(7.3.3) sends $[1_{\mathcal{R}_{n^{-}}}]\mapsto[1-n\tau_{m}]$.

###### Sketch of the proof of Theorem 5.6.15.

Since any $\mathbb{Z}$-graded homomorphism of $\mathbb{Z}$-graded algebras is
also $\mathbb{Z}/2\mathbb{Z}$-graded, we may assume that $m\geq 2$. In view of
Example 7.2.6 and Lemma 7.3.1, it suffices to show that there are no
$\mathbb{Z}[\tau_{m}]$-linear homomorphisms between the right-hand sides of
(7.3.2) and (7.3.3) preserving the distinguished element. This is done in
[ac1]*Theorem 5.1 and Theorem 6.3. ∎

##  8\. Graded classification and symbolic dynamics

Since Leavitt path algebras are graded algebras, it is natural to take into
account the grading of these algebras when trying to determine an invariant
for them. In this section we will look at the graded Grothendieck group
$K_{0}^{\operatorname{gr}}$ as such an invariant. The main conjectures
pertaining to Leavitt path algebras we consider in this section state that the
graded Grothendieck group, along with its positive cone (and the position of
the identity) is a complete invariant (Conjectures 8.1.2, 8.3.1 and 8.8.2).

###  8.1. Polycephaly graphs and the graded classification conjecture

We write $C_{n}$ for the graph consisting of a single cycle of length $n$. A
graph having only one cycle $C$ such that all vertices are connected to some
vertex in the support of $C$ is called a _$C_{n}$ -comet_ if $|C|=n$. A
_multiheaded comet_ is a graph consisting of a finite number of comets such
that cycles are mutually disjoint and every vertex connects to at least one
cycle. A finite graph $E$ is _polycephaly_ if each vertex connects to disjoint
cycles, or to sinks or roses. For formal definitions and more details of
multiheaded comets and polycephaly graphs, see [haz2013]*page 296.

######  Theorem 8.1.1 (Hazrat [haz2013]*Theorem 9).

Let $E$ and $F$ be polycephaly graphs. Then
$L(E)\cong_{\operatorname{gr}}L(F)$ if and only if there is an order
preserving $\mathbb{Z}[x,x^{-1}]$-module isomorphism

$\big{(}K_{0}^{\operatorname{gr}}(L(E)),[L(E)]\big{)}\cong\big{(}K_{0}^{\operatorname{gr}}(L(F)),[L(F)]\big{)}.$

###### Strategy of the proof.

It was shown in [haz2013]*Theorem 2 that if $E$ is polycephaly, then $L(E)$
decomposes as a direct sum of matrix algebras, graded by certain sequences of
weights as in (7.1.1), with one matrix algebra over $\mathsf{k}$ for each
sink, one over $\mathsf{k}[x^{n},x^{-n}]$ for each comet $C_{n}$ and one over
$L_{n}$ for each $n$-petal rose in $E$ $(n\geq 2)$. Then
$K_{0}^{\operatorname{gr}}$ was computed for these three types of algebras
[haz2013]*Theorems 6, 7 and 8 as a pointed, partially ordered
$\mathbb{Z}[x,x^{-1}]$-module. Finally the theorem was proved in
[haz2013]*Theorem 9 using those calculations. ∎

Theorem 8.1.1 motivated the following conjecture.

######  Conjecture 8.1.2 (_Graded Classification Conjecture_ [haz2013]).

Let $E$ and $F$ be finite graphs. There is an order preserving
$\mathbb{Z}[x,x^{-1}]$-module isomorphism
$\phi:K_{0}^{\operatorname{gr}}(L(E))\overset{\sim}{\longrightarrow}K_{0}^{\operatorname{gr}}(L(F))$
with $\phi([L(E)])=[L(F)]$ if and only if there exists a unital
$\mathbb{Z}$-graded $\mathsf{k}$-isomorphism $\psi:L(E)\rightarrow L(F)$ such
that $K^{\operatorname{gr}}_{0}(\psi)=\phi$.

######  Remark 8.1.3.

One can also formulate a version of Conjecture 8.1.2 for row-finite graphs
with an infinite number of vertices, with the condition that $\phi$ preserves
the order unit replaced by the condition that it preserves the generating
interval [hazlia]*p. 450. Then one may further strengthen the conjecture by
requiring that $\psi$ be a $\ast$-homomorphism. This strong form of the
conjecture was established by Hazrat and Vaš in [hazlia]*Theorem 5.5 for a
certain class $\mathcal{C}$ of row-finite countable graphs and under some
hypothesis on the ground field $\mathsf{k}$. A countable graph $E$ belongs to
$\mathcal{C}$ if and only if $L(E)$ decomposes as a direct sum of countably
many summands, each of them a finite or countably infinite matrix algebra with
coefficients in either $\mathsf{k}$ or $\mathsf{k}[x,x^{-1}]$. A graph
theoretic description of such graphs is given in [chain]*Theorem 3.7. The
condition on ground field $\mathsf{k}$ is that it be $2$-proper and
$\ast$-Pythagorean (see [hazlia]*p. 420 for a definition of these terms); for
example $\mathbb{C}$, $\mathbb{R}$ and, in general, any quadratically closed
$\ast$-subfield of $\mathbb{C}$ satisfy these conditions.

###  8.2. The graded classification conjecture up to a twist

Recall that an automorphism $g$ of a unital algebra $\mathcal{A}$ is said to
be locally inner if given any finite number of elements $x_{1},\dots,x_{n}$ in
$\mathcal{A}$ there is an inner automorphism $h$ of $\mathcal{A}$ such that
$h(x_{i})=\varphi(x_{i})$ for $i=1,\dots,n$. It is well-known
([goodearl]*Lemma 15.23(b)) that any automorphism $\varphi$ of a unital
ultramatricial algebra $\mathcal{A}$, such that $K_{0}(\varphi)=\text{id}$, is
locally inner.

Let $R$ be a ring with identity and $p$ an idempotent of $R$. Observe that if
$\phi:R\to R$ is a not-necessarily unital ring homomorphism then for
$p=\phi(1)$ we have $\phi(R)\subset pRp$. We call $\phi$ a _corner
isomorphism_ if the induced homomorphism $\phi:R\longrightarrow pRp$ is an
isomorphism. The _corner skew Laurent polynomial ring_ associated to
$(R,\phi)$, denoted by $R[t_{+},t_{-},\phi]$, is a unital ring constructed as
follows. For $n\geq 0$, set $p_{n}=\phi^{n}(p)$. The elements of
$R[t_{+},t_{-},\phi]$ are the formal expressions

$t^{j}_{-}r_{-j}+t^{j-1}_{-}r_{-j+1}+\dots+t_{-}r_{-1}+r_{0}+r_{1}t_{+}+\dots+r_{i}t^{i}_{+},\,\,(i,j\geq
0).$

Addition is componentwise, and multiplication is determined by the
distributive law and the following rules.

$t_{-}t_{+}=1,\qquad t_{+}t_{-}=p,\qquad rt_{-}=t_{-}\phi(r),\qquad
t_{+}r=\phi(r)t_{+}.$ (8.2.1)

Corner skew Laurent polynomial rings are studied in [arabrucom]; their
$K$-groups were calculated in [abc]*Theorem 3.6. Assigning degrees $-1$ to
$t_{-}$ and $1$ to $t_{+}$ makes $A:=R[t_{+},t_{-},\phi]$ a
$\mathbb{Z}$-graded ring with $A=\bigoplus_{i\in\mathbb{Z}}A_{i}$, where for
$p_{i}=\phi^{i}(1)$ $(i\geq 0)$ we have

$A_{i}=\left\\{\begin{matrix}Rp_{i}t^{i}_{+}&\text{ for }i>0,\\\
t^{-i}_{-}p_{-i}R&\text{ for }i<0,\\\ R&\text{ for }i=0.\end{matrix}\right.$

Setting $p=1$ and $\phi$ the identity map, $R[t_{+},t_{-},\phi]$ reduces to
the familiar Laurent polynomial ring $R[t,t^{-1}]$.

######  Theorem 8.2.2 (Ara, González-Barroso, Goodearl, Pardo [skew]*Lemma
2.4).

Let $A$ be a $\mathbb{Z}$-graded unital ring possessing a left invertible
element $t_{+}\in A_{1}$. Then $t_{+}$ has a left inverse $t_{-}\in A_{-1}$,
and for $p=t_{+}t_{-}$ and

$\displaystyle\phi:A_{0}$ $\displaystyle\longrightarrow pA_{0}p,$ (8.2.3)
$\displaystyle a$ $\displaystyle\longmapsto t_{+}at_{-}$

we have a ring isomorphism $A\cong A_{0}[t_{+},t_{-},\phi]$.

Let $E$ be a finite graph without sources. Using Theorem 8.2.2, one can
present $L(E)$ as a corner skew Laurent polynomial ring as follows. For each
$v\in E^{0}$, choose an edge $e_{v}$ such that $r(e_{v})=v$ and consider

$t_{+}=\sum_{v\in E^{0}}e_{v}\in L(E)_{1}.$

Then $t_{-}=t_{+}^{*}$ satisfies $t_{-}t_{+}=1$; set $p=t_{+}t_{-}$ and let

$\displaystyle\alpha:L(E)_{0}$ $\displaystyle\longrightarrow pL(E)_{0}p$
$\displaystyle a$ $\displaystyle\longmapsto t_{+}at_{-}$

One checks that $\alpha$ is an isomorphism. Then thanks to Theorem 8.2.2, we
have

$L(E)=L(E)_{0}[t_{+},t_{-},\alpha].$

Now let $g$ be a locally inner automorphism of $L(E)_{0}$. Then we define
$L^{g}(E)$ by

$L^{g}(E)=L(E)_{0}[t_{+},t_{-},g\circ\alpha].$

The graded algebra $L^{g}(E)$ has the same graded $K$-theory as $L(E)$. Indeed
the graded $K$-groups are the same because both are strongly graded with the
same $0$-degree component, and the $\mathbb{Z}$-actions on them coincide
because $g$ induces the identity in $K$-theory. We are now in a position to
describe the following result of Ara and Pardo.

######  Theorem 8.2.4 (Ara-Pardo [apgrad]*Theorem 4.1).

If $E,F$ are finite essential graphs, then the following are equivalent:

  1. (1)

There is an order preserving $\mathbb{Z}[x,x^{-1}]$-module isomorphism

$\big{(}K_{0}^{\operatorname{gr}}(L(E)),[L(E)]\big{)}\cong\big{(}K_{0}^{\operatorname{gr}}(L(F)),[L(F)]\big{)};$

  2. (2)

There exists a locally inner automorphism $g$ of $L(E)_{0}$ such that
$L^{g}(E)\cong_{\operatorname{gr}}L(F)$.

###  8.3. The fullness conjecture

One of the conjectures raised in [haz2013] is that the graded Grothendieck
group $K_{0}^{\operatorname{gr}}$, as a functor between the category of
Leavitt path algebras and the category of preordered
$\mathbb{Z}[x,x^{-1}]$-modules with order unit is full.

######  Conjecture 8.3.1 (Hazrat, [haz2013]).

For any order preserving $\mathbb{Z}[x,x^{-1}]$-module homomorphism
$\phi:K^{\operatorname{gr}}_{0}(L(E))\rightarrow
K^{\operatorname{gr}}_{0}(L(F))$ with $\phi([L(E)])=[L(F)]$, there exists a
unital $\mathbb{Z}$-graded $\mathsf{k}$-homomorphism $\psi:L(E)\rightarrow
L(F)$ such that $K^{\operatorname{gr}}_{0}(\psi)=\phi$.

G. Arnone in [guido2023] and L. Vaš in [vas] proved Conjecture 8.3.1,
independently and with different approaches. In fact Vaš considers graphs with
finitely many vertices and infinitely many edges while Arnone requires that
the graphs to be finite, but shows that the lifting map can be chosen to be a
diagonal preserving graded $*$-homomorphism.

######  Theorem 8.3.2 (Arnone [guido2023], Vaš [vas]).

Let $E$ and $F$ be finite graphs and
$\phi:K^{\operatorname{gr}}_{0}(L(E))\rightarrow
K^{\operatorname{gr}}_{0}(L(F))$ a preordered $\mathbb{Z}[x,x^{-1}]$-module
homomorphism with $\phi([L(E)])=[L(F)]$. Then there exists a unital
$\mathbb{Z}$-graded homomorphism $\psi:L(E)\rightarrow L(F)$ such that
$K^{\operatorname{gr}}_{0}(\psi)=\phi$. Furthermore if $\phi$ is injective, so
is $\psi$.

###  8.4. $C^{*}$-version of the classification conjecture

Write ${\mathbb{T}}$ for the circle group, $\gamma_{E}$ for the gauge circle
action on $C^{*}(E)$, and $K^{\mathbb{T}}_{0}(C^{*}(E))$ for
${\mathbb{T}}$-equivariant $K$-theory [chrisp]. There are canonical order
preserving isomorphisms of $\mathbb{Z}[x,x^{-1}]$-modules (see [haz2013]*p.
275, [haziso] and [eilers2]*Proof of Theorem A)

$K^{\operatorname{gr}}_{0}(L(E))\cong K_{0}(L(E\times_{1}\mathbb{Z}))\cong
K_{0}(C^{*}(E\times_{1}\mathbb{Z}))\cong K_{0}^{\mathbb{T}}(C^{*}(E)).$
(8.4.1)

Thus one can pose the analytic version of Conjecture 8.1.2 as follows.

######  Conjecture 8.4.2.

Let $E$ and $F$ be finite graphs. Then there is an order preserving
$\mathbb{Z}[x,x^{-1}]$-module isomorphism

$\phi:K^{\mathbb{T}}_{0}(C^{*}(E))\overset{\sim}{\longrightarrow}K_{0}^{\mathbb{T}}(C^{*}(F)),$

with $\phi([C^{*}(E)])=[C^{*}(F)]$ if and only if there is a
${\mathbb{T}}$-equivariant $\ast$-isomorphism
$C^{*}(E)\overset{\sim}{\longrightarrow}C^{*}(F)$.

In fact in Conjecture 8.1.2 if the graded Grothendieck groups are isomorphic,
one expects to have a graded $\ast$-isomorphism between the Leavitt path
algebras. This stronger version of Conjecture 8.1.2 implies Conjecture 8.4.2.
For, if $K^{\mathbb{T}}_{0}(C^{*}(E))\cong K_{0}^{\mathbb{T}}(C^{*}(F))$, then
by (8.4.1), $K_{0}^{\operatorname{gr}}(L_{\mathbb{C}}(E))\cong
K_{0}^{\operatorname{gr}}(L_{\mathbb{C}}(F))$, so there is a graded
$\ast$-isomorphism
$L_{\mathbb{C}}(E)\cong_{\operatorname{gr}}L_{\mathbb{C}}(F)$. Passing to the
completion, we get a ${\mathbb{T}}$-equivariant $\ast$-isomorphism
$C^{*}(E)\overset{\sim}{\longrightarrow}C^{*}(F)$ (see
[abramstomforde]*Theorem 4.4).

###  8.5. The graded classification conjecture for amplified graphs

A directed graph $E$ is called an _amplified graph_ if for any $v,w\in E^{0}$,
the set of edges from $v$ to $w$ is either empty or infinite. The following
theorem shows that the graded Grothendieck group precisely pinpoints the
isomorphism class of amplified graphs; namely if the graded Grothendieck
groups are isomorphic, then the associated graphs are isomorphic.

######  Theorem 8.5.1 (Eilers, Ruiz, Sims [eilers2]*Theorem A).

Let $E$ and $F$ be countable amplified graphs and let $\mathsf{k}$ be a field.
Then the following are equivalent:

  1. (1)

$E\cong F$;

  2. (2)

there is an order preserving $\mathbb{Z}[x,x^{-1}]$-module isomorphism
$K_{0}^{\operatorname{gr}}(L_{\mathsf{k}}(E))\cong
K_{0}^{\operatorname{gr}}(L_{\mathsf{k}}(F))$;

  3. (3)

there is an order preserving $\mathbb{Z}[x,x^{-1}]$-module isomorphism
$K_{0}^{\mathbb{T}}(C^{*}(E))\cong K_{0}^{\mathbb{T}}(C^{*}(F))$ of
$\mathbb{T}$-equivariant $K_{0}$-groups.

###  8.6. Graded Morita equivalence

The notion of graded Morita theory plays an important role in this subject. It
relates the theory of Leavitt path algebras to the theory of symbolic
dynamics. One of the main conjectures in the theory is around the notion of
graded Morita equivalence (Conjecture 8.8.2).

######  Definition 8.6.1.

Let $A$ and $B$ be $\Gamma$-graded rings.

  1. (1)

A functor $\phi:{\operatorname{{Gr}-}}A\rightarrow{\operatorname{{Gr}-}}B$ is
called a _graded functor_ if
$\phi\mathcal{T}_{\alpha}=\mathcal{T}_{\alpha}\phi$ for all $\alpha\in\Gamma$.

  2. (2)

A graded functor
$\phi:{\operatorname{{Gr}-}}A\rightarrow{\operatorname{{Gr}-}}B$ is called a
_graded equivalence_ if there is a graded functor
$\psi:{\operatorname{{Gr}-}}B\rightarrow{\operatorname{{Gr}-}}A$ such that
$\psi\phi\cong 1_{{\operatorname{{Gr}-}}A}$ and $\phi\psi\cong
1_{{\operatorname{{Gr}-}}B}$.

  3. (3)

If there is a graded equivalence between ${\operatorname{{Gr}-}}A$ and
${\operatorname{{Gr}-}}B$, we say $A$ and $B$ are graded equivalent or graded
Morita equivalent and write
${\operatorname{{Gr}-}}A\approx_{\operatorname{gr}}{\operatorname{{Gr}-}}B$.

  4. (4)

A functor $\phi:\operatorname{Mod-}A\rightarrow\operatorname{Mod-}B$ is called
a _graded functor_ if there is a graded functor
$\phi^{\prime}:{\operatorname{{Gr}-}}A\rightarrow{\operatorname{{Gr}-}}B$ such
that the following diagram, where the vertical functors are forgetful
functors, commutes.

$\textstyle{{\operatorname{{Gr}-}}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi^{\prime}}$$\scriptstyle{U}$$\textstyle{{\operatorname{{Gr}-}}B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{U}$$\textstyle{\operatorname{Mod-}A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{\operatorname{Mod-}B.}$
(8.6.2)

The functor $\phi^{\prime}$ is called an _associated graded functor_ of
$\phi$.

  5. (5)

A functor $\phi:\operatorname{Mod-}A\rightarrow\operatorname{Mod-}B$ is called
a _graded equivalence_ if it is graded and it is an equivalence.

Recall that for any ring $R$ we let $\operatorname{\mathbb{M}}_{\infty}(R)$
denote the (nonunital) ring consisting of those countably infinite square
matrices over $R$ that contain at most finitely many nonzero entries (see
Section 4.1). Furthermore, we say an idempotent $e$ in a ring $R$ is _full_ ,
if the two-sided ideal generated by $e$ is the whole ring $R$.

The equivalence between statements (1)-(5) in the following theorem can be
found in [hazbook]*Theorem 2.3.8 and the equivalence of the latter with (6) is
a recent result of Abrams, Ruiz and Tomforde [abramsmori]. They are all
parallel to the classical results in the non-graded Morita theory, with the
added complexity of shifts being present in the graded theory. Once we assume
the grade group $\Gamma$ is trivial, we recover the classical results.

######  Theorem 8.6.3.

Let $A$ and $B$ be two $\Gamma$-graded rings. The following are equivalent:

  1. (1)

${\operatorname{{Gr}-}}A$ is graded equivalent to ${\operatorname{{Gr}-}}B$;

  2. (2)

$\operatorname{Mod-}A$ is graded equivalent to $\operatorname{Mod-}B$;

  3. (3)

$B\cong_{\operatorname{gr}}\operatorname{End}_{A}(P)$ for a graded
$A$-progenerator $P$;

  4. (4)

$B\cong_{\operatorname{gr}}e\operatorname{\mathbb{M}}_{n}(A)(\overline{\delta})e$
for a full homogeneous idempotent
$e\in\operatorname{\mathbb{M}}_{n}(A)(\overline{\delta})$, where
$\overline{\delta}=(\delta_{1},\dots,\delta_{n})$, $\delta_{i}\in\Gamma$.

  5. (5)

There are graded finitely generated projective $A-B$ bimodule $P$ and
$B-A$-bimodule $Q$, such that $P\otimes_{B}Q\cong_{\operatorname{gr}}A$ and
$Q\otimes_{A}P\cong_{\operatorname{gr}}B$ as $A-A$ and $B-B$ bimodules,
respectively.

  6. (6)

There exists a sequence $(\gamma_{m})_{m\in\mathbb{N}}$ in $\Gamma$ such that
$\operatorname{\mathbb{M}}_{\infty}(A)$ is graded isomorphic to
$\operatorname{\mathbb{M}}_{\infty}(B)((\gamma_{m})_{m\in\mathbb{N}})$.

###  8.7. Graded Morita equivalence and shift equivalence

Recall the notion of shift equivalence from Section 6. The theorem below
provides a bridge between the theory of symbolic dynamics and Leavitt path
algebras through graded Grothendieck groups.

######  Theorem 8.7.1.

Let $E$ and $F$ be finite regular graphs and let $A_{E}$ and $A_{F}$ be their
adjacency matrices. Then $A_{E}$ is shift equivalent to $A_{F}$ if and only if
there is an order preserving $\mathbb{Z}[x,x^{-1}]$-module isomorphism
$K_{0}^{\operatorname{gr}}(L(E))\cong K_{0}^{\operatorname{gr}}(L(F))$.

###### Proof.

Observe that two matrices are shift equivalent if and only if their transposes
are. Hence in view of Remark 6.6, the theorem follows from Theorem 6.5. ∎

The following Proposition heavily relies on Williams’ Theorem 6.1 on graph
moves. One can check that these graph moves preserve the associated Leavitt
path algebras up to graded Morita equivalence. Thus we have the following
proposition.

######  Proposition 8.7.2 (Hazrat [hazd]*Proposition 15).

Let $E$ and $F$ be finite graphs.

  1. (1)

If $E$ is essential and $F$ is obtained from an in-splitting or out-splitting
of the graph $E$, then $L(E)$ is graded Morita equivalent to $L(F)$.

  2. (2)

For essential $E$ and $F$, if the adjacency matrices $A_{E}$ and $A_{F}$ are
strongly shift equivalent then $L(E)$ is graded Morita equivalent to $L(F)$.

  3. (3)

If $E$ and $F$ are regular and $L(E)$ is graded Morita equivalent to $L(F)$,
then the adjacency matrices $A_{E}$ and $A_{F}$ are shift equivalent.

######  Remark 8.7.3.

Conjecture 8.8.2 says that the converse of assertion (3) in Proposition 8.7.2
also holds.

######  Example 8.7.4.

Consider the graph below on the left. William’s out/in splitting starts with a
partition $E^{1}=\bigsqcup_{i=1}^{n}E^{1}_{i}$ of the set of edges. The latter
induces, for each $v\notin\operatorname{sink}(E)$, a partition
$\mathfrak{s}^{-1}\\{v\\}=\bigsqcup_{i=1}^{n}(E^{1}_{i}\cap s^{-1}(v))$ of the
set of outgoing edges, and similary, if $v\notin\operatorname{sour}(E)$, a
partition on the set $r^{-1}\\{v\\}$ incoming edges. If we start with the
partition of $E^{1}$ which consists of one edge in each term, and apply
Williams’ outsplitting change of the graph, we obtain the graph on the right.
Then our result implies that the Leavitt path algebras associated to these
graphs are graded Morita equivalent. In fact it was proved in
[question]*Theorem 2.8 that out-splitting yields graded isomorphic Leavitt
path algebras.

###  8.8. Graded Morita equivalence, derived equivalence and the singularity
category

Let $\mathsf{k}$ be a field and $E$ a finite graph. The _path algebra_
$\mathsf{k}E$ is the tensor algebra of the $\mathsf{k}^{E^{0}}$-bimodule
$\mathsf{k}^{E^{1}}$. Let $\mathsf{k}E\vartriangleright J$ be the two-sided
ideal generated by paths of length greater or equal to $1$. Set
$A(E)=\mathsf{k}E/J^{2}$. There are close relations between Leavitt path
algebras and the representation theory of $A(E)$. For a unital algebra $A$,
the singularity category of $A$, introduced by Buchweitz and Orlov, is defined
as the quotient category

$\operatorname{\bf
D}_{\operatorname{sgn}}(A)=D^{b}(\operatorname{Mod-}A)/\operatorname{Perf}(A),$

where $D^{b}(\operatorname{Mod-}A)$ is the bounded derived category of $A$ and
$\operatorname{Perf}(A)$ is the subcategory of perfect complexes. One can show
that the global dimension of $A$ is finite if and only if $\operatorname{\bf
D}_{\operatorname{sgn}}(A)$ is trivial.

The following theorem, due to Chen and Yang [chen], uses results of Paul Smith
[smith3] and Hazrat [haz2013]. Recall that two differential graded algebras
(dgas) are _derived equivalent_ if the derived categories of their categories
of differential graded modules are equivalent. Any $\mathbb{Z}$-graded algebra
can be considered as a dga with trivial differential. In particular this
applies to Leavitt path algebras; in part (2) of Theorem 8.8.1, derived
equivalence should be understood in this sense.

######  Theorem 8.8.1 (Chen, Yang [chen]).

Let $E$ and $F$ be finite regular graphs. Then the following statements are
equivalent.

  1. (1)

The Leavitt path algebras $L(E)$ and $L(F)$ are graded Morita equivalent.

  2. (2)

The Leavitt path algebras $L(E)$ and $L(F)$, regarded as dgas with trivial
differential, are derived equivalent.

  3. (3)

The singularity categories $\operatorname{\bf D}_{\operatorname{sgn}}(A(E))$
and $\operatorname{\bf D}_{\operatorname{sgn}}(A(F))$ are triangulated
equivalent.

Assembling all the results so far, we have the following conjecture:

######  Conjecture 8.8.2.

Let $E$ and $F$ be finite regular graphs. Then the following are equivalent.

  1. (1)

There is an isomorphism of partially ordered $\mathbb{Z}[x,x^{-1}]$-modules
$K_{0}^{\operatorname{gr}}(L(E))\overset{\sim}{\longrightarrow}K_{0}^{\operatorname{gr}}(L(E))$;

  2. (2)

There is a gauge-preserving Morita equivalence between $C^{*}(E)$ and
$C^{*}(F)$;

  3. (3)

There is a graded Morita equivalence between $L(E)$ and $L(F)$;

  4. (4)

The singularity categories $\operatorname{\bf D}_{\operatorname{sgn}}(A(E))$
and $\operatorname{\bf D}_{\operatorname{sgn}}(A(F))$ are triangulated
equivalent;

  5. (5)

The adjacency matrices $A_{E}$ and $A_{F}$ are shift equivalent.

######  Remark 8.8.3.

By Theorem 8.8.1, conditions (3) and (4) of Conjecture 8.8.2 are equivalent.
By Proposition 8.7.2, (3) implies (5). Assuming the response to Question 6.4
is positive, (5) would further be equivalent to the existence of a finite
sequence of moves that carries $E$ to $F$.

###  8.9. Talented monoids and the Morita equivalence conjecture for meteor
graphs

For a finite essential graph $E$, the Krieger monoid $\Delta_{A_{E}^{t}}^{+}$
is the positive cone of $K_{0}^{\operatorname{gr}}(L(E))$ (this follows by the
same argument as in 8.7.1). The latter is the _talented monoid_ $T_{E}$;
$K_{0}^{\operatorname{gr}}(L(E))$ is its group completion. Therefore by
Theorem 6.5, from a shift equivalence of finite essential graphs $E$ and $F$
we obtain a $\mathbb{Z}$-monoid isomorphism of talented monoids $T_{E}\cong
T_{F}$. This isomorphism gives us control over elements of the monoids (such
as minimal elements, atoms, etc.) and consequently on the geometry of the
graphs (the number of cycles, their lengths, etc.).

We start with a definition of talented monoids via generators and relations.

######  Definition 8.9.1.

For a row-finite graph $E$, the _talented monoid_ $T_{E}$ is the commutative
monoid generated by $\\{v(i)\mid v\in E^{0},i\in\mathbb{Z}\\}$ subject to

$v(i)=\sum_{e\in s^{-1}(v)}r(e)(i+1),$ (8.9.2)

for every $v\in\operatorname{reg}(E)$ and $i\in\mathbb{Z}$.

The above relations define a congruence relation which also respects the
action of $\mathbb{Z}$. Namely, the monoid $T_{E}$ is equipped by a natural
$\mathbb{Z}$-action:

${}^{n}v(i)=v(i+n)$

for $n,i\in\mathbb{Z}$. Proposition 5.7 of [ahls] relates this monoid to the
theory of Leavitt path algebras: there is a $\mathbb{Z}$-module isomorphism
$T_{E}\cong\mathcal{V}^{\operatorname{gr}}(L(E))$. In fact we have

$\displaystyle T_{E}$
$\displaystyle\cong\,\mathcal{V}^{\operatorname{gr}}(L(E))\cong\mathcal{V}(L(\overline{E})),$
$\displaystyle v(i)$
$\displaystyle\longmapsto\big{(}L(E)v\big{)}(i)\longmapsto
L(\overline{E})v_{i}$

where $\overline{E}$ is the covering graph of $E$. Thus the talented monoid
$T_{E}$ is conical and cancellative ([ahls]*Section 5). Note that, we use left
$L(E)$-modules to construct $\mathcal{V}^{\operatorname{gr}}(L(E))$ so that it
matches the definition of our talented monoid.

The following theorem shows how the geometry of graphs translates into
properties of their talented monoids. A graph $E$ is said to satisfy
_Condition (L)_ if every cycle in $E$ has an exit. A closed path
$\alpha=e_{1}\cdots e_{n}$ based at a vertex $v$ is _simple_ if
$s(e_{i})=v\Rightarrow i=1$. We say that $E$ satisfies _Condition (K)_ if for
every vertex $v$ in the support of a closed simple path there exist at least
two distinct closed simple paths based at $v$.

######  Theorem 8.9.3 ([hazli]).

Let $E$ be a row-finite graph and $T_{E}$ the talented monoid associated to
$E$.

  1. (1)

The graph $E$ satisfies Condition (L) if and only if $\mathbb{Z}$ acts freely
on $T_{E}$.

  2. (2)

The graph $E$ satisfies Condition (K) if and only if $\mathbb{Z}$ acts freely
on any quotient of $T_{E}$ by an order-ideal.

A _meteor graph_ is an essential connected graph consisting of two disjoint
cycles and the paths connecting these cycles. We can give a positive answer to
Conjecture 8.8.2 for the case of meteor graphs.

######  Theorem 8.9.4 (Cordeiro, Gillaspy, Gonçalves, Hazrat [oberwol]).

Let $E$ and $F$ be essential graphs, where $E$ is a meteor graph. Then the
following are equivalent.

  1. (1)

The Leavitt path algebras $L(E)$ and $L(F)$ are graded Morita equivalent;

  2. (2)

The graph $C^{*}$-algebras $C^{*}(E)$ and $C^{*}(F)$ are equivariantly Morita
equivalent;

  3. (3)

There is an order-preserving $\mathbb{Z}[x,x^{-1}]$-module isomorphism
$K_{0}^{\operatorname{gr}}(L(E))\overset{\sim}{\longrightarrow}K_{0}^{\operatorname{gr}}(L(F))$;

  4. (4)

The talented monoids $T_{E}$ and $T_{F}$ are $\mathbb{Z}$-isomorphic;

  5. (5)

The graphs $E$ and $F$ are shift equivalent;

  6. (6)

The graphs $E$ and $F$ are strongly shift equivalent.

###### Sketch of the proof.

An order preserving $\mathbb{Z}[x,x^{-1}]$-module isomorphism
$K_{0}^{\operatorname{gr}}(L(E))\overset{\sim}{\longrightarrow}K_{0}^{\operatorname{gr}}(L(F))$,
gives an isomorphism on the level of talented monoids $T_{E}\cong T_{F}$.
Then, a careful analysis of the monoid isomorphism allows us to show that $E$
and $F$ can be transformed, via in- and out-splitting and their inverses, into
each other. Applying Williams’ Theorem 6.1, we obtain that the graphs $E$ and
$F$ are strongly shift equivalent and, as a consequence of Proposition 8.7.2,
we get that $L(E)$ is graded Morita equivalent to $L(F)$. ∎

###  8.10. Graded homotopy classification

Let $B=\bigoplus_{i\in\mathbb{Z}}B_{i}$ be a $\mathbb{Z}$-graded algebra and
$B[t]$ the polynomial ring graded so that $B_{n}[t]$ is the graded component
of degree $n$. Let $\phi,\psi:A\to B$ be graded homomorphisms of
$\mathbb{Z}$-graded algebras. We say that $\phi$ and $\psi$ are _graded
homotopic_ and write $\phi\sim_{\operatorname{gr}}\psi$, if there is a finite
sequence of elementary homotopies
$H_{0}:\phi\to\phi_{1},\dots,H_{n}:\phi_{n}\to\psi$ such that each $H_{i}$ is
a graded homomorphism. The following theorem is concerned with primitive
graphs. A finite graph $E$ is _primitive_ if there is an $N$ such that for
every pair of vertices $(v,w)$ of $E$ there is a path $\alpha$ of length $N$
with $s(\alpha)=v$ and $r(\alpha)=w$.

The talented monoid $T_{E}$ (Section 8.9) of a finite graph $E$ is pointed by
$1_{E}=\sum_{v\in E^{0}}v(0)$ .

######  Theorem 8.10.1 (Arnone [guidotopy]*Theorem 8.1).

Let $E$ and $F$ be two finite, primitive graphs. If there exists an
isomorphism of $\mathbb{Z}$-monoids
$T_{E}\overset{\sim}{\longrightarrow}T_{F}$ sending $1_{E}\mapsto 1_{F}$ then
the algebras $L(E)$ and $L(F)$ are graded homotopy equivalent.

The proof of Theorem 8.10.1 uses graded bivariant algebraic $K$-theory
techniques [kkg, ac]. The hypothesis that $E$ and $F$ be primitive is needed
so that certain idempotents of $L(E)$ arising from edges are full.

###  8.11. Compatibility conditions on shift equivalence that imply graded
Morita equivalence

In what follows, if $X$ is finite set we write $\mathsf{k}X$ for the
$\mathsf{k}$-vector space with basis $X$. We will occasionally consider
$\mathsf{k}X$ as a $\mathsf{k}$-algebra with the product $x\cdot
y=\delta_{x,y}x$. For example $\mathsf{k}E^{0}$ is isomorphic to the
subalgebra of $L(E)$ generated by $E^{0}$.

Let $E$ and $F$ be finite regular graphs. If $A=A_{E}$ and $B=A_{F}$ are shift
equivalent with lag $\ell$, then by definition there are matrices
$R\in\mathbb{N}_{0}^{E^{0}\times F^{0}}$ and $S\in\mathbb{N}_{0}^{F^{0}\times
E^{0}}$ satisfying the identities (6.2) and (6.3). Let $G=G(R)$ be a set with

$|R|_{1}=\sum_{(v,w)\in E^{0}\times F^{0}}R_{v,w}$

elements, and let $s:G\to E^{0}$ and $r:G\to F^{0}$ be such that

$R_{v,w}=\\#\big{(}s^{-1}(v)\cap r^{-1}(w)\big{)}.$

Next consider the $\mathsf{k}$-vector space $M_{R}=\mathsf{k}G(R)$. Give
$M_{R}$ a $(\mathsf{k}E^{0},\mathsf{k}F^{0})$-bimodule structure as follows:
for each $g\in G=G(E)$, $v\in E^{0}$ and $w\in F^{0}$, put

$vg=\delta_{v,s(g)}g,\,gw=\delta_{r(g),w}g$

Similarly we define the $\mathsf{k}$-vector space $M_{S}$ and make it into a
$(\mathsf{k}{F^{0}},\mathsf{k}{E^{0}})$-bimodule. Let

$\displaystyle s,r:G(R)\times_{F^{0}}G(S)=\\{(g,h)\in G(R)\times
G(S)\,\colon\,r(g)=s(h)\\}\to E^{0}\sqcup F^{0}$ $\displaystyle
s(g,h)=s(g),\,\,r(g,h)=r(h).$

Then for each pair $(v_{1},v_{2})\in E^{0}\times E^{0}$

$\\#\\{(g,h)\in
G(R)\times_{F^{0}}G(S)\,\colon\,s(g,h)=v_{1},\,r(g,h)=v_{2}\\}=(RS)_{v_{1},v_{2}}=A^{\ell}_{v_{1},v_{2}}=\\#v_{1}E^{\ell}v_{2}.$

Hence from $RS=A^{\ell}$ we obtain

$M_{R}\otimes_{\ell{F^{0}}}M_{S}=\bigoplus_{(g,h)\in
G(R)\times_{F^{0}}G(S)}\mathsf{k}\cdot(g,h)\cong\mathsf{k}E^{\ell}\cong(\mathsf{k}{E^{1}})^{\otimes_{\mathsf{k}{E^{0}}}^{\ell}}.$
(8.11.1)

The isomorphism 8.11.1 translates one of the conditions in the definition of
shift equivalence to bimodule terms. We may similarly translate all of them;
setting $M=M_{R}$, $N=M_{S}$ we obtain bimodule isomorphisms

$\displaystyle
M\otimes_{\mathsf{k}{F^{0}}}N\overset{\omega_{E}}{\overset{\sim}{\longrightarrow}}(\mathsf{k}{E^{1}})^{\otimes_{\mathsf{k}{E^{0}}}^{\ell}},\,\,N\otimes_{\mathsf{k}{E^{0}}}M\overset{\omega_{F}}{\overset{\sim}{\longrightarrow}}(\mathsf{k}{F^{1}})^{\otimes_{\mathsf{k}{F^{0}}}^{\ell}},$
(8.11.2)
$\displaystyle\mathsf{k}{E^{1}}\otimes_{\mathsf{k}{E^{0}}}M\overset{\sigma_{M}}{\overset{\sim}{\longrightarrow}}M\otimes_{\mathsf{k}{F^{0}}}\mathsf{k}{F^{1}},\,\,\mathsf{k}{F^{1}}\otimes_{\mathsf{k}{F^{0}}}N\overset{\sigma_{N}}{\overset{\sim}{\longrightarrow}}N\otimes_{\mathsf{k}{E^{0}}}\mathsf{k}{E^{1}}.$

######  Proposition 8.11.3 ([recast]*Theorem 3.9).

Let $E$ and $F$ be regular graphs. The following are equivalent.

  1. (1)

$E$ and $F$ are shift equivalent;

  2. (2)

There exist bimodules ${}_{\mathsf{k}{E^{0}}}M_{\mathsf{k}{F^{0}}}$ and
${}_{\mathsf{k}{F^{0}}}N_{\mathsf{k}{E^{0}}}$ and bimodule isomorphisms
(8.11.2).

######  Remark 8.11.4.

Both the construction of $M$ and $N$ and the isomorphisms (8.11.2) were first
done in [lens] in the setting of $C^{*}$-algebras, where the analogue of a
bimodule is a $C^{*}$-correspondence. The analogue of Proposition 8.11.3 in
that context is [lens]*Proposition 3.5.

To state the next theorem we need more notation. Let $E$, $F$ and $G$ be
finite graphs and $\sigma:\mathsf{k}{E^{1}}\otimes_{\mathsf{k}{E^{0}}}M\to
M\otimes_{\mathsf{k}{F^{0}}}\mathsf{k}{F^{1}}$ and
$\psi:\mathsf{k}{F^{1}}\otimes_{\mathsf{k}{F^{0}}}N\to
N\otimes_{\mathsf{k}{G^{0}}}\mathsf{k}{G^{1}}$ bimodule homormophisms. The
composite

$\sigma\\#\psi:\mathsf{k}{E^{1}}\otimes_{\mathsf{k}{E^{0}}}M\otimes_{\mathsf{k}{F^{0}}}N\overset{\sigma\otimes\operatorname{id}_{N}}{\longrightarrow}M\otimes_{\mathsf{k}{F^{0}}}\mathsf{k}{F^{1}}\otimes_{\mathsf{k}{F^{0}}}N\overset{\operatorname{id}_{M}\otimes\psi}{\longrightarrow}M\otimes_{\mathsf{k}{F^{0}}}N\otimes_{\mathsf{k}{G^{0}}}\mathsf{k}{G^{1}}$

is a homomorphism of $(\mathsf{k}{E^{0}},\mathsf{k}{G^{0}})$-bimodules. (Here
and elsewhere, we omit the associativity isomorphisms of the tensor product.)

######  Theorem 8.11.5 (Abrams, Ruiz and Tomforde [recast]*Theorem 6.7).

Let $E$ and $F$ be finite regular graphs. Assume there are bimodules and
isomorphisms as in (8.11.2) such that the following diagrams commute

$\textstyle{\mathsf{k}{E^{1}}{\otimes}_{\mathsf{k}{E^{0}}}M\otimes_{\mathsf{k}{F^{0}}}N\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma_{M}\\#\sigma_{N}}$$\scriptstyle{\operatorname{id}\otimes\omega_{E}}$$\textstyle{M\otimes_{\mathsf{k}{F^{0}}}N\otimes_{\mathsf{k}{E^{0}}}\mathsf{k}{E^{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\omega_{E}\otimes\operatorname{id}}$$\textstyle{\mathsf{k}{E^{1}}^{\otimes_{\mathsf{k}{E^{0}}}m+1}}$
(8.11.10)
$\textstyle{\mathsf{k}{F^{1}}{\otimes}_{\mathsf{k}{F^{0}}}N\otimes_{\mathsf{k}{E^{0}}}M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma_{N}\\#\sigma_{M}}$$\scriptstyle{\operatorname{id}\otimes\omega_{F}}$$\textstyle{N\otimes_{\mathsf{k}{E^{0}}}M\otimes_{\mathsf{k}{F^{0}}}\mathsf{k}{F^{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\omega_{F}\otimes\operatorname{id}}$$\textstyle{\mathsf{k}{F^{1}}^{\otimes_{\mathsf{k}{F^{0}}}m+1}}$
(8.11.15)

Then $L(E)$ and $L(F)$ are graded Morita equivalent.

###### Sketch of the proof.

Consider the $(\mathsf{k}E^{0},L(F))$-bimodule
$P=M\otimes_{\mathsf{k}F^{0}}L(F)$ and the $(\mathsf{k}F^{0},L(E))$-bimodule
$Q=N\otimes_{KE^{0}}L(E)$. Both of these are $\mathbb{Z}$-graded, with the
gradings induced by those of $L(E)$ and $L(F)$. Following an idea of Eryüzlü
[eryu] in the $C^{*}$-algebra context, one uses $\sigma_{M}$ to extend the
left (graded) $\mathsf{k}E^{0}$-module structure on $P$ to a left graded
$L(E)$-module structure. For example multiplication by edges is defined by the
composite of the maps induced by $\sigma_{M}$ and by multiplication in $L(F)$

$\mathsf{k}{E^{1}}\otimes_{\mathsf{k}{E^{0}}}M\otimes_{\mathsf{k}{F^{0}}}L(F)\overset{\sigma_{M}\otimes\operatorname{id}}{\overset{\sim}{\longrightarrow}}M\otimes_{\mathsf{k}{F^{0}}}\mathsf{k}{F^{1}}\otimes_{\mathsf{k}{F^{0}}}L(F)\overset{\operatorname{id}\otimes\cdot}{\longrightarrow}M\otimes_{\mathsf{k}{F^{0}}}L(F)$

Similarly, the left $\mathsf{k}F^{0}$-module structure on $Q$ is promoted to a
left $L(F)$ module structure using $\sigma_{N}$. The composite

$P\otimes_{L(F)}Q=M\otimes_{\mathsf{k}{F^{0}}}N\otimes_{\mathsf{k}{E^{0}}}L(E)\overset{\omega_{E}\otimes\operatorname{id}_{L(E)}}{\longrightarrow}\mathsf{k}{E^{m+1}}\otimes_{\mathsf{k}{E^{0}}}L(E)\overset{\cdot}{\to}L(E).$

is clearly an ungraded isomorphism $(\mathsf{k}E^{0},L(E))$-bimodules.
Furthermore, using the commutativity of (8.11.10) one checks that it is also
left $L(E)$-linear. By symmetry, $\omega_{F}$ and (8.11.15) give an ungraded
$(L(F),L(F))$-bimodule isomorphism $Q\otimes_{L(E)}P\cong L(F)$. It follows
that $P\otimes_{L(F)}:\operatorname{Mod}L(F)\to\operatorname{Mod}L(E)$ is a
graded equivalence. By Theorem 8.6.3, this implies that
$L(E)\sim_{\operatorname{gr}}L(F)$. ∎

######  Proposition 8.11.16 ([recast]*Proposition 6.9).

Let $E$ and $F$ be finite regular graphs. If $E$ and $F$ are strong shift
equivalent, then the hypotheses of Theorem 8.11.5 are satisfied.

###### Proof.

We may assume there are matrices $R\in\mathbb{N}_{0}^{E^{0}\times F^{0}}$ and
$S\in\mathbb{N}_{0}^{F_{0}\times E^{0}}$ such that $RS=A$ and $SR=B$. Let
$M=M_{R}$ and $N=M_{S}$ as in the discussion above (8.11.2). Choose bimodule
isomorphisms

$\omega_{E}:M\otimes_{\mathsf{k}{F^{0}}}N\overset{\sim}{\longrightarrow}\mathsf{k}{E^{1}},\,\,\omega_{F}:N\otimes_{\mathsf{k}{E^{0}}}M\overset{\sim}{\longrightarrow}\mathsf{k}{F^{1}}.$

Set

$\sigma_{M}=(\operatorname{id}_{M}\otimes\omega_{F})\circ(\omega_{E}^{-1}\otimes\operatorname{id}_{M}),\,\sigma_{N}=(\operatorname{id}_{N}\otimes\omega_{E})\circ(\omega_{F}^{-1}\otimes\operatorname{id}_{F}).$

A straightforward calculation shows that

$\sigma_{M}\\#\sigma_{N}=\omega_{E}^{-1}\otimes\omega_{E},\,\,\sigma_{N}\\#\sigma_{M}=\omega_{F}^{-1}\otimes\omega_{F}.$

The proposition is now immediate. ∎

######  Remark 8.11.17.

A different compatibility condition for shift equivalence was considered by
Carlsen, Dor-On and Eilers in [lens]. They expressed their condition in terms
of the choices of bijections between $\mathsf{k}$-linear bases used above to
define the isomorphisms (8.11.2). In terms of the isomorphisms (8.11.2)
themselves, it boils down to requiring that

|  |
---|---|---
$\textstyle{\mathsf{k}E^{\ell}\otimes_{\mathsf{k}E^{0}}M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}\otimes\sigma_{M}}$$\textstyle{\mathsf{k}E^{\ell-1}\otimes_{\mathsf{k}E^{0}}M\otimes_{\mathsf{k}F^{0}}\mathsf{k}F^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}\otimes\sigma_{M}\otimes\operatorname{id}}$$\scriptstyle{\sigma_{M}\otimes\operatorname{id}}$$\textstyle{M\otimes_{\mathsf{k}F^{0}}\mathsf{k}F^{\ell}}$$\textstyle{M\otimes_{\mathsf{k}F^{0}}N\otimes_{\mathsf{k}E^{0}}M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\omega_{E}\otimes\operatorname{id}}$$\scriptstyle{\operatorname{id}\otimes\omega_{F}}$

(where $\ell$ is the lag of the shift equivalent) and the analogous triangle
with $F$ and $E$ and $N$ and $M$ switched, both commute. The authors show in
[lens]*Theorem 7.3 that the latter, as well as another related condition, are
both equivalent to requiring that $E$ and $F$ be strongly shift equivalent.

##  9\. Filtered $K$-theory, shift equivalence, and $C^{*}$-Morita equivalence

Following the early work of Rørdam [ror] and Restorff [restorff], it became
clear that one way to preserve enough information in the presence of ideals in
a $C^{*}$-algebra, is to further consider the $K$-groups of the ideals, their
subquotients, and how they are related to each other via the six-term
sequence. Over the next ten years since [ror, restorff] appeared, this
approach, which is now called _filtered $K$-theory_, was subsequently
investigated and further developed by Eilers, Restorff, Ruiz and Sørensen
[errs4, errs2], who showed that the lattice of gauge invariant prime ideals
and their subquotient $K$-groups can be used as an invariant. For a graph
$C^{*}$-algebra $C^{*}(E)$, we denote this invariant by
$\operatorname{FK}_{0,1}(C^{*}(E))$. In their major work [errs3], Eilers,
Restorff, Ruiz and Sørensen proved that filtered $K$-theory is a complete
invariant for unital graph $C^{*}$-algebras. Thus for finite graphs $E$ and
$F$,

$\operatorname{FK}_{0,1}(C^{*}(E))\cong\operatorname{FK}_{0,1}(C^{*}(F))\Longrightarrow
C^{*}(E)\cong_{\text{Morita eq.}}C^{*}(F).$ (9.1)

In [errs] the four authors introduced filtered $K$-theory in the purely
algebraic setting and showed that if two Leavitt path algebras over
$\mathbb{C}$ have isomorphic filtered algebraic $K$-theory then the associated
graph $C^{*}$-algebras have isomorphic filtered $K$-theory.

$\operatorname{FK}_{0,1}(L_{\mathbb{C}}(E))\cong\operatorname{FK}_{0,1}(L_{\mathbb{C}}(F))\Longrightarrow\operatorname{FK}_{0,1}(C^{*}(E))\cong\operatorname{FK}_{0,1}(C^{*}(F)).$
(9.2)

In [arahazli] it was shown that graded $K$-theory determines filtered
$K$-theory up to a certain precisely defined quotient.

$K_{0}^{\operatorname{gr}}(L_{\mathbb{C}}(E))\cong
K_{0}^{\operatorname{gr}}(L_{\mathbb{C}}(F))\Longrightarrow\operatorname{\mathrm{F}\overline{\mathrm{K}}}_{0,1}(L_{\mathbb{C}}(E))\cong\operatorname{\mathrm{F}\overline{\mathrm{K}}}_{0,1}(L_{\mathbb{C}}(F)).$
(9.3)

This shows the richness of the graded Grothendieck group as an invariant.
Namely, the single group $K_{0}^{\operatorname{gr}}(L(E))$ of the Leavitt path
algebra $L(E)$ associated to a graph $E$, with coefficients in a field
$\mathsf{k}$, contains all the information about the $K_{0}$ groups and the
quotient groups $\overline{K}_{1}$ of $K_{1}$ and how they are related via the
long exact sequence of $K$-theory. The important link established in
[arahazli], was that the existence of an isomorphism at the level of this
quotient filtered $K$-theory of Leavitt path algebras still implies the
existence of an isomorphism at the level of the corresponding filtered
$K$-theory of $C^{*}$-algebras (compare this with (9.2)):

$\operatorname{\mathrm{F}\overline{\mathrm{K}}}_{0,1}(L_{\mathbb{C}}(E))\cong\operatorname{\mathrm{F}\overline{\mathrm{K}}}_{0,1}(L_{\mathbb{C}}(F))\Longrightarrow\operatorname{FK}_{0,1}(C^{*}(E))\cong\operatorname{FK}_{0,1}(C^{*}(F)).$
(9.4)

Using these connections one can prove the following proposition.

######  Proposition 9.5 ([arahazli]).

Let $E$ and $F$ be finite regular graphs. If $E$ and $F$ are shift equivalent,
then the $C^{*}$-algebras $C^{*}(E)$ and $C^{*}(F)$ are Morita equivalent.

###### Proof.

Since $E$ and $F$ are shift equivalent, we have an isomorphism of Krieger’s
dimension groups

$(\Delta_{E},\Delta_{E}^{+},\delta_{E})\cong(\Delta_{F},\Delta_{F}^{+},\delta_{F}).$

But by Corollary 8.7.1, Krieger’s dimension group for the graph $E$ coincides
with the graded Grothendieck group $K_{0}^{\operatorname{gr}}(L(E))$. We thus
obtain an order-preserving $\mathbb{Z}[x,x^{-1}]$-module isomorphism
$K_{0}^{\operatorname{gr}}(L(E))\cong K_{0}^{\operatorname{gr}}(L(F))$. By
(9.2) and (9.4), the filtered $K$-theory of the corresponding graph
$C^{*}$-algebras are isomorphic. Now (9.1) gives that the $C^{*}$-algebras
$C^{*}(E)$ and $C^{*}(F)$ are Morita equivalent. ∎

The following diagram summarizes what is known and what could be considered as
conjectural links.

|  |  |  |  |  |  |  |
---|---|---|---|---|---|---|---|---
|  |  |  |  |  |  |  |
|  |  |  |
|  |  |  |
$\textstyle{A_{E}\cong_{\text{shift
eq.}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces
A_{F}}$$\textstyle{{\color[rgb]{1,0,0}L(E)\approx_{\text{gr Morita
eq}}L(F)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{\bf
D}_{\operatorname{sgn}}(A(E))\cong\operatorname{\bf
D}_{\operatorname{sgn}}(A(F))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\color[rgb]{1,0,0}K_{0}^{\operatorname{gr}}(L(E))\cong\ignorespaces\ignorespaces\ignorespaces\ignorespaces
K_{0}^{\operatorname{gr}}(L(F))}{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bf
Conjecture~{}\ref{conjmogr}}$$\textstyle{\Delta_{E}\approx\Delta_{F}{\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}$$\textstyle{T_{E}\approx_{\mathbb{Z}}T_{F}}$$\textstyle{\operatorname{\mathrm{F}\overline{\mathrm{K}}}_{0,1}(L(E))\cong\ignorespaces\ignorespaces\ignorespaces\ignorespaces\operatorname{\mathrm{F}\overline{\mathrm{K}}}_{0,1}(L(F))}$$\textstyle{\operatorname{FK}_{0,1}(C^{*}(E))\cong\ignorespaces\ignorespaces\ignorespaces\ignorespaces\operatorname{FK}_{0,1}(C^{*}(F))}$$\textstyle{C^{*}(E)\cong_{\text{Morita
eq}}C^{*}(F)}$

## References