Proposition 4.5. Fix
$\xi \in E^0/{\sim }$
. Let
$\{S_e, U_g\}$
be a Cuntz–Krieger
$(\mathcal {G}, E)$
-family. For
$\mu , \beta \in E^k \xi $
, let
$e_{\mu }\otimes e_{\beta }^*$
denote the rank-one operator on the Hilbert space
$l^2(\{E^k \xi \})$
, and let
$\Theta _{\mu , \beta }:=S_{\mu } U_{g_{s(\,\mu )}}U^*_{g_{s(\,\beta )}}S^*_{\beta } \in C^*(S, U)$
. Then there is an injective homomorphism
$$ \begin{align*}\theta: \mathcal{K}(l^2(\{E^k \xi\})) \rightarrow \overline{\mathrm{span}}\{\Theta_{\mu, \beta} \mid \mu, \beta \in E^k \xi\}\end{align*} $$
such that
$\theta (e_{\mu }\otimes e_{\beta }^*)=\Theta _{\mu , \beta }$
.
Proof. We claim that the elements
$\Theta _{\mu , \beta }$
are matrix units. Let
$\mu , \beta , \alpha , \rho \in E^k \xi $
. Then,
$$ \begin{align*} \Theta_{\mu, \beta}\Theta_{\alpha, \rho}&= (S_{\mu} U_{g_{s(\,\mu)}}U^*_{g_{s(\,\beta)}}S^*_{\beta})(S_{\alpha} U_{g_{s(\alpha)}}U^*_{g_{s(\rho)}}S^*_{\rho})\\ &= \begin{cases} S_{\mu} U_{g_{s(\,\mu)}} U^*_{g_{s(\rho)}} S_{\rho}^* & \mathrm{if} \ \beta=\alpha, \\ 0 & \mathrm{otherwise,} \end{cases} \end{align*} $$
and
$(\Theta _{\mu , \beta })^*=S_{\beta } U_{g_{s(\,\beta )}}U^*_{g_{s(\,\mu )}}S^*_{\mu }=\Theta _{\beta , \mu }$
. Hence,
$\{\Theta _{\mu , \beta } \mid \mu , \beta \in E^k \xi \}$
is a family of matrix units. Since
$$ \begin{align*} \lVert S_{\mu} U_{g_{s(\,\mu)}}U^*_{g_{s(\,\beta)}}S^*_{\beta}\rVert^2 = \lVert S_{\beta} U_{g_{s(\,\beta)}}U^*_{g_{s(\,\mu)}}S^*_{\mu} S_{\mu} U_{g_{s(\,\mu)}}U^*_{g_{s(\,\beta)}}S^*_{\beta}\rVert =\lVert P_v\rVert = 1, \end{align*} $$
by Lemma 3.3, these are nonzero matrix units. Hence, by Corollary A.9 of [Reference Raeburn6], we get the injective homomorphism
$\theta $
as claimed.
Proposition 4.6. Fix
$\xi \in E^0/{\sim }$
and
$v \in \xi $
. Let
$\{S_e, U_g\}$
be a Cuntz–Krieger
$(\mathcal {G}, E)$
-family. Let
$\pi _{\overline {V}}$
and
$\theta $
be as in Propositions 4.4 and 4.5, respectively. Then, there exists a homomorphism
$$ \begin{align*} \theta \otimes \pi_{\overline{V}}: \mathcal{K}(l^2(\{E^k \xi \})) \otimes C^*(\mathcal{G}^v_v) \rightarrow \mathcal{F}_k(S, U, \xi) \end{align*} $$
such that
$$ \begin{align*}\theta \otimes \pi_{\overline{V}}((e_{\mu}\otimes e_{\beta}^*) \otimes \delta_h)=\theta (e_{\mu}\otimes e_{\beta}^*) \pi_{\overline{V}}(\delta_h)=\pi_{\overline{V}}(\delta_h)\theta (e_{\mu}\otimes e_{\beta}^*)\end{align*} $$
for all
$e_{\mu }\otimes e_{\beta }^* \in \mathcal {K}(l^2(\{E^k \xi \}))$
and for all
$\delta _h \in C^*(\mathcal {G}^v_v)$
.
Proof. We have
$\theta (e_{\mu }\otimes e_{\beta }^*)=\Theta _{\mu , \beta }$
and
$\pi _{\overline {V}}(\delta _h)=\overline {V}_h$
in
$\mathcal {F}_k(\xi )$
for all
$e_{\mu }\otimes e_{\beta }^* \in \mathcal {K}(l^2(\{E^k \xi \}))$
and for all
$\delta _h \in C^*(\mathcal {G}^v_v)$
. Then,
$$ \begin{align*} \Theta_{\mu, \beta}\overline{V}_h &= S_{\mu} U_{g_{s(\,\mu)}}U^*_{g_{s(\,\beta)}}S^*_{\beta} \overline{V}_h \\ &= S_{\mu} U_{g_{s(\,\mu)}}U^*_{g_{s(\,\beta)}}S^*_{\beta} \sum_{\gamma \in E^k \xi}S_\gamma U_{g_{s(\gamma)}}U_h U^*_{g_{s(\gamma)}}S^*_\gamma \\ &= \sum_{\gamma \in E^k \xi }S_{\mu} U_{g_{s(\,\mu)}}U^*_{g_{s(\,\beta)}}S^*_{\beta} S_\gamma U_{g_{s(\gamma)}}U_h U^*_{g_{s(\gamma)}}S^*_\gamma \\ &= S_{\mu} U_{g_{s(\,\mu)}}U_h U^*_{g_{s(\,\beta)}}S^*_{\beta}. \end{align*} $$
A similar calculation gives
$\overline {V}_h \Theta _{\mu , \beta }=S_{\mu } U_{g_{s(\,\mu )}}U_h U^*_{g_{s(\,\beta )}}S^*_{\beta }$
. Hence,
$\Theta _{\mu , \beta } \overline {V}_h=\overline {V}_h \Theta _{\mu , \beta }$
. We claim that
$$ \begin{align*} \overline{\mathrm{span}}\{\Theta_{\mu, \beta}\overline{V}_h \mid \mu, \beta \in E^k\xi, h \in \mathcal{G}^v_v\} =\mathcal{F}_k(S, U, \xi). \end{align*} $$
Let
$\mu , \beta , \alpha , \rho \in E^k\xi $
and
$h_1, h_2 \in \mathcal {G}^v_v$
. Then,
$$ \begin{align*} \Theta_{\mu, \beta}\overline{V}_{h_1} \Theta_{\alpha, \rho}\overline{V}_{h_2}=\Theta_{\mu, \beta}\Theta_{\alpha, \rho}\overline{V}_{h_1}\overline{V}_{h_2}= \delta_{\beta, \alpha}\Theta_{\mu, \rho}\overline{V}_{h_1h_2} \end{align*} $$
and
$(\Theta _{\mu , \beta }\overline {V}_h)^*=\overline {V}_{h^{-1}}\Theta _{\beta , \mu }=\Theta _{\beta , \mu }\overline {V}_{h^{-1}}$
. So,
$\overline {\mathrm { span}}\{\Theta _{\mu , \beta }\overline {V}_h \mid \mu , \beta \in E^k\xi , h \in \mathcal {G}^v_v\}$
is a
$C^*$
-subalgebra of
$\mathcal {F}_k(S, U, \xi )$
. Moreover, it contains the generators of
$\mathcal {F}_k(S, U, \xi )$
, so it is all of
$\mathcal {F}_k(S, U, \xi )$
.
Now the universal property of the (maximal) tensor product gives the desired homomorphism
$\theta \otimes \pi _{\overline {V}}$
.
We show next the homomorphism
$\theta \otimes \pi _{\overline {V}}$
is faithful. To show this, we need to verify that both
$\theta $
and
$\pi _{\overline {V}}$
are injective. From Proposition 4.5, we already know that
$\theta $
is injective, so it suffices to show that
$\pi _{\overline {V}}$
is injective as well.
Lemma 4.7. Fix
$\xi \in E^0/{\sim }$
and
$v \in \xi $
. Let
$\{S_e, U_g\}$
be a Cuntz–Krieger
$(\mathcal {G},E)$
-family. Suppose that the homomorphism
$\pi _U: C^*(\mathcal {G}^v_v) \rightarrow C^*(S, U)$
that maps
$\delta _h$
to
$U_h$
is injective. Fix
$k \in \mathbb {N}$
and
$v\in E^0$
. Let
$\overline {V}_h$
be as in Proposition 4.3. Then, the homomorphism
$\pi _{\overline {V}}^{(v,k)}: C^*(\mathcal {G}^v_v) \rightarrow \mathcal {F}_k(S, U, \xi )$
that maps
$\delta _h$
to
$\overline {V}_h$
is injective.
Proof. Fix
$\lambda \in E^k \xi $
and let
$Y_{\lambda }=S_{\lambda } U_{g_{s(\lambda )}}$
. Then,
$$ \begin{align*}Y^*_{\lambda} \overline{V}_h Y_{\lambda} = \sum_{\mu \in E^k \xi} U^*_{g_{s(\lambda)}}S^*_{\lambda} S_{\mu} U_{g_{s(\,\mu)}}U_h U^*_{g_{s(\,\mu)}}S^*_{\mu} S_{\lambda} U_{g_{s(\lambda)}} = U_h. \end{align*} $$
Define
$\mathrm {Ad}_{Y_{\lambda }}: \mathcal {F}_k(S, U, \xi ) \rightarrow C^*(S, U)$
by
$\mathrm {Ad}_{Y_{\lambda }}(a)=Y^*_{\lambda } a Y_{\lambda }$
. By linearity and continuity,
$\mathrm {Ad}_{Y_{\lambda }} \circ \pi _{\overline {V}}^{(v,k)}=\pi _U$
. Hence,
$\mathrm {Ad}_{Y_{\lambda }} \circ \pi _{\overline {V}}^{(v,k)}$
is injective, so
$\pi _{\overline {V}}^{(v,k)}$
is also injective.
Since
$\mathcal {K}(l^2(\{E^k \xi \}))$
is simple and nuclear, Proposition 4.5 and Lemma 4.7 show that if
$\pi _U$
is injective on
$C^*(\mathcal {G}^v_v)$
, then the homomorphism of Proposition 4.6 is an isomorphism. So,
$$ \begin{align} \mathcal{F}_k(\xi)\cong \mathcal{K}(l^2(\{E^k \xi \}))\otimes C^*(\mathcal{G}^v_v). \end{align} $$
Moreover, we obtain the following corollary. Recall that
$$ \begin{align*} \mathcal{F}_k=\overline{\mathrm{span}}\{s_{\mu} u_g s^*_{\beta} \mid s(\,\mu)= g \cdot s(\,\beta), \,\mathrm{and}\, \lvert \,\mu \rvert=\lvert \,\beta \rvert=k \}. \end{align*} $$
Corollary 4.8. Let
$(\mathcal {G},E)$
be a self-similar groupoid. Fix
$\xi \in E^0/{\sim }$
and
$v \in \xi $
. Suppose that for each
$k \in \mathbb {N}$
, there exists
$\lambda \in vE^k$
such that the map
$g\mapsto (g\cdot \lambda )\varphi (g,\lambda )$
is injective. Then,
$$ \begin{align*} \mathcal{F}_k \cong \bigoplus_{\xi \in E^0/\sim}\mathcal{F}_k(\xi)\cong\bigoplus_{\xi \in E^0/\sim}\mathcal{K}(l^2(\{E^k \xi \}))\otimes C^*(\mathcal{G}^v_v). \end{align*} $$
Proof. For
$\mu , \beta , \alpha , \rho \in E^k$
with
$s(\,\mu )=g \cdot s(\,\beta ) \in \xi _1$
and
$s(\alpha )= h \cdot s(\rho ) \in \xi _2$
, the equation of Lemma 3.3 gives
$$ \begin{align*}(s_{\mu} u_g s^*_{\beta})(s_{\alpha} u_h s^*_{\rho})=\begin{cases} s_{\mu} u_{g}u_{h} s^{*}_{\rho} & \mathrm{if} \ \beta=\alpha, \\ 0 & \mathrm{otherwise}. \end{cases} \end{align*} $$
Hence,
$\mathcal {F}_k(\xi _1) \mathcal {F}_k(\xi _2)=0$
, when
$\xi _1 \neq \xi _2$
, so Corollary A.11 of [Reference Raeburn6] combined with (4.2) gives an isomorphism of
$\bigoplus _{\xi \in E^0/\sim } \mathcal {F}_k(\xi )$
onto
$\mathcal {F}_k$
. Equation (4.2) gives the second isomorphism.
Corollary 4.9. Let
$(\mathcal {G},E)$
be a self-similar groupoid. Then,
$$ \begin{align*}C^*(\mathcal{G}, E)^{\gamma} =\overline{\bigcup_k \mathcal{F}_k}=\overline{\bigcup_k \bigg(\bigoplus_{\xi \in E^0/\sim}\mathcal{F}_k(\xi)\bigg)}.\end{align*} $$
Proof. For any k, we claim that
$\mathcal {F}_k \subset \mathcal {F}_{k+1}$
. Fix
$\mu , \beta \in E^k,g \in \mathcal {G}$
with
$s(\,\mu )=g \cdot s(\,\beta )$
. We have
$$ \begin{align*}s_{\mu} u_g s^*_{\beta} = s_{\mu} u_g u_{s(g)}s^*_{\beta} = \sum_{e \in s(g)E^1}s_{\mu} u_g s_e s^*_e s^*_{\beta} = \sum_{e \in s(g)E^1}s_{\mu (g\cdot e)}u_{\varphi(g,e)}s^*_{\beta e} \in \mathcal{F}_{k+1}.\end{align*} $$
Hence,
$\mathcal {F}_k \subset \mathcal {F}_{k+1}$
for all k. By Corollary 4.1, the claim follows.
Lemma 4.10. Let
$(\mathcal {G}, E)$
be a self-similar groupoid. Suppose that
$\{T_e,W_g\}$
is a
$(\mathcal {G},E)$
-family in a
$C^*$
-algebra B. Let
$$ \begin{align*}\pi_{T,W} : C^*(\mathcal{G},E)\rightarrow C^*(T,W)\end{align*} $$
be the homomorphism induced by the universal property. Suppose that for each
$v \in E^0$
, the homomorphism
$\pi _{v,W}: C^*(\mathcal {G}^v_v) \rightarrow C^*(T,W)$
such that
$\pi _{v,W}(\delta _g)=W_g$
for all g is injective. Then,
$\pi _{T,W}$
is isometric on
$C^*(\mathcal {G},E)^{\gamma }$
.
Proof. Fix
$\xi \in E^0/{\sim }$
and
$ v \in \xi $
. Choose elements
$g_w\in \mathcal {G}^w_v$
for
$w\in \xi $
with
$g_v=v$
. For
$h \in \mathcal {G}^v_v$
and
$k\in \mathbb {N}$
, let
$\overline {W}_h=\sum _{\mu \in E^k \xi } T_{\mu } W_{g_{s(\,\mu )}}W_h W^*_{g_{s(\,\mu )}}T^*_{\mu }$
as in Proposition 4.3. Lemma 4.7 shows that the homomorphism
$\pi _{\overline {W}}: C^*(\mathcal {G}^v_v) \rightarrow \mathcal {M}C^*(T,W)$
is injective.
Let
$\theta \otimes \pi _{\overline {W}}$
be as in Proposition 4.6. Since
$\mathcal {K}(l^2(E^k\xi ))$
is simple and nuclear, and since each
$T_{\mu } T^*_{\beta } \neq 0$
, the map
$\pi _{T,W} \circ (\theta \otimes \pi _{\overline {u}})=\theta \otimes \pi _{\overline {W}}$
is injective on each
$\mathcal {F}_k(\xi )$
. Therefore, it is also injective on
$\mathcal {F}_{k}=\bigoplus _{\xi \in E^0/\sim }\mathcal {F}_{k}(\xi )$
. Because every injective
$C^*$
-algebra homomorphism is isometric,
$\pi _{T,W}$
is isometric on
$\mathcal {F}_{k}$
. Hence,
$\pi _{T, W}$
is isometric on
$\bigcup _k\mathcal {F}_{k}$
and hence on
$\overline {U_k\mathcal {F}_k}=C^*(\mathcal {G},E)^{\gamma }$
.
5 The gauge-invariant uniqueness theorem
Theorem 5.1. Let
$(\mathcal {G}, E)$
be a self-similar groupoid. Suppose that
$(T,W)$
is a
$(\mathcal {G}, E)$
-family in a
$C^*$
-algebra B. The universal property of
$C^*(\mathcal {G}, E)$
gives a homomorphism
$$ \begin{align*} \pi_{T,W} : C^*(\mathcal{G},E)\rightarrow C^*(T,W). \end{align*} $$
If there is a continuous action
$\eta : \mathbb {T} \rightarrow \mathrm {Aut} B$
such that
$\eta _z(T_e)=zT_e$
and
$\eta _z(W_g)=W_g$
for all
$e \in E^1$
and
$g \in G$
, and if the homomorphism
$\pi _{v,W}$
is injective for each
$v \in E^0$
, then
$\pi _{T,W}$
is an isomorphism of
$C^*(\mathcal {G},E)$
onto
$C^*(T,W)$
.
Proof. Let
$\Phi : C^*(\mathcal {G}, E) \rightarrow C^*(\mathcal {G}, E)^{\gamma }$
be the faithful conditional expectation of Corollary 4.1. Let
$\Psi :C^*(T,W)\rightarrow C^*(T,W)^{\eta }$
be the corresponding expectation obtained from
$\eta $
. Since
$\eta _z \circ \pi _{T,W}$
and
$\pi _{T,W} \circ \gamma _z$
agree on generators, they are equal. Hence,
$\Psi \circ \pi _{T,W} =\pi _{T,W} \circ \Phi $
. By [Reference Sims, Whitehead and Whittaker8, Lemma 3.14],
$\pi _{T,W}$
is injective if it is injective on
$C^*(\mathcal {G}, E)^{\gamma }$
, which it is by Lemma 4.10.
Acknowledgements
The author would like to thank her supervisors, Aidan Sims and Anna Duwenig, for their careful reading and their helpful feedback which improved the manuscript. The author would like also to thank the Ministry of Education, Culture, Research and Technology of the Republic of Indonesia who provided their sponsorship for the PhD study of the author at University of Wollongong.
