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\title[Status of graded conjecture of graphs algebras] {The Graded Grothendieck group as a classification tool for algebras,\\ current status}

\begin{document}
\maketitle

It is conjectured \cite{hazrat2} that the graded Grothendieck group is a complete invariant for Leavitt path algebras. This was established for the class of Leavitt path algebras associated to polycephaly graphs \cite{hazrat2}. One can write the analytic version of this conjecture and it is hoped that it would be easier to check the analytic version for graph $C^*$-algebras. In following we give a brief summary of the current status of the program. All Leavitt path algebras are defined over an arbitrary field. 


\begin{conjecture}\label{conjalg}
Let $E$ and
$F$ be finite graphs. Then there is an order preserving $\mathbb Z[x,x^{-1}]$-module
isomorphism \[\phi: K_0^{\gr}(\L(E)) \longrightarrow K_0^{\gr}(\L(F))\] with 
$\phi([\L(E)]=[\L(F)]$ 
if and only if
$\L(E)\cong_{\gr} \L(F).$
\end{conjecture}

Here the ordered $\mathbb Z[x,x^{-1}]$-module isomorphism $K_0^{\gr}(\L(E)) \cong K_0^{\gr}(\L(F))$ should give that these algebras are graded Morita equivalent (see the diagram below). 

\medskip 

Denote by $\gamma_E$ the gauge circle actions on $C^*(E)$ and $K^\T_0(C^*(E))$ the equivaraint $K$-theory of $C^*(E)$.  There are canonical ordered isomorphisms  of $\Z[x,x^{-1}]$-modules (see~\cite{haziso})

\begin{equation}\label{copenhag}
K^{\gr}_0(L(E)) \cong K_0(L(E \times \mathbb Z)) \cong  K_0(C^*(E \times \mathbb Z))\cong K_0^{\mathbb T}(C^*(E)).
\end{equation}

Thus one can pose the analytic version of Conjecture~\ref{conjalg} as follows. 

\begin{conjecture}\label{conjanal}
Let $E$ and $F$ be finite graphs. Then there is an order preserving $\mathbb Z[x,x^{-1}]$-module
isomorphism
\[ \phi: K^\T_0(C^*(E)) \longrightarrow K_0^\T(C^*(F)),\] 
with 
$\phi([C^*(E)])=[C^*(F)]$ 
 if and only if
 $C^*(E) \cong C^*(F)$ which respect the gauge action. 
\end{conjecture}

In fact in Conjecture~\ref{conjalg} if the graded Grothendieck groups are isomorphic, one should have the isomorphism between the Leavitt path algebras are indeed $*$-isomorphism. If this is the case, then Conjecture~\ref{conjalg} implies Conjecture~\ref{conjanal}. For, if $K^\T_0(C^*(E)) \cong K_0^\T(C^*(F))$, then by (\ref{copenhag}), $K_0^{\gr}(\L(E)) \cong K_0^{\gr}(\L(F))$, so $\L(E)\cong_{\gr} \L(F)$ as $*$-isomorphism. This implies that $C^*(E) \cong C^*(F)$ which respects the gauge action. (\cite[Theorem~4.4]{abramstomforde}).

\medskip 

\noindent We can summarise the current status of the conjectures in the following diagram: 


Notations:

$\bullet$ $E$ and $F$ are finite graphs, $A_E$ and $A_F$ the adjacency matrices and $X_E$ and $X_F$ are associated shift of finite types. 

$\bullet$  SSE stands for  {\it strongly shift equivalent}; 
SE stands for {\it shift equivalent}. 

$\bullet$ $D(X(E))=(\Delta_A, \Delta_A^+,\delta_A)$ is the Krieger's dimension group associated to the matrix $A$. 

$\bullet$  $\cong_{\gr}$ denotes the graded isomorphism; $\approx_{\gr}$ denoted the graded Morita equivalent. 
\begin{equation*}
\xymatrix@=3pt{
 &&&&&&  \scriptstyle{
 D(X(E)) \approx   D(X(F))} \ar@{<->}[ddd]^{\text{\tiny [1], [4]}}    \ar@{<->}[rr]^{\text{\tiny{Krieger}}} && \scriptstyle{A_E \approx_{\operatorname{SE}} A_F} \\ 
 \\
 \\
\scriptstyle{X_E \cong X_F} \ar@{<->}[rr]^{\text{\tiny Williams}}  &&  \scriptstyle{A_E \approx_{\operatorname{SSE}} A_F}  \ar@{->}[rr]^{\text{\tiny in/out splitting}}  \ar@{->}[ddd]|X^{\text{does not imply}} && \scriptstyle{\LL(E) \approx_{\gr} \LL(F)}  \ar@{->}[rr]
 &&  \scriptstyle{\textcolor{blue}{K_0^{\gr}(\LL(E)) \cong K_0^{\gr}(\LL(F))}}  \ar@/^1.5pc/@{.>}[ll]_{\bf ?}
\\
 \\
 \\
 &&\scriptstyle{\LL(E)  \cong_{\gr} \LL(F)}  &  &&& \scriptstyle{QGr P(E) \approx   QGr P(F) \ar@{<->}[uuu]_
 {\text{\tiny{as ordered group, [4]}}}}
 }
\end{equation*}

\medskip 

In the diagram above, it should be possible to replace $K_0^{\gr}$ by $K_0^{\T}$ and $L(E)$ and $L(F)$ by $C^*(E)$ $C^*(F)$ and all statements would still hold (see Conjecture 2). 



\begin{thebibliography}{10}


\bibitem{abramstomforde}  G. Abrams,  M. Tomforde, {\it Isomorphism and Morita equivalence of graph algebras}, Trans. Amer. Math. Soc. {\bf 363} (2011), no. 7, 3733--3767. 

\bibitem{arapardo} P. Ara, E. Pardo, {\it Towards a K-theoretic characterization of graded isomorphisms between Leavitt path algebras,}  J. K-Theory {\bf 14} (2014), no. 2, 203--245.

\bibitem{hazrat2} R. Hazrat, {\it The graded Grothendieck groups and the classification of Leavitt path algebras,}
Math Annalen, {\bf 355} (2013), no. 1, 273--325.


\bibitem{hazrat1} R. Hazrat, {\it The dynamics of Leavitt path algebras}, Journal of Algebra, {\bf 384} (2013) 242--266.


\bibitem{haziso} R. Hazrat, {\it A note on the isomorphism conjectures for Leavitt path algebras,}
Journal of Algebra $\mathbf{375}$ (2013) 33--40. 

\bibitem{krieger} W. Krieger, {\it On dimension functions and topological Markov chains}, Invent. Math. {\bf 56} (1980), no. 3, 239--250.



\end{thebibliography}

\end{document}