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\begin{document}

 \PageNum{957}

 \Volume{2012}{May}{28}{5}
 \OnlineTime{December 7, 2011}
 \DOI{10.1007/s10114-011-0106-8}
 \EditorNote{Received February 5, 2010, accepted February 22, 2011}

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\AuthorMark{Aranda Pino G., et al.}
               %%%  appear on the head of even pages  %%%

 \TitleMark{$\ast$-Regular Leavitt Path Algebras of Arbitrary Graphs}
           %%%  appear on the head of odd pages  %%%

\title{$\bm{\ast}$-Regular Leavitt Path Algebras of Arbitrary Graphs        %%%   Main Title of your paper  %%%
\footnote{The first author is partially supported by the Spanish
MEC and Fondos FEDER through project MTM2007-60333, and by the
Junta de Andaluc\'{\i}a and Fondos FEDER, jointly, through projects FQM-336 and FQM-2467}}                                      %%%   the Fund which you are supported by  %%%

\author{Gonzalo \uppercase{Aranda Pino}}
         %%%  1st Author's information   %%%
    {Departamento de \'Algebra, Geometr\'\i a y Topolog\'\i a, Universidad de M\'alaga,
 $29071$ M{\'a}laga, Spain\\
    E-mail\,$:$ g.aranda@uma.es}

\author{Kulumani \uppercase{Rangaswamy}}     %%%  2nd Author's info, if exists, or you may delete this part directly  %%%
    {Department of Mathematics, University of Colorado, Colorado Springs,
 CO $80933$, USA\\
    E-mail\,$:$ ranga@uccs.edu}

\author{Lia \uppercase{Va\v{s}}}     %%%  3rd Author's info,
  {Department of Mathematics, Physics and Statistics, University of the Sciences in
Philadelphia, Philadelphia, PA $19104$, USA\\
    E-mail\,$:$ l.vas@usp.edu}

\maketitle%


\Abstract{If $K$ is a field with involution and $E$ an arbitrary
graph, the involution from $K$ naturally induces an involution of
the Leavitt path algebra $L_K(E).$ We show that the involution on
$L_K(E)$ is proper if the involution on $K$ is positive-definite,
even in the case when the graph $E$ is not necessarily finite or
row-finite. It has been shown that the Leavitt path algebra
$L_K(E)$ is regular if and only if $E$ is acyclic. We give
necessary and sufficient conditions for $L_{K}(E)$ to be
$\ast$-regular (i.e., regular with proper involution). This
characterization of $\ast$-regularity of a Leavitt path algebra is
given in terms of an algebraic property of $K,$ not just a
graph-theoretic property of $E.$ This differs from the known
characterizations of various other algebraic properties of a
Leavitt path algebra in terms of graph-theoretic properties of $E$
alone. As a corollary, we show that Handelman's conjecture
(stating that every $\ast$-regular ring is unit-regular) holds for
Leavitt path algebras. Moreover, its generalized version for rings
with local units also continues to hold for Leavitt path algebras
over arbitrary graphs.}
       % the abstract




 \Keywords{Leavitt path algebra, $*$-regular, involution,
arbitrary graph}        % the keywords

 \MRSubClass{16D70, 16W10, 16S99}      % MR(2000) Subject Classification

%\baselineskip 15pt

\section{Introduction}

Leavitt path algebras can be regarded as the algebraic
counterparts of the graph $C^*$-algebras, the descendants from the
algebras investigated by  Cuntz in \cite{Cuntz}. Leavitt path
algebras can also be viewed as a broad generalization of the
algebras constructed by  Leavitt in \cite{Leavitt} to produce
rings without the Invariant Basis Number property.


The Leavitt path algebra $L_K(E)$ was introduced in the papers
\cite{AA1} and \cite{AMP}. The algebra  $L_K(E)$ was first defined
for a row-finite graph $E$ (countable graph such that every vertex
emits only a finite number of edges) and a field $K$. Although
their history is very recent, a flurry of activity has followed
the papers \cite{AA1} and \cite{AMP}. The main directions of
research include: characterization of algebraic properties of a
Leavitt path algebra $L_K(E)$ in terms of graph-theoretic
properties of $E$; study of the modules over $L_K(E)$; computation
of various substructures (such as the Jacobson radical, the
center, the socle and the singular ideal); investigation of the
relationship and connections with $C^*(E)$ and general
$C^*$-algebras; classification programs; study of the $K$-theory;
and generalization of the constructions and results first from
row-finite to countable graphs and finally, from countable to
completely arbitrary graphs.  For examples of each of these
directions see \cite{AA1, APS, AMMS1, AT, AbAnhLouP, ABC,
Tomforde, G}.

The base field $K$ is naturally endowed with an involution $^-$
(the identity involution can always be considered in the absence
of other possibilities). A given involution on $K$ naturally
induces an involution of a Leavitt path algebra $L_K(E).$ The
presence of an involution on a ring yields some favorable
features: the ring is isomorphic to its opposite ring and a
certain dose of symmetry is present. For example, a left Rickart
$\ast$-ring is also a right Rickart $\ast$-ring while this is not
the case for one-sided Rickart rings. Also, consideration of a
complex Leavitt path algebra $L_{\mathbb{C}}(E)$ as an algebra
with involution (induced from the complex-conjugate involution on
$\mathbb{C}$), brings $L_{\mathbb{C}}(E)$ a step closer to its
analytic counterpart $C^*(E).$ These facts justify our interest in
the study of the involution on a Leavitt path algebra.

Raeburn \cite{Iain} showed that the involution on $L_K(E)$ is
proper if the involution on $K$ is positive-definite and $E$ is a
row-finite countable graph without sinks. We extend this result to
arbitrary graphs (Proposition
\ref{involutionproperarbitrarygraphs}). We also show that the
converse holds: if the induced involution on $L_K(E)$ is
positive-definite for every (equivalently some) graph $E$, then
the involution on $K$ is positive-definite (Proposition
\ref{moreimplications}).

In \cite{AR}, Abrams and Rangaswamy characterized the (von
Neumann) regular Leavitt path algebras $L_K(E)$ as precisely those
with acyclic underlying graphs $E.$ In light of our consideration
of the involution on $L_K(E),$ we wonder when is $L_K(E)$
$*$-regular (regular with proper involution). In Theorem
\ref{characterizationtheorem}, we characterize $*$-regular Leavitt
path algebras as exactly those with $E$ acyclic and $K$ that is
proper up to a certain extent (determined by the least upper bound
of the numbers of all paths that end at any given vertex of $E$).
Note that we do not impose any conditions on the cardinality of
$E$: we work with completely arbitrary graphs.

Most existing characterization theorems for Leavitt path algebras have the following form:
$$
\mbox{\em $L_K(E)$ has $($ring-theoretic$)$ property $(P)$ if and
only if $E$ has $($graph-theoretic$)$ property $(P').$}
$$
Such theorems have been formulated and proven for a good number of
ring-theoretic properties. For example, simple, purely infinite
simple, exchange, semisimple, regular and other Leavitt path
algebras have been characterized. It is interesting that the
underlying field $K$ did not play any role in those
characterization theorems. Theorem \ref{characterizationtheorem},
however, has a different form:
$$
\mbox{\em $L_K(E)$ has property $(P)$ if and only if  $E$ has
property $(P')$ {\em and} $K$ has property $(P'').$ }
$$
In other words, Theorem \ref{characterizationtheorem} is the first
characterization theorem that involves a ring-theoretic property
of the field $K$ as well. Moreover, the characterization of
$L_K(E)$ that is positive-definite (Proposition
\ref{moreimplications}) also has the above form that features the
field $K$ as well.

The paper is organized as follows. In Section 2 we recall the
basic definitions, examples and properties of Leavitt path
algebras, whereas in Section 3 we focus on the involution of
$L_K(E)$ and prove Propositions
\ref{involutionproperarbitrarygraphs} and \ref{moreimplications}.
We devote Section 4 to the proof of the characterization theorem
for the $\ast$-regular Leavitt path algebras (Theorem
\ref{characterizationtheorem}). Finally, in Section 5 we consider
\cite[Problem 48, p.\,380]{G} that we shall refer to ``Handelman's
conjecture''. This conjecture is stating that every $\ast$-regular
ring is unit-regular. We prove that Handelman's conjecture holds
for Leavitt path algebras. In fact, we formulate a generalized
version of the conjecture for rings with local units and show that
it holds for Leavitt path algebras over arbitrary graphs.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Definitions and Preliminaries}

We recall some graph-theoretic concepts, the definition and standard examples of Leavitt path algebras.

A (\emph{directed}) \emph{graph} $E=(E^0,E^1,r,s)$ consists of two sets $E^0$ and $E^1$ (with no restriction
on their cardinals) together with maps $r,s:E^1 \to E^0$. The elements of $E^0$ are called \emph{vertices}
and the elements of $E^1$ \emph{edges}. For $e\in E^1$, the vertices $s(e)$ and $r(e)$ are called the
\emph{source} and \emph{range} of $e$. If $s^{-1}(v)$ is a finite set for every $v\in E^0$, then the graph is
called \emph{row-finite}. If $E^0$ is finite and $E$ is row-finite, then $E^1$ must necessarily be finite as
well; in this case we say simply that $E$ is \emph{finite}.

A vertex which emits (receives) no edges is called a \emph{sink
$($source$)$}. A vertex $v$ is called an \emph{infinite emitter}
if $s^{-1}(v)$ is an infinite set. A \emph{path} $\mu$ in a graph
$E$ is a finite sequence of edges $\mu=e_1\cdots e_n$ such that
$r(e_i)=s(e_{i+1})$ for $i=1,\ldots,n-1$. In this case,
$s(\mu)=s(e_1)$ and $r(\mu)=r(e_n)$ are the \emph{source} and
\emph{range} of $\mu$, respectively, and $n$ is the \emph{length}
of $\mu$. We view the elements of $E^{0}$ as paths of length $0$.



If $\mu$ is a path in $E$, with $v=s(\mu)=r(\mu)$ and $s(e_i)\neq s(e_j)$ for every $i\neq j$, then $\mu$ is
called a \emph{cycle}. A graph which contains no cycles is called \emph{acyclic}.


Let $K$ denote an arbitrary base field and $E$ an arbitrary graph.
The {\em Leavitt path $K$-algebra} $L_K(E)$ is the $K$-algebra
generated by the set $E^0\cup E^1\cup \{e^*\,|\, e\in E^1\}$ with
the following relations:

{\rm(V)} $vw= \delta_{v,w}v$ for all $v,w\in E^0$.

{\rm(P1)} $s(e)e=er(e)=e$ for all $e\in E^1$.

{\rm(P2)} $r(e)e^*=e^*s(e)=e^*$ for all $e\in E^1$.

{\rm(CK1)} $e^*f=\delta _{e,f}r(e)$ for all $e,f\in E^1$.

{\rm(CK2)} $v=\sum _{e\in s^{-1}(v)}ee^*$ for every $v\in E^0$ that is neither a sink nor an infinite
emitter.

The first three relations are the path algebra relations. The last
two are the so-called Cuntz--Krieger relations.

We let $r(e^*)$ denote $s(e)$, and we let $s(e^*)$ denote $r(e)$.
If $\mu = e_1 \cdots e_n$ is a path in $E$, we write $\mu^*$ for
the element $e_n^* \cdots e_1^*$ of $L_{K}(E)$. With this
notation, the Leavitt path algebra $L_{K}(E)$ can be viewed as a
$K$-vector space span of $\{pq^{\ast } \,|\,p,q\,  \hbox{are paths
in} \,  E\}$. (Recall that the elements of $E^0$ are viewed as
paths of length $0$, so that this set includes elements of the
form $v$ with $v\in E^0$.)

If $E$ is a finite graph, then $L_{K}(E)$ is unital with $\sum_{v\in E^0} v=1_{L_{K}(E)}$; otherwise,
$L_{K}(E)$ is a ring with a set of local units consisting of sums of distinct vertices of the graph.

Many well-known algebras can be realized as the Leavitt path algebra of a graph. The most basic graph
configurations are shown below (the isomorphisms for the first three can be found in \cite{AA1}, the fourth
in \cite{S}, and the last one in \cite{AA2}).

\begin{example}\label{examples}{\rm The ring of Laurent polynomials $K[x,x^{-1}]$ is the Leavitt path
algebra of the graph given by a single loop graph. Matrix algebras
${\mathbb M}_n(K)$ can be realized by the line graph with $n$
vertices and $n-1$ edges. Classical Leavitt algebras $L(1,n)$ for
$n\geq 2$ can be obtained by the $n$-rose --- a graph with a
single vertex and $n$ loops. Namely, these three graphs are
\vspace{3mm}$$\begin{matrix} \xymatrix{{\bullet} \ar@(ur,ul)}
\hskip10mm & \xymatrix{{\bullet} \ar [r]  & {\bullet} \ar [r]  &
{\bullet} \ar@{.}[r] & {\bullet} \ar [r]  & {\bullet} } \hskip10mm
& \xymatrix{{\bullet} \ar@(ur,dr)  \ar@(u,r) \ar@(ul,ur)  \ar@{.}
@(l,u) \ar@{.} @(dr,dl) \ar@(r,d) & }
\end{matrix}$$

\vspace{4mm}

The algebraic counterpart of the Toeplitz algebra $T$ is the
Leavitt path algebra of the graph having one loop and one exit:
\vspace{0.8mm}$$\xymatrix{{\bullet} \ar@(dl,ul) \ar[r] &
{\bullet}}$$

Combinations of the previous examples are possible. For instance, the Leavitt path algebra of the graph

$$\xymatrix{{\bullet} \ar [r]  & {\bullet} \ar [r]  & {\bullet}
\ar@{.}[r] & {\bullet} \ar [r]  & {\bullet}
 \ar@(ur,dr)  \ar@(u,r)  \ar@(ul,ur)  \ar@{.} @(l,u) \ar@{.} @(dr,dl) \ar@(r,d) & }$$
\vspace{0.2mm}

\noindent is ${\mathbb M}_n(L(1,m))$, where $n$ denotes the number
of vertices in the graph and $m$ denotes the number of loops. }
\end{example}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The Involution on a Leavitt Path Algebra}

We recall some standard definitions first.

An \emph{involution} $\ast$ on a ring $R$ is an additive map
$^\ast:R\rightarrow R$ that satisfies $(ab)^{\ast
}=b^{\ast}a^{\ast }$ and $(a^{\ast })^{\ast }=a$ for all $a,b\in
R.$ For any $a\in R$, the element $a^{\ast }$ is called the
\emph{adjoint} of $a$. An element $p$ in a ring with involution
$\ast$ is called a \emph{projection} if $p$ is a self-adjoint
$(p^{\ast }=p)$ idempotent $(p^{2}=p)$.

If there is an involution defined on a ring $R,$ then $R$ is said
to be a {\em $\ast$-ring}. If $R$ is also an algebra over $K$ with
an involution $^-,$ then $R$ is a {\em $\ast$-algebra} if
$(ax)^*=\overline{a}x^*$ for $a\in K,$ and $x\in R.$

Let $n$ be a positive integer. An involution $\ast$ on a ring $R$
is said to be \emph{n-proper} if
$$x_{1}^{\ast}x_{1}+\cdots+x_{n}^{\ast}x_{n}=0 \text{ implies } x_{1}=\cdots =x_{n}=0$$ for any $n$
elements $x_{1},\ldots,x_{n}$ in $R$. A ring with an $n$-proper
involution will be referred to as an $n$-\emph{proper} ring. This
property is clearly left-right symmetric, since each element
$x_{i}=a_{i}^{\ast}$ for some $a_{i}\in R$. A 1-proper involution
is simply said to be {\em proper}.

The involution $\ast$ is said to be \emph{positive-definite} if it
is $n$-proper for every positive integer $n.$ A $\ast$-ring with a
positive-definite involution will be referred to as a {\em
positive-definite} ring.

A field can have both an $n$-proper involution and an involution
that is not $n$-proper. For example, consider the field
$\mathbb{C}$: it is 2-proper (in fact positive-definite) for the
conjugate involution ($a+{\rm i}b\mapsto a-{\rm i}b$) and not
2-proper for the identity involution.  Also, the same involution
can be $n$-proper and not $(n+1)$-proper (identity involution on
$\mathbb{C}$ for $n=1$).

Before we turn to Leavitt path algebras, let us recall one last
fact about general $\ast$-rings. Recall that if $R$ is a ring with
involution $^-$, then the involution $^-$ induces the
$\ast$-\emph{transpose involution} on the ring ${\mathbb
M}_{n}(R)$ of $n\times n $ matrices over $R$ given by
$$A=(a_{ij})\mapsto A^*=(\overline{a_{ji}}).$$

We believe that the following lemma is well known but we are not aware of a reference for it. For
completeness, we provide a proof here.

\begin{lemma}\label{R_n-proper}
Let $n$ be a positive integer and let $R$ be a ring with
involution $^-$. Then the $\ast$-transpose involution on ${\mathbb
M}_{n}(R)$ is proper if and only if the involution $^-$ in $R$ is
$n$-proper.
\end{lemma}
\begin{prof}
Assume that the involution $^-$ is $n$-proper in $R$. Suppose
$A^*A=0$ for some matrix $A=(a_{ij})\in {\mathbb M}_{n}(R)$.  Then
the diagonal entries of the product $A^*A$ are zero and so
$\sum_{j=1}^{n}\overline{a_{ji}}a_{ji}=0$ for every
$i=1,\ldots,n$. Since $^-$ is $n$-proper, $a_{ij}=0$ for all
$i,j$. Hence $A=0.$

Conversely, suppose $\sum_{i=1}^{n}\overline{a_{i}}a_{i}=0$ for
$a_{i}\in R $. Consider $A$ to be the matrix of ${\mathbb
M}_{n}(R)$ that has the elements $a_{1},\ldots,a_{n}$ in its first
column and zeroes in the rest of its columns. Then $A^*A=0$. Since
the $\ast$-transpose involution is proper, $A=0.$ So $a_{i}=0$ for
every $i$.
\end{prof}

We turn now to Leavitt path algebras. Let $K$ be a field with
involution $^-$ and let $E$ be an arbitrary graph. Recall that a
typical element of the Leavitt path algebra $L_{K}(E)$ can be
written as $\sum_{i=1}^{n}k_{i}p_{i}q_{i}^{\ast}$ where $p_{i}$
and $q_{i}$ are paths and $k_i\in K.$ It is straightforward to see
that the map $\ast$ given by
$$\bigg(\sum_{i=1}^{n}k_{i}p_{i}q_{i}^{\ast}\bigg)^{\ast}=
\sum_{i=1}^{n}\overline{k_i}q_{i}p_{i}^{\ast}$$ defines the
involution on $L_{K}(E)$ making it into a $\ast$-algebra.

If $K$ is the field of complex numbers $\mathbb{C}$ and we
consider the conjugate involution, the $\ast$-algebra structure of
$L_{\mathbb{C}}(E)$ agrees with the $^{\ast}$-algebra structure of
$L_{\mathbb{C}}(E)$ (that is used for instance to see
$L_{\mathbb{C}}(E)$ as a dense $*$-subalgebra of $C^*(E)$ as shown
in \cite[Theorem 7.3]{Tomforde}).

As we will see in the next section, the $*$-regularity of a
Leavitt path algebra $L_K(E)$ is closely related to the condition
stating that the involution of $L_K(E)$ is $n$-proper or
positive-definite. In this section, we characterize the
positive-definiteness of $L_K(E)$.

The following proposition can be proved by easily adapting
Raeburn's result \cite[Lemma 1.3.1]{Iain} to our notation and
context.

\begin{proposition}\label{Iain} Let $E$ be a row-finite, countable graph without sinks. If the involution
$^-$ on $K$ is positive-definite, then the involution $\ast$ on
$L_K(E)$ is proper.
\end{proposition}
\begin{prof}
The proof follows completely \cite[Lemma 1.3.1]{Iain}. One only
needs to take into account that the axioms in \cite{Iain} are
given so that a path $e_1e_2\cdots e_n$ in  \cite{Iain}
corresponds to the path $e_ne_{n-1}\cdots e_1$ here (i.e., the
edges in \cite{Iain} are multiplied so that the action of $f$
precedes the action of $e$ in the product $ef,$ contrary to the
action we consider here). Because of this difference, the
assumptions of \cite[Lemma 1.3.1]{Iain} that $L_K(E)$ is
column-finite with no sources, correspond exactly to our
assumptions that $L_K(E)$ is row-finite with no sinks.
\end{prof}

It is noted in \cite{Iain} that the condition that $E$ does not
have sinks (sources, in the terminology of that paper) can be
avoided by using the so-called Yeend's trick. This assumption can
also be avoided using a technique called the Desingularization
Process. The added benefit of the desingularization is that it can
help us also get rid of the row-finiteness assumption. The
desingularization of a graph $E$ is a new graph $F$ obtained by
adding a tail (more details can be found in  \cite{DT} or
\cite{AA3}) to every sink or infinite emitter. The resulting graph
$F$ is a row-finite graph without sinks such that $L_K(E)$ embeds
in $L_K(F)$ via an embedding that is a $*$-algebra homomorphism
(see \cite[Proposition 5.1]{AA3} for more details).

Finally, the last remaining assumption (that the graph is countable) in Proposition \ref{Iain} can be avoided
by means of the Subalgebra Construction (see \cite{AR} for more details). We recall here the relevant concept
$E_F$ used in this construction. We shall use $E_F$ in our main theorem too.

Let $F$ be a finite set of edges in $E$. We define $s(F)$ (resp. $r(F)$) to be the sets of those vertices in
$E$ that appear as the source (resp. range) vertex of at least one element of $F$. The graph $E_F$ is then
defined as follows (see \cite[Definition 2]{AR}):
\begin{align*}
&E_F^0=F\cup (r(F)\cap s(F)\cap s(E^1\setminus F))\cup
(r(F)\setminus s(F)),\\
&E_F^1=\{(e,f)\in F\times E^0_F \,|\,r(e)=s(f)\}\cup
\{(e,r(e))\,|\,e\in F \text{ with } r(e)\in (r(F)\setminus
s(F))\},\end{align*} where $s((x,y))=x \text{ and }
r((x,y))=y\text{ for any }(x,y)\in E_F^1.$

 The graph $E_F$ is
finite (see the comment after \cite[Definition 2]{AR}). Also, by
\cite[Proposition~1]{AR}, there is an algebra homomorphism
$\theta: L_K(E_F)\rightarrow L_K(E).$ Furthermore,
\cite[Proposition~2]{AR} shows that for every finite set of
elements $S$ of $L_K(E),$ there is a subalgebra $B(S)$ of $L_K(E)$
containing $S.$ The subalgebra $B(S)$ is of the form
$L_{K}(E_{F})\oplus (\bigoplus_{i=1}^m Kx_{i})$ where $F$ is a
finite set of edges defined using $S$ (see \cite[p.\,7]{AR}) and
$x_i, i=1,\ldots, m$ is a finite set of vertices (defined   in
\cite[p.\,8]{AR}). By \cite[Proposition 2]{AR}, $L_{K}(E)$ is a
directed union of subalgebras $B(S)$, where the $S$ varies over
all finite subsets of $L_K(E)$. Furthermore, and key to our
current discussion, $\theta$ preserves the involution by the
construction (as it can be seen from the proof of
\cite[Proposition~1]{AR}) so it is a $\ast$-algebra homomorphism.

\begin{proposition}\label{involutionproperarbitrarygraphs} Let $E$ be an arbitrary graph. If the involution
$^-$ on $K$ is positive-definite, then the involution $*$ on
$L_{K}(E)$ is proper.
\end{proposition}
\begin{prof}
We prove the claim first for the case when $E$ is a countable. Let
$F$ be a desingularization of $E.$ By the Desingularization
Process, $F$ is a row-finite graph without sinks and there is a
$\ast$-algebra monomorphism $\phi:L_{K}(E)\to L_{K}(F).$ Now,
suppose that $a^*a=0$ in $L_{K}(E)$. Apply $\phi$ to get that
$\phi(a)^*\phi(a)=0$ in $L_{K}(F).$ Since $F$ is row-finite and
does not contain sinks, Proposition~\ref{Iain} can be applied and
so $\phi(a)=0.$ Then $a=0$ since $\phi$ is a monomorphism.

\par
Now suppose that $E$ is arbitrary and let $a\in L_{K}(E)$ be such
that $a^*a=0$. By the Subalgebra Construction, for the finite set
$S=\{a, a^*\},$ there is a finite set of edges $F$ and a finite
number of vertices $x_i, i=1,\ldots, m$ such that the subalgebra
$B(S)$ of $L_K(E)$ is of the form $L_{K}(E_{F})\oplus
(\bigoplus_{i=1}^m Kx_{i})$ and $a,a^*\in B(S)$. Since
$\bigoplus_{i=1}^m Kx_{i}$ is a direct summand in the previous
equation for $B(S)$, we can actually attach a finite number of
isolated vertices $v_{1},\ldots,v_{m}\not\in E_{F}^0$ to the graph
$E_{F}$  so that we obtain a new finite graph $G$ such that
$$L_K(G)\cong L_{K}(E_{F})\oplus \bigg(\bigoplus_{i=1}^m Kv_{i}\bigg)$$ via a $*$-algebra isomorphism.

Since $B(S)$ is a subalgebra of $L_{K}(E)$, the equation $a^*a=0$
holds in $B(S)\cong L_{K}(G)$. Apply the previous case to
$L_{K}(G)$ in order to deal with possible sinks in $G.$ Then we
have $a=0$. This finishes the proof.
\end{prof}



The last result of this section is a characterization of Leavitt
path algebras which have positive-definite involutions in terms of
the corresponding property in the field $K$.

\begin{proposition}\label{moreimplications} Let $K$ a field with involution. The following conditions
are equivalent\,$:$

{\rm(i)} The involution on $K$ is positive-definite.

{\rm(ii)} The involution on $L_{K}(E)$ is positive-definite for
every graph $E$.

{\rm(iii)} The involution on $L_{K}(E)$ is positive-definite for
some graph $E$.

Thus, if $E$ is an arbitrary graph, $L_{K}(E)$ is
positive-definite if and only if $K$ is positive-definite.
\end{proposition}
\begin{prof}
(i) $\Rightarrow$ (ii) Given $E,$ let us consider the graph $M_nE$
obtained from $E$ by attaching a line of length $n-1$ to every
vertex of $E$ so that each line ends at the given vertex of the
graph (see more details in \cite{AT}). The graph $M_nE$ has the
property that ${\mathbb M}_n(L_K(E))$ is isomorphic to $L_K(M_nE)$
as $\ast$-algebras by \cite[Proposition 9.3]{AT}.

By Proposition \ref{involutionproperarbitrarygraphs}, we know that
$L_K(M_nE)$ is $*$-proper. So, ${\mathbb M}_n(L_K(E))$ is
$*$-proper. Now apply Lemma \ref{R_n-proper} to get that $L_K(E)$
is $n$-proper.

\par

(ii) $\Rightarrow$ (iii) is a tautology.

\par

(iii) $\Rightarrow$ (i) Suppose that $\sum_{i=1}^n
\overline{k_{i}}k_{i}=0$ for $k_{i}\in K$. Let $E$ be a graph such
that the involution on $L_{K}(E)$ is positive-definite. Let $v\in
E^0$. Since $v$ is a projection then $0=(\sum_{i=1}^n
\overline{k_{i}}k_{i})v=\sum_{i=1}^n(k_{i}v)^* (k_{i}v) $ and
therefore $k_{i}v=0$ for all $i$ by hypothesis. But $E^0$ is a set
of linearly independent elements in $L_{K}(E)$ by \cite[Lemma
1.5]{G}, so that $k_{i}=0$ for all $i$, as needed.
\end{prof}




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{$\bm{*}$-Regular Leavitt Path Algebras}


The (von Neumann) regular Leavitt path algebras $L_K(E)$ were
characterized in \cite{AR} as those whose graphs $E$ are acyclic.
In light of the consideration of $L_K(E)$ as a ring with
involution, we wonder which acyclic graphs have $L_K(E)$ that is
$*$-regular. We provide an answer to this question in this
section.

Recall that a ring $R$ is (von Neumann) regular if for every $a\in R$ there exists $b\in R$ such that
$aba=a$, or equivalently \cite[Theorem 4.23]{Lam}, every right (resp. left) principal ideal is generated by
an idempotent. This statement continues to hold if $R$ is a ring with local units since $b\in bR$ (and $b\in
Rb$) for all $b$ in $R$ so the principal right (and left) ideals of $R$ have the same form as the principal
right (left) ideals of a unital ring.

If $R$ is a $\ast$-ring, the projections take over the role of
idempotents. Thus, the concept of regularity for rings corresponds
to $\ast$-regularity for $*$-rings: a $\ast$-ring $R$ is said to
be $\ast$\emph{-regular} if every principal right ideal is
generated by a projection. This definition naturally extends to
rings with local units. Note that the condition of
$\ast$-regularity is left-right symmetric since $aR=pR$ implies
that $Ra^{\ast}=Rp$ for any $a\in R$ and a projection $p\in R.$ So
every principal left ideal of $R$ is also generated by a
projection in the case when every principal right ideal is.

A $\ast$-ring is $\ast$-regular if and only if it is regular and
the involution $*$ is proper (see \cite[Exercise 6A, \S 3]{Be}).
In the next proposition we give a proof of this fact for rings
with local units.

\begin{proposition}\label{regularproper}
Let $R$ be a ring with local units and with an involution $\ast$.
Then $R$ is $\ast$-regular if and only if $R$ is regular and
$\ast$ is proper.
\end{proposition}
\begin{prof}
If $R$ is $\ast$-regular, then it is also regular because every
projection is an idempotent. Now assume that $a^{\ast}a=0$ for
some $a\in R.$ Then $aR=pR$ for some projection $p$ so $a=pa$
($a=px$ for some $x\in R$ implies that $pa=ppx=px=a$) and $p=ay$
for some $y\in R.$ So, $a^{\ast}=a^{\ast}p=a^{\ast}ay=0$. Hence
$a=0.$ Thus $\ast$ is proper.

Conversely, suppose that $R$ is regular and $\ast$ is proper.
Since every principal right ideal is generated by an idempotent,
it is enough to show that for an arbitrary idempotent $x$ in $R$,
$Rx^{\ast}=Rp $ for some projection $p\in R$. First observe that
for any $x\in R$, $r_{R}(x)=r_{R}(x^{\ast}x)$ where $r_{R}(b)$
denotes the right annihilator of the element $b\in R$. This is
because $x^{\ast}xy=0$ implies that
$(xy)^{\ast}xy=y^{\ast}(x^{\ast}xy)=0$ so that $xy=0$ for any
$y\in R$. By the regularity of $R$, $Rx^{\ast}x=Rf$ for some
idempotent $f\in R$. Thus $r_{R}(x)=r_{R}(x^{\ast}x)=r_{R}(f)$ and
so the left annihilators $l_{R}(r_{R}(x))$ and $l_{R}(r_{R}(f))$
are also equal.

We claim that $Rx=l_{R}(r_{R}(x))$. To see this, first note that,
since $x$ is an idempotent, $r_{R}(x)=\{a-xa\,|\,a\in R\} $. So if
$y\in l_{R}(r_{R}(x))$, then $y(a-xa)=0$ for all $a\in R$, that is
$ya=yxa$ for all $a\in R$. Since $R$ is a ring with local units,
there is an idempotent $u\in R$ such that $yu=y$ and $xu=x$. Hence
$y=yu=yxu=yx\in Rx$. Thus $l_{R}(r_{R}(x))\subseteq Rx$. Since the
reverse inclusion is obvious, $Rx=l_{R}(r_{R}(x))$.

Similarly, $Rf=l_{R}(r_{R}(f))$. Thus $Rx=Rf=Rx^{\ast}x $. Hence $x=ax^{\ast}x$ for some $a\in R$. Let
$p=ax^{\ast}$. We claim that $p$ is a projection with $Rx^{\ast}=Rp$. To see this, note that $x=px$ and so
$pp^{\ast}=pxa^{\ast}=xa^{\ast}=p^{\ast}$. Since $(pp^{\ast})^{\ast}=pp^{\ast}$, we get $p=p^{\ast}$. From
$pp^*=p^*$ and $p=p^{\ast}$ we have $p^{2}=p$ and so $p$ is a projection. From $p=ax^{\ast}$ and
$x^*=x^{\ast}p$ we have that $Rx^{\ast}=Rp.$
\end{prof}

We turn to Leavitt path algebras now. For any vertex $v$ in a
graph $E$, let $\mu (v)$ denote the cardinality of the set of all
the paths $\alpha$ in $E$ with $r(\alpha )=v$ (including the
trivial path $v$). With this notation, we recall the statement of
\cite[Lemma 3.4 and Proposition 3.5]{AAS1}. Let $E$ be a finite
acyclic graph and $v$ a sink in $E.$ The set
$$I_{v}=\bigg\{\sum_{i}k_{i}\alpha _{i}\beta _{i}^{\ast}\,\bigg|\,\alpha _{i},\beta _{i}\text{ paths in }E\text{ with }r(\alpha
_{i})=r(\beta _{i})=v,k_{i}\in K\bigg\}$$ is an ideal of
$L_{K}(E)$ isomorphic to the matrix ring ${\mathbb
M}_{\mu(v)}(K).$ If $\{v_{1},\ldots,v_{m}\}$ is the set of all
sinks in $E,$ then $L_{K}(E)=\bigoplus _{i=1}^{m}I_{v_{i}}\cong
\bigoplus _{i=1}^{m}{\mathbb M}_{\mu (v_{i})}(K)$. Let us denote
this isomorphism by $\phi$ and let us call it the \emph{canonical
isomorphism}.

From \cite[Lemma 3.4 and Proposition 3.5]{AAS1} it can be seen
that the restriction of $\phi$ on a direct summand $I_v,$ for a
vertex $v,$ is given by $\phi(\sum_{i,j}k_{ij}\alpha _{i}\alpha
_{j}^{\ast})=(k_{ij})\in {\mathbb M}_{\mu (v)}(K)$ where
$i,j=1,\ldots,\mu(v),$ $\alpha_i$ and $\alpha_j$ are paths ending
at $v$ and $k_{ij}\in K.$

\begin{lemma}\label{canonicalisomorphism} Let $E$ be a finite acyclic graph and let $\{v_{1},\ldots,v_{m}\}$
be all the sinks in $E$. The canonical isomorphism
$\phi:L_{K}(E)=\bigoplus _{i=1}^{m}I_{v_{i}}\rightarrow \bigoplus
_{i=1}^{m}{\mathbb M}_{\mu (v_{i})}(K)$ is a $*$-algebra
isomorphism $($with the standard involution on $L_{K}(E)$ and the
$\ast$-transpose involution on the matrix algebras$)$.
\end{lemma}
\begin{prof}
Since $\phi$ maps direct summands $I_{v_i}$ on direct summands
${\mathbb M}_{\mu (v_{i})}(K),$ it is enough if we prove the
statement when $E$ has only one sink $v$. If $\alpha
_{1},\ldots,\alpha _{\mu (v)}$ are all the different paths
(including the trivial path) ending in $v,$ then a typical element
of $L_K(E)=I_{v}$ has the form $\sum_{i,j}k_{ij}\alpha _{i}\alpha
_{j}^{\ast}$ for  $i,j=1,\ldots,\mu(v),$ $\alpha_i$ and $\alpha_j$
paths ending at $v$ and $k_{ij}\in K.$ Then we have
$$\phi \bigg(\bigg(\sum_{i,j}k_{ij}\alpha _{i}\alpha_{j}^{\ast}\bigg)^{\ast}\bigg)=
\phi \bigg(\sum_{i,j}\overline{k_{ij}}\alpha
_{j}\alpha_{i}^{\ast}\bigg)= \phi
\bigg(\sum_{i,j}\overline{k_{ji}}\alpha
_{i}\alpha_{j}^{\ast}\bigg) =(\overline{k_{ji}})=(k_{ij})^*.$$
This proves the claim since $(k_{ij})^*=\big(\phi
\big(\sum_{i,j}k_{ij}\alpha _{i}\alpha_{j}^{\ast}\big)
\big)^{\ast}.$
\end{prof}





We finally have all the ingredients in hand to prove the main result of the paper.


\begin{theorem}\label{characterizationtheorem}
Let $E$ be an arbitrary graph, $K$ be a field with involution $^-$
and let $\sigma=\sup \{\mu (v):v\in E^{0}\}$ in case the supremum
is finite or $\sigma=\omega$ otherwise. The following conditions
are equivalent\,$:$

 {\rm(i)} $L_{K}(E)$ is $\ast$-regular.

{\rm(ii)} $L_{K}(E)$ is regular and proper.

{\rm(iii)} $E$ is acyclic and $K$ is $n$-proper for every finite $n\leq \sigma$.

\end{theorem}
\begin{prof}
(i) $\Leftrightarrow$ (ii) is Proposition \ref{regularproper}.



(ii) $\Leftrightarrow$ (iii)\q By \cite[Theorem 1]{AR}, $L_{K}(E)$
is  regular if and only if $E$ is acyclic. So it is enough if we
show, under the assumption that $L_{K}(E)$ is regular
(equivalently, $E$ is acyclic), that the involution $^-$ in $K$ is
$n$-proper for every finite $n\leq \sigma$ if and only if the
involution $\ast$ in $L_{K}(E)$ is proper.

Now \cite[Proposition 2 and Theorem 1]{AR} also state that, when
$E$ is acyclic, $L_{K}(E)$ is a directed union of subalgebras
$B(S)$ where each $B(S)\overset{\theta }{\cong }L_{K}(E_{F})\oplus
(\bigoplus _{i=1}^{m}Kx_{i})$ with $E_{F}$ a finite acyclic graph
constructed corresponding to various non-empty finite subsets $F$
of edges in $E$. Moreover $\theta$ is a $\ast$-algebra isomorphism
as we noted before. For a fixed $F$, $E_F$ has a finite number of
sinks. Let us denote them by $v_{1},\ldots, v_k$. Then
$L_{K}(E_{F})\overset{\phi }{\cong }\bigoplus _{i=1}^{k}{\mathbb
M}_{\mu _{E_{F}}(v_{i})}(K)$ as $\ast$-algebras by Lemma
\ref{canonicalisomorphism}. Thus, the involution $\ast$ in
$L_{K}(E)$ is proper if and only if the $\ast$-transpose
involution is proper in ${\mathbb M}_{\mu _{^{E_{F}}}(v_{i})}(K)$
with $v_{i}\in E_{F}$ for all the various graphs $E_{F}$
corresponding to each $B(S)$ in the stated directed system of
subalgebras of $L_{K}(E)$.

We distinguish two situations.



\th{Case 1} Suppose that $\sigma $ is infinite. Then either $\mu
(v)$ is infinite for some vertex $v$ or for every positive integer
$n$, there is a vertex $v_{n}$ with $\mu _{E}(v_{n})$ an integer
larger than $n$. In either case for each integer $n>1,$ we can
choose a vertex $v_{n}$ and a finite subset $F_{n}$ of edges that
appear in the $n-1$ distinct paths (other than the trivial path
$v_{n}$) ending in $v_{n}$. The vertex $v_{n}$ is a sink in
$E_{F_{n}}$ by the definition of the graph $E_{F_{n}}.$

Moreover, $e_{1}\cdots e_{k}$ is a path of length $k$ in $E$
ending in $v_{n}$ if and only if the path in $E_{F_n}$ given by
$(e_{1},e_{2})(e_{2},e_{3})\cdots (e_{k},v_{n})$  has length $k$
and ends in $v_{n}$. Thus, $v_{n}$ is a sink in $E_{F_{n}}$ with
$\mu_{E_{F_{n}}}(v_{n})=n.$ The graph $E_{F_{n}}$ is finite
acyclic (by \cite[Lemma 1]{AR}). So, $L_{K}(E_{F_{n}})$ contains
the ideal $I_{v_{n}}\cong {\mathbb M}_{n}(K)$.

Since this holds for every $n$, the involution $\ast$ is proper in
each subalgebra $B(S)$ if and only if the $\ast$-transpose
involution is proper in ${\mathbb M}_{n}(K)$ for each positive
integer $n$. This is equivalent, by Lemma \ref{R_n-proper}, to the
statement that the involution $^-$ in $K$ is $n$-proper for every
$n$, that is, that $K$ is positive-definite.



\th{Case 2} Suppose that $\sigma$ is finite, say $\sigma =n$ for
some positive integer $n$. If $n=1$, every vertex in $E$ is
isolated and $L_{K}(E)$ is isomorphic to $\bigoplus _{v\in
E^{0}}Kv$ where $Kv\cong K.$ Both of those algebras are proper, so
we are done.

Suppose $n>1$ and let $v$ be a vertex for which $\mu (v)=n$. Let $F_{v}$ be the non-empty finite set of edges
in all the $\mu _{E}(v)-1$ nontrivial paths ending in $v$. As noted in Case 1, $v$ is a sink (and in this
case, the only sink) in the finite acyclic graph $E_{F_{v}}$ and, moreover, $\mu _{E_{F_{v}}}(v)=\mu_{E}(v)$.

So by Lemma \ref{canonicalisomorphism}, we have $L_{K}(E_{F_{v}})$
$\cong {\mathbb M}_{\mu _{E}(v)}(K)$ as $*$-algebras. Moreover, as
$n$ is the least upper bound of $\{\mu (v):v\in E^{0}\}$, then all
matrices ${\mathbb M}_{\mu_{E_{F_i}}(v_i)}(K)$ appearing in all
other subalgebras $B(S)$ for various other finite subsets of edges
$F_i$ and sinks $v_i$ will all have order that is less than or
equal to $n$.

Therefore the involution $\ast$ is proper in each $B(S)$ if and
only if the $\ast$-transpose involution is proper in ${\mathbb
M}_{\mu _{E}(v)}(K)$. Since $\mu (v)=n$, the last statement holds
exactly when the involution $^-$ in $K$ is $n$-proper, again by
Lemma \ref{R_n-proper}. This finishes the proof.
\end{prof}

It is interesting to point out that the presence of involution gives a more prominent role to the field $K$
than it had in the previous characterization theorems (e.g., simplicity \cite{AA1}, purely infinite
simplicity \cite{AA2}, finite-dimensionality \cite{AAS1}, just to cite a few). In particular, Theorem
\ref{characterizationtheorem} also contrasts the characterization of regularity from \cite{AR} that was
independent of the field $K.$

We further illustrate this behavior with an easy example. If $E$ is the graph
$$\xymatrix{{\bullet} \ar [r]  & {\bullet}},$$
then $L_K(E)\cong {\mathbb M}_2(K)$ as $\ast$-algebras for any
field $K$. If $K={\mathbb R}$ with the identity involution,
$L_{\mathbb{R}}(E)$ is $*$-regular because ${\mathbb R}$ is
positive-definite. However if $K={\mathbb C}$ with the identity
involution, then $L_{\mathbb{C}}(E)\cong {\mathbb M}_{2}(
\mathbb{C})$ is regular but it is \emph{not} $\ast$-regular (since
the identity involution in $\mathbb{C}$ is not $2$-proper).
Furthermore, if $K={\mathbb C}$ with the conjugate involution,
then $L_{\mathbb{C}}(E)$ \emph{is} $*$-regular, because the
conjugation of complex numbers is positive-definite.

Also, since the identity involution on a field of characteristic
$n>0$ is not $n$-proper, the properness (thus also $*$-regularity)
of a Leavitt path algebra over such field depends on the
characteristic of the field. This fact also brings the field
characteristic into spotlight.

Let us note the following corollary of Theorem \ref{characterizationtheorem}.

\begin{corollary}\label{starregularequivalences} Let $K$ be a field with involution. The following conditions
are equivalent\,$:$

{\rm(i)} The involution on $K$ is positive-definite.

{\rm(ii)} $L_{K}(E)$ is $\ast$-regular for every acyclic graph
$E$.

\end{corollary}
\begin{prof}
(i) $\Rightarrow$ (ii) follows directly from Theorem
\ref{characterizationtheorem}.

\par

(ii) $\Rightarrow$ (i) Let us assume that $L_K(E)$ is
$\ast$-regular for every acyclic $E$. Consider the line of length
$n-1$ (see the second graph in Examples \ref{examples}). The
Leavitt path algebra of this graph is isomorphic to ${\mathbb
M}_{n}(K).$ From the assumption that this algebra is $*$-regular,
we obtain that $K$ is $n$-proper by Lemma \ref{R_n-proper}. Since
this holds for every $n$, $K$ is positive-definite.
\end{prof}

It is also interesting to note that the two equivalences of
Corollary \ref{starregularequivalences} parallel the first two
equivalences of Proposition \ref{moreimplications}. The last
equivalence of Proposition \ref{moreimplications} in the
$*$-setting would have the form: ``$L_{K}(E)$ is $\ast$-regular
for some acyclic graph $E$''. However, this statement is weaker
than the other two equivalences in Corollary
\ref{starregularequivalences} so we do not have complete analogy
with Proposition \ref{moreimplications}. To see this, consider a
graph consisting of a single vertex and the complex numbers with
the identity involution. The Leavitt path algebra of this graph is
$\ast$-regular but the field is not positive-definite.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Handelman's Conjecture for Leavitt Path Algebras}


We close this paper by pointing out that Handelman's conjecture has a positive answer for the family of
Leavitt path algebras of arbitrary graphs. The conjecture can be stated as follows.

\begin{conjecture} [{Handelman \cite[Problem 48, p.\,380]{G2}}] Every $\ast$-regular ring is
unit-regular.
\end{conjecture}

This conjecture assumes that the ring is unital. First, we note
that it is true for unital Leavitt path algebras. Let us assume
that a unital $L_K(E)$ is $*$-regular. Then $E$ is acyclic by
Theorem \ref{characterizationtheorem}. Then we have that $L_K(E)$
is unit-regular by \cite[Theorem 2]{AR}.

To prove that the conjecture remains true for Leavitt path algebras of arbitrary graphs, we adapt the notion
of unit-regularity for rings with local units, as was done in \cite{AR} for instance.

Recall that a ring $R$ with identity is said to be \emph{unit-regular} if for each $a\in R$, there is a unit
(an invertible element) $u$ such that $aua=a$. If $R$ is a ring with local units, then $R$ is called
\emph{locally unit-regular} if for each $a\in R$ there is an idempotent (a local unit) $v$ and local inverses
$u,u'$ such that $uu^{\prime }=v=u^{\prime }u$, $va=av=a$ and $aua=a$.

Clearly, a unit-regular (unital) ring is locally unit-regular
(take the idempotent $v$ from the definition of locally
unit-regular to be the identity). Conversely, if a ring with
identity is locally unit-regular, then it is unit-regular (see
also \cite[Lemma 3\,(1)]{AR}). To see this, let $a\in R$. Then
there is an idempotent $v$ and local inverses $u,u'$ in $vRv$ such
that $uu^{\prime }=v=u^{\prime }u$, $va=av=a$ and $aua=a$. Then
$w=u+(1-v)$ and $w^{\prime }=u^{\prime }+(1-v)$ satisfy
$ww^{\prime }=1=w^{\prime }w$ and $a=awa$. Hence $R$ is
unit-regular.


\begin{corollary}
Let $E$ be an arbitrary graph and let $K$ be a field with
involution. Suppose $L_{K}(E)$ is $\ast$-regular. Then

{\rm(i)} $L_{K}(E)$ is locally unit-regular.

{\rm(ii)} If $L_K(E)$ is a unital ring, then $L_{K}(E)$ is unit-regular.

\end{corollary}
\begin{prof}
(i) If $L_{K}(E)$ is $\ast$-regular, we have that $E$ is acyclic
by Theorem \ref{characterizationtheorem}. Then $L_{K}(E)$ is
locally unit-regular by \cite[Theorem 2]{AR}.


(ii) is a consequence of the fact that every unital locally
unit-regular ring is unit-regular.~\end{prof}

Let us also note that the converse of Handelman's conjecture is
not true. The examples of unit-regular and not $*$-regular rings
can be found in the class of Leavitt path algebras as well. For
instance, ${\mathbb M}_2({\mathbb C})$ with the identity
involution on ${\mathbb C}$ is such an example: we know that it is
not $*$-regular but it is unit-regular (as a semisimple ring, see
\cite[p.\,38]{G2}).

\acknowledgements{\rm The authors thank Gene Abrams for his valuable discussions during the preparation of
this paper.}



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\end{document}
