Quantum semigroups generated by locally compact semigroups
M.A.\
Aukhadiev
Yu.N.\
Kuznetsova
Let $S$ be a subsemigroup of a locally compact group $G$, such that $S^{-1S=G$. We consider the $C^*$-algebra $C^*_\delta(S)$ generated by the operators of translation by all elements of $S$ in $L^2(S)$. We show that this algebra admits a comultiplication which turns it into a compact quantum semigroup. For $S$ with nonempty interior, this comultiplication can be extended to the von Neumann algebra $VN(S)$ generated by $C^*_\delta(S)$.
1504.00407.tex
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\begin{document}

\title[Quantum semigroups generated by locally compact semigroups] {Quantum semigroups generated by locally compact semigroups}


\author{M.A.\,Aukhadiev, Yu.N.\,Kuznetsova}
\address[M.A.\,Aukhadiev]{Mathematisches Institut, Westf{\"a}lische Wilhelms-Universit{\"a}t M{\"u}nster,  Einsteinstra{\ss}e 62, 48149 M{\"u}nster, Germany; \newline
Kazan State Power Engineering University\\
        Kazan, Russia, 420066 }
        \address[Yu.N.\,Kuznetsova]{Laboratoire de Math\'ematiques\\
        Universit\'e de Franche-Comt\'e, 16 route de Gray, 25030 Besan\c con, France}
%% Note the doubled @@:
\email[M.A.\,Aukhadiev]{m.aukhadiev@wwu.de}
\email[Yu.N.\,Kuznetsova]{yulia.kuznetsova@univ-fcomte.fr}

\thanks{The first author is supported in part by the Alexander von Humboldt Foundation and by the Russian Foundation for Basic
Research (Grant 14-01-31358).}

\date{August 28, 2015}

\subjclass{Primary 81R15; 20G42; 16T10; 22A20}
\begin{abstract}
Let $S$ be a subsemigroup of a locally compact group $G$, such that $S^{-1}S=G$. We consider the $C^*$-algebra $C^*_\delta(S)$ generated by the operators of translation by all elements of $S$ in $L^2(S)$. We show that this algebra admits a comultiplication which turns it into a compact quantum semigroup. For $S$ with nonempty interior, this comultiplication can be extended to the von Neumann algebra $VN(S)$ generated by $C^*_\delta(S)$.
\end{abstract}
\maketitle


\section{Introduction}

The notion of a quantum semigroup, as a $C^*$- or von Neumann algebra with a comultiplication, appeared well before the term and before the notion of a locally compact quantum group. But it is especially these last years that substantial examples of quantum semigroups are considered; we would like to mention families of maps on finite quantum spaces \cite{soltan}, quantum semigroups of quantum partial permutations \cite{banica}, quantum weakly almost periodic functionals \cite{daws}, quantum Bohr compactifications \cite{salmi, soltan2}.

In this article we construct a rather ``classical'' family of compact quantum semigroups, which are associated to sub-semigroups of locally compact groups.
The interest of our objects is in fact that they provide natural examples of  $C^*$-bialgebras which are co-commutative and are not however duals of functions algebras. Recall that the classical examples of quantum groups belong to one of the two following types: they are either function algebras, such as the algebra $C_0(G)$ of continuous functions vanishing at infinity on a locally compact group $G$, or their duals, such as the reduced group $C^*$-algebras $C^*_r(G)$. In the semigroup situation one can go beyond this dichotomy.

If $S$ is a discrete semigroup, then the algebra $C^*_\delta(S)$ which we consider coincides with the reduced semigroup $C^*$-algebra $C^*_r(S)$ which has been known since long ago \cite{coburn1,coburn2,barnes, word, paterson}.
If $S=G$ is a locally compact group, then $C^*_\delta(S)=C^*_\delta(G)$ is the $C^*$-algebra generated by all left translation operators in $B(L^2(G))$ \cite{kodaira, bedos}. If $G$ is moreover abelian, then $C^*_\delta(G)$ equals to the algebra $C(\widehat G_d)$ of continuous functions on the dual of the discrete group $G_d$ \cite{kodaira}.

The new case considered in this paper concerns non-discrete nontrivial subsemigroups of locally compact groups, and our objective is to show that they admit a natural coalgebra structure.
Let $G$ be a locally compact group, and let $S$ be its sub-semigroup such that $S^{-1}S=G$. Set $H_S=\{f\in L^2(G): \supp f\subset S\}$; let $E_S$ be the orthogonal projection of $L^2(G)$ onto $H_S$ and let $J_S$ be the right inverse of $E_S$, so that $E_SJ_S = {\rm Id}_{H_S}$.
After the study of semigroup ideals in Section 2, in Section 3 we define $C^*_\delta(S)$ as the $C^*$-algebra generated in $B(H_S)$ by the operators $T_a = E_S L_a J_S$ over all $a\in S$, where $L_a$ is the operator of the left translation by $a$ on $L^2(G)$. In Section 4 we show that $C^*_\delta(S)$ admits a comultuplication $\Delta$ such that $\Delta(T_a)=T_a\otimes T_a$. The same facts are derived in parallel for the universal semigroup $C^*$-algebra $C^*(S)$ defined axiomatically via defining relations.

In the discrete case, the construction was carried out by X.~Li \cite{Li} (see also \cite{CEL}). The discrete abelian case was studied in detail in \cite{AGL}.

In Section 5, we show that for $S$ with nonempty interior, the comultiplication can be extended to the von Neumann algebra $VN(S)$ generated by $C^*_\delta(S)$ in $B(H_S)$. In the case $S=G$, this is the classical group von Neumann algebra. We can thus call $VN(S)$ the semigroup von Neumann algebra.

\section{Semigroup ideals}

Let $G$ be a locally compact group, $S$ a closed subsemigroup of $G$ containing the identity $e$ of $G$ and such that $G=S^{-1}S$. Denote by $\mu$ the left Haar measure on $G$.

For any subset $X\subset S$ and any $p\in S$, define the {\it translations in $S$}:
\begin{equation}\label{px}
p\cc X=\{pq \colon q\in X\},\ p^{-1}\cc X=\{q\in S\colon pq\in X\}.
\end{equation}
Obviously, $p\cc S$ is a right ideal in $S$, $e\cc S=e^{-1}\cc S=S$ and $p^{-1}\cc S=S$ for any $p\in S$.
For the usual translations in $G$, we use the notation $g\cdot_G X = \{gh: h\in G\}$, so that $p^{-1}X =S\cap p^{-1}\cdot_G X$.
It is also easy to see that $p\cc (q\cc X)=(pq)\cc X$ and $p^{-1}\cc (q^{-1}\cc X) = (qp)^{-1}\cc X$ for any $X\subset S$ and all $p,q\in S$. We will omit parentheses in the products of this type. Moreover, $p^{-1}p X=X$,
but in general, the products $pq^{-1}$ or $p^{-1}q$ should be viewed purely formally, and $pp^{-1}\cc X$ might differ from $X$ (see, for example, Lemma \ref{s2}).


More precisely,
denote by $\mathcal{F}=\mathcal{F}(S)$ the free monoid generated by $S$ and $S^{-1} \setminus S $. Any element in $\mathcal{F}$ is a finite word with alternating symbols in  $S$ and $S^{-1}$. The operation of taking inverse in $G$ induces the operation $w\mapsto w^{-1}$ on the monoid $\mathcal{F}$, by $\big(p_1^{\pm1}\cdots p_n^{\pm1}\big)^{-1} = p_n^{\mp1}\cdots p_1^{\mp1}$.

For every $w=p_1^{\pm1}\cdots p_n^{\pm1}\in\mathcal{F}$ and $X\subset S$, define by induction $wX = p_1^{\pm1}(\dots(p_n^{\pm1}X)\dots)$. If $X=S$, then $wS$ is a right ideal in $S$. Define the family of all \emph{constructible right ideals} in $S$ \cite{Li}:
$$\mathcal{J}=\{\bigcap\limits_{i=1}^{n} w_iS\colon w_i\in\mathcal{F} \}\cup\{\emptyset\}.$$

Suppose that $w\in F$ has the form $w=p^{-1}_1q_1p^{-1}_2q_2\dots p^{-1}_nq_n$ with $p_j,q_j\in S$, maybe with $p_1=e$ or $q_n=e$. Then it follows from the definition that $ wS$ is the set of elements $x$ satisfying
\begin{equation}\label{xf} \begin{array}{c}
x=p^{-1}_1q_1\dots p^{-1}_nq_n r_{n+1},\\
 \mbox{ where } r_{n+1}\in S \mbox{ and } \\ r_k=p^{-1}_kq_k\dots p^{-1}_nq_n r_{n+1} \in S \mbox{ for all } k=1,\dots,n.
\end{array}\end{equation}


Define a homomorphism $\mathcal{F}\to G$: $w\mapsto (w)_G$, by $(p^{\pm1})_G=p^{\pm1}$ for $p\in S$.
We fix also an injection  $\imath: G\hookrightarrow \mathcal{F}$ which might not be a homomorphism: for any element $g\in G$ we fix one of its representations $g=p^{-1}q$ and set $\imath(g)=p^{-1}q\in \mathcal{F}$. The notation $gX$, where $g\in G$ and $X\subset S$, is understood in the sense $gX=\imath(g)X$.


\begin{lemma}\label{s1}

For any $w_1,w_2\in\mathcal{F}$, we have $w_1w_2S\subset w_1S$.
\end{lemma}
\begin{proof}
Follows immediately from the facts that $pS\subset S, p^{-1}S= S$ for any $p\in S$.
\end{proof}




\begin{lemma}\label{s2}

For any $w\in\mathcal{F}$, $wS=ww^{-1}S$.
\end{lemma}
\begin{proof}
We can assume that $w$ has the form $w=p^{-1}_1q_1p^{-1}_2q_2\dots p^{-1}_nq_n$ with $p_j,q_j\in S$, maybe with $p_1=e$ or $q_n=e$. Then every $x\in wS$ has the form (\ref{xf}) with $r_{n+1}\in S$ and $r_k=p^{-1}_kq_k\dots p^{-1}_nq_n r_{n+1} \in S$ for all $k=1,\dots,n$.

Now write $x=p^{-1}_1q_1\dots p^{-1}_nq_nq_n^{-1}p_n\dots q_1^{-1}p_1x$. In this product, $x\in S$ and $r_{k+1}=q_k^{-1}p_k\dots q_1^{-1}p_1x\in S$ for $k=1,\dots,n$, as well as $r_k=p^{-1}_kq_k\dots p^{-1}_nq_n q_n^{-1}p_n\dots q_1^{-1}p_1x\in S$ for $k=1,\dots, n$. It follows that $x\in ww^{-1}S$, so $wS\subset ww^{-1}S$. The inverse inclusion follows from Lemma \ref{s1}.
\end{proof}


\begin{lemma}\label{s3}
Let a word $w\in\mathcal{F}$ have the form $w=w_1w_2$, where $w_1,w_2\in\mathcal{F}$. Then $wS=w_1S\cap (w_1)_Gw_2S$.
\end{lemma}
\begin{proof}
Suppose $w_1=p^{-1}_1q_1p^{-1}_2q_2\dots p^{-1}_iq_i$, $w_2=p^{-1}_{i+1}q_{i+1}p^{-1}_{i+2}q_{i+2}\dots p^{-1}_nq_n$ with $p_j,q_j\in S$. Then every $x\in wS$ satisfies \eqref{xf}. This implies directly that $x\in w_1S$. If we denote $(w_1)_G=p^{-1}q$, $p,q\in S$, then in the notations \eqref{xf} we have also $x=p^{-1}qr_{i+1}$ what implies that $x\in p^{-1}qw_2S = wS$.

Conversely, if $wS=w_1S\cap (w_1)_Gw_2S$, then

$$x=p^{-1}_1q_1p^{-1}_2q_2\dots p^{-1}_iq_ir_{i+1}',$$
$$r_{i+1}'\in S,\ r_k'= p^{-1}_kq_k\dots p^{-1}_iq_ir_i'\in S\ \mbox{ for }k=1,\dots,i,$$
and $$ x=p^{-1}qp^{-1}_{i+1}q_{i+1}p^{-1}_{i+2}q_{i+2}\dots p^{-1}_nq_n r_{n+1},$$ $$x\in S,\  r_{n+1}\in S,\ r_k= p^{-1}_kq_k\dots p^{-1}_nq_nr_{n+1}\in S\ \mbox{ for }k=i+1,\dots,n.$$
By cancellation, it follows that $r_{i+1}' = p^{-1}_{i+1}q_{i+1}p^{-1}_{i+2}q_{i+2}\dots p^{-1}_nq_n r_{n+1}$, thus in fact the condition \eqref{xf} holds for $x$.
\end{proof}

\begin{corollary}\label{s4}
For any $v,w\in\mathcal{F}$, we have $vS\cap wS=ww^{-1}vS$.
\end{corollary}
\begin{proof}
Since $(ww^{-1})_G=e$, by Lemmas \ref{s2} and \ref{s3} we have
$$wS\cap vS=ww^{-1}S\cap vS=ww^{-1}S\cap (ww^{-1})_GvS=ww^{-1}vS.$$
\end{proof}

It follows that $$\mathcal{J}=\{wS|\ w\in F\}\cup \{\emptyset\} .$$

\section{The semigroup C*-algebras}

In what follows we assume the following property for $S$. If $X=\cup^n_{j=1}X_j$ up to a null set for $X,X_1,\dots ,X_n\in\mathcal{J}$, then $X=X_j$, also up to a null set, for some $1\leq j\leq n$. We call this property \emph{topological independence of constructible right ideals} in $S$, the algebraic one being introduced in \cite{Li}. It is exactly this property which will guarantee that our comultiplication is well defined. Note that the topological independence implies the algebraic one.

Recall the definition of the full semigroup C*-algebra \cite{Li}. Consider a family of isometries $\{v_p| p\in S\}$ and a family of projections
$\{e_X| X\in\mathcal{J}\}$ satisfying the following relations for any $p,q\in S$, and $X,Y\in\mathcal{J}$:
\begin{equation}\label{e3}
 v_{pq}=v_pv_q,\ v_pe_Xv_p^*=e_{pX},
\end{equation}

\begin{equation}\label{e4}
e_S=1,\ e_{\emptyset}=0,\ e_{X\cap Y}=e_Xe_Y.
\end{equation}

The universal C*-algebra $C^*(S)$ of the semigroup $S$ is by definition generated by $\{v_p| p\in S\}\bigcup \{e_X| X\in\mathcal{J}\}$ with the relations above.

The $C^*$-algebra $C^*(S)$ contains a commutative $C^*$-algebra $D(S)$ generated by the family of projections $\{e_X| X\in\mathcal{J}\}$.

Consider the Hilbert space $L^2(G)$ with respect to $\mu$. For any measurable subset $X\subset G$ set $H_X=\{f\in L^2(G): {\rm ess}\supp f\subset X\}$; this subspace is isomorphic to $L^2(X,\mu)$. Let $I_X\in L^2(G)$ be the characteristic function of $X$ and $E_X$ the orthogonal projection of $L^2(G)$ onto $H_X$, which is just the multiplication by $I_X$. Let $L\colon G\to B(L^2(G)) $ be the left regular representation of $G$, i.e. for any $a,b\in G,\ f\in L^2(G)$
\begin{equation}\label{la} (L_af)(b)=f(a^{-1}b).
\end{equation}
We define the left regular representation $T\colon S\to B(H_S)$ of the semigroup $S$ analogously to $L$. For any $a,b\in S,\ f\in H_S$ we set
\begin{equation}\label{e5}
(T_af)(b)=f(a^{-1}b),
\end{equation}
so that $T_p=E_SL_pE_S$; then
\begin{equation}\label{e6}
(T_a^*f)(b)=I_S(b)f(ab).
\end{equation}

One can verify that $T_a$ is an isometry, $T_a^*T_a=I$, and that for any $f\in H_S$  for $a,b\in S$ we have
$$ (T_aT_a^*f)(b)=I_S(a^{-1}b)f(b).$$
Clearly, $a^{-1}b\in S$ if and only if $b\in aS$, where $aS$ is a constructible right ideal defined in the previous section. Hence the projection   $T_aT_a^*$ is an operator of multiplication by $I_{aS}$. The map $T\colon S\to B(H_S)$ is obviously a representation of $S$. Let $C^*_\delta(S)$ be the $C^*$-subalgebra in $B(H_S)$ generated by the operators $\{T_a: a\in S\}$.

If $S=G$, then $C^*_\delta(S)=C^*_\delta(G)$ is the $C^*$-algebra generated by all left translation operators in $B(L^2(G))$ \cite{kodaira, bedos}. If $S$ is discrete, then $C^*_\delta(S)=C^*_r(S)$ is the reduced semigroup $C^*$-algebra \cite{paterson}.

A finite product of the generators $T_a, T_b^*$ for any $a,b\in S$ is called \emph{a monomial}. We will use also the notation $T_{a^{-1}}=T_a^*$, what does not create confusion in the case $a^{-1}\in S$. Generally, for every $w=p_1^{\pm1}\dots p_n^{\pm1}\in\mathcal F$ we denote $T_w = T_{p_1^{\pm1}}\dots T_{p_n^{\pm1}}$, and clearly every monomial has this form.

\begin{lemma}\label{l5}
For any monomial $T_w$, function $f\in H_S$ and $x\in G$ we have
\begin{equation}\label{e7}
(T_wf)(x)=I_{wS}(x)\cdot f((w^{-1})_Gx)
\end{equation}
\end{lemma}
\begin{proof}
Let $k$ be the length of the word $w$. For $k=1$, either $w=a$ or $w=a^{-1}$ with some $a\in S$. If $w=a$ then for $f\in H_S$ we have
$f(a^{-1}x) = f(a^{-1}x)I_S(a^{-1}x) = I_{aS}(x)f(a^{-1}x)$, thus the expressions \eqref{e5} and \eqref{e7} are equal. If $w=a^{-1}$, then, due to the fact that $a^{-1}S=S$, the formula (\ref{e6}) implies (\ref{e7}).

 Suppose (\ref{e7}) is proved for $k\leq n$ and $w=vw'$ is a word in $\mathcal{F}$ with the length $k+1$, where the length of $v$ and $w'$ is 1 and $k$ respectively. First assume that $v=a\in S$ and denote $g=T_{w'}f$. Then for any $x\in G$ we have
 $$(T_wf)(x)=(T_aT_{w'}f)(x)=(T_ag)(x)=$$
 $$=g(a^{-1}x)=(T_{w'}f)(a^{-1}x)=$$
$$=I_{w'S}(a^{-1}x)f((w'^{-1})_Ga^{-1}x)=$$
 $$=I_{aw'S}(x)f(((aw')^{-1})_Gx).$$
Now assume that $v=a^{-1}\in S^{-1}$. Then for any $x\in G$ we have

 $$(T_wf)(x)=(T_a^*T_{w'}f)(x)=(T_a^*g)(x)=$$
 $$=I_S(x)g(ax)=I_S(x)(T_{w'}f)(ax)=I_S(x)I_{w'S}(ax)f((w'^{-1})_Gax)=$$
 Note that $x\in S$ and $ax\in w'S$ if and only if $x\in a^{-1}w'S$.
 $$=I_{a^{-1}w'S}(x)f(((a^{-1}w')^{-1})_Gx)=I_{wS}(x)\cdot f((w^{-1})_Gx).$$
And the formula (\ref{e7}) follows.
\end{proof}

\begin{lemma}\label{l2}
The $C^*$-algebra $C^*_\delta(S)$ is isomorphic to the $C^*$-subalgebra in $B(L^2(G))$ generated as a \textbf{linear space} by
\begin{equation}\label{EwSLwES}
E_{wS}L_{(w)_G}E_S, \quad w\in\mathcal{F},
\end{equation}
and equivalently by
\begin{equation}\label{EwSLw}
E_{wS}L_{(w)_G}, \quad w\in\mathcal{F}.
\end{equation}
\end{lemma}
\begin{proof}
It follows directly from \eqref{e7} that $T_w E_S = E_{wS} L_{(w)_G} E_S$ for every $w\in\mathcal F$. At the same time, $E_STE_S=TE_S$ for every $T\in C^*_\delta(S)$. Thus, the mapping $T\mapsto TE_S=E_STE_S$ is a *-homomorphism from $C^*_\delta(S)$ to $B(L^2(G))$, and its image is generated exactly by the operators \eqref{EwSLw}. Moreover, this mapping is clearly isometric and thus it is an isomorphism.

To arrive at the second description, one calculates that $L_g E_S = E_{g\cdot_G S} L_g$ for every $g\in G$. Thus,
$$
E_{wS}L_{(w)_G}E_S = E_{wS\cap ((w)_G\cdot_G S)}L_{(w)_G}.
$$
By definition, $wS\subset (w)_G\cdot_G S$, and \eqref{EwSLw} follows.
\end{proof}

For a monomial $T_w$ define its \emph{index} by $(w)_G\in G$. We have $\mathrm{ind}T_w^*=(w)_G^{-1}$ and $\mathrm{ind}(T_vT_w)=(v)_G(w)_G$. Recall that $E_X\in B(L^2(G))$ is the operator of multiplication by $I_X$.

\begin{corollary}\label{c2} A monomial $T_w$ in $C^*_\delta(S)$ is an orthogonal projection if and only if $\mathrm{ind}T_w=e$. In this case $T_w=E_{wS}$.

\end{corollary}
\begin{proof}
Let $T_w$ be an orthogonal projection. Then $(ww)_G=(w)_G^2=(w)_G$ and $w_G^{-1}=w_G$. Hence, $(w)_G=\mathrm{ind}T_w=e$.

Suppose that $\mathrm{ind}T_w=e$. Then due to Lemma \ref{l5}, $T_w=E_{wS}$ which is an orthogonal projection.
\end{proof}

\begin{lemma}\label{pro} Every projection $E_X$ for $X\in\mathcal{J}$ is contained in $C^*_\delta(S)$ and equals $T_{ww^{-1}}$ for some $w\in\mathcal{F}$.\end{lemma}
\begin{proof}
By Corollary \ref{s4}, $X=wS$ for some $w\in F$. Due to Corollary \ref{c2}, if $(w)_G=e$ then $E_{wS}\in C^*_\delta(S)$.

Suppose $w$ is an arbitrary element in $\mathcal{F}$. By Lemma \ref{s2}, $wS=ww^{-1}S$ and $E_{wS}=E_{ww^{-1}S} $. Since $(ww^{-1})_G=e$, by Corollary \ref{c2} we have that $E_{wS}=T_{ww^{-1}}\in C^*_\delta(S)$.
\end{proof}

\begin{lemma}\label{left}
There exists a surjective *-homomorphism $\lambda \colon C^*(S)\to C_\delta^*(S)$ such that $\lambda(v_p)=T_p$, $\lambda(e_X)= E_X$. It will be called the left regular representation of $C^*(S)$.
\end{lemma}
\begin{proof}
One can easily verify that the operators $T_p$ and $E_X$ satisfy the equations (\ref{e3}) and (\ref{e4}) for all $p\in S,\ X\in\mathcal{J}$. The universality of $C^*(S)$ implies the existence of the homomorphism $\lambda$.
\end{proof}

Denote by $D_\delta(S)$ the $C^*$-subalgebra in $C^*_\delta(S)$ generated by monomials with index equal to $e$. By Corollary \ref{c2} and Lemma \ref{pro} $D_\delta(S)$ is generated by projections $\{E_X| \ X\in\mathcal{J}\}$, and is obviously commutative.


\begin{lemma}\label{D}
The algebras $D(S)$ and $D_\delta(S)$ are isomorphic.
\end{lemma}
\begin{proof}
The left regular representation restricted to $D(S)$ is surjective. Applying Lemma 2.20 in \cite{Li} and using the independence of constructible right ideals in $S$ we obtain injectivity of $\lambda|_{D(S)}$.
\end{proof}

There exists a natural action of the semigroup $S$ on the $C^*$-algebra $D_\delta(S)$.
\begin{equation}\label{act}
\tau_p(A)=T_p AT_p^*,\ p\in S,\ A\in D_\delta(S).
\end{equation}
Using the formula (\ref{e7}), we obtain for $A=E_X$, $X\in\mathcal{J}$:
\begin{equation}
\label{actex}\tau_p(E_X)=E_{pX}.
\end{equation}

\section{The universal and reduced compact quantum semigroups}

Consider the $C^*$-subalgebra $\mathcal{A}$ in $C^*(S)\otimes_{max}C^*(S)$ generated by the elements
$$\{  v_p\otimes v_p, \ e_X\otimes e_X\colon p\in S,\ X\in\mathcal{J}\}.$$

Clearly, these elements satisfy relations (\ref{e3}), (\ref{e4}). The universal property of $C^*(S)$ implies the existence of a unital $*$-homomorphism $\Delta_u\colon C^*(S)\to\mathcal{A}$, such that
$$\Delta_u(v_p)=v_p\otimes v_p,\ \Delta_u(e_X)=e_X\otimes e_X.$$
The map $\Delta_u$ admits a restriction $\Delta_u|_{D(S)}: D(S)\to D(S)\otimes_{max} D(S)$ which is also a unital *-homomorphism.

The pair $\mathbb{Q}(S)=(C^*(S),\Delta_u)$ is a compact quantum semigroup \cite{AGL}. We call the algebra $C^*(S)$ with this structure the \emph{universal algebra of functions on the compact quantum semigroup} $\mathbb{Q}(S)$ associated with the semigroup $S$.

We recall that a semigroup is called right reversible if every pair of non-empty left ideals has a non-empty intersection. The following theorem by Ore can be found in \cite{Cl}.

\begin{theorem}\label{Clif}
A cancellative semigroup $S$ can be embedded into a group $G$ such that $G=S^{-1}S$ if and only if it is right reversible.
\end{theorem}

Define a partial order on $S$: $p\leq q$ if $qp^{-1}\in S$, or equivalently if $q\in Sp$. Due to the assumption $G=S^{-1}S$, we obtain by Theorem \ref{Clif} that for any $p,q\in S$ the left ideals $\{xp\colon x\in S\}$, $\{yq\colon y\in S\}$ have a non-empty intersection. Hence $S$ is upwards directed with respect to this partial order.

Consider the directed system of $C^*$-algebras $\mathcal{A}_p$ indexed by $p\in S$, where every $\mathcal{A}_p=D_\delta(S)$. For $p,q\in S$ such that $p\leq q$ we have $qp^{-1}\in S$ and the action (\ref{act}) generates a $*$-homomorphism $\tau_{qp^{-1}}\colon \mathcal{A}_p\to\mathcal{A}_q$:
$$\tau_{qp^{-1}}(A)=T_{qp^{-1}}AT_{qp^{-1}}^*.$$
Clearly, $\tau_{qp^{-1}} = \tau_{qr^{-1}}\tau_{rp^{-1}}$ for $p\le r\le q$. Let $D_\delta^{(\infty)}(S)$ denote the $C^*$-inductive limit of the directed system $\{\mathcal{A}_p,\tau_{qp^{-1}}\}$.

Recall the notation $q^{-1}\cdot_G X=\{q^{-1}x\colon x\in X\}\subset G$ for $q\in S$ and $X\in\mathcal{J}$.

\begin{lemma}\label{ind}The $C^*$-algebra $D_\delta^{(\infty)}(S)$ is isomorphic to $$D=C^*(\{E_{q^{-1}\cdot_G X}\colon q\in S,\ X\in \mathcal{J}\})\subset B(L^2(G)).$$ \end{lemma}

\begin{proof}
By definition, $D_\delta(S)\subset B(H_S)$. Recall that we denote $J_S:H_S\to L^2(G)$ the canonical imbedding; denote by $\pi\colon D_\delta(S)\to B(L^2(G))$ the lifting $\pi(A)= J_S AE_S$.

For any $p\in S$, the map
$$\phi_p(A)=L_p^*\pi(A)L_p,\ \ A\in D_\delta(S),$$
is a $*$-homomorphism $\phi_p\colon D_\delta(S)\to D$, such that
$\phi_p(E_X)= E_{p^{-1}\cdot_G X}$ for all $X\in\mathcal{J}$.

Then for $p\leq q$ and $X\in\mathcal{J}$ we have:
$$\phi_q \tau_{qp^{-1}}(E_X)=%L_q^*(\pi(T_{qp^{-1}} E_X T_{qp^{-1}}^*))L_q= $$$$ =
L_q^*E_{qp^{-1}X}L_q=E_{q^{-1}\cdot (qp^{-1}X)}=E_{p^{-1}\cdot X}=\phi_p(E_X).$$
So the maps $\phi_p$ agree with $\tau_{qp^{-1}}$. The homomorphisms $\phi_p$ are injective since $\pi$ is obviously injective and $L_p$ is a unitary operator. It follows that the limit map $\Phi=\lim\phi_p\colon D^{(\infty)}_\delta(S)\to D $ is injective.

To prove surjectivity of $\Phi$ it suffices to show that for any $q_1,\dots,q_n\in S$, $X_1,\dots, X_n\in\mathcal{J}$ and $\lambda_1,\dots,\lambda_n\in\mathbb{C}$ we have $$\sum\limits_{i}\lambda_i E_{q_i^{-1}\cdot_G X_i}\in\Phi(D^{(\infty)}_\delta(S)).$$

Since the system $\{\mathcal{A}_p,\tau_{qp^{-1}}\}$ is upwards directed,
 there exists $s\in S$ such that $q_i\leq s $, $i=1,2,\dots n$, and it implies that $sq_i^{-1}X_i\in\mathcal J$ and $q_i^{-1}\cdot_G X_i=s^{-1}\cdot_G(sq_i^{-1}X_i)\in \phi_s(D_\delta(S))$. Hence $$\sum\limits_{i}\lambda_i E_{q_i^{-1}\cdot_G X_i}\in\phi_s(D_\delta(S))$$  and we obtain
$$D=\overline{\bigcup\limits_{p\in S}\phi_p(D_\delta(S))}$$
Therefore $\Phi$ is surjective and we get the isomorphism $D_\delta^{(\infty)}(S)\cong D$.\end{proof}


On $B(L^2(G))$, we have the adjoint action of $G$ generated by the left regular representation: $A\mapsto L_gAL_g^*$.
Let us show that $D$ is invariant under this action. For $g\in G$
$$
L_g E_{q^{-1}\cdot_G X} L_g^* = E_{(gq^{-1})\cdot_G X}.
$$
Since $G=S^{-1}S$, we can write $gq^{-1} = t^{-1}s$ with some $s,t\in S$. Then $s\cdot_G X = s\cdot X\in \mathcal J$, and $E_{(gq^{-1})\cdot_G X} = E_{(t^{-1}s)\cdot_G X}\in D$.

The isomorphism $\Phi$ defined above allows then to define an action of $G$ on $D_\delta^{(\infty)}(S)$: for $u\in D_\delta^{(\infty)}(S)$, $g\in G$ we set
\begin{equation}\label{taug}
\tau(g) u = \Phi^{-1}( L_g\Phi(u)L_g^*).
\end{equation}
$(D_\delta^{(\infty)}(S),G,\tau)$ is a $C^*$-dynamical system, and by definition the pair  $(\Phi,L)$ is a covariant representation of this system.

\begin{lemma}\label{cros} The reduced crossed product $D_\delta^{(\infty)}(S)\rtimes_{r,\tau}G $ of the commutative $C^*$-algebra $D_\delta^{(\infty)}(S)$ and the group $G$ by the action $\tau$, generated by the covariant representation $(\Phi,L)$, is isomorphic to  $C^*(\{E_X,L_g\colon X\in S^{-1}\cdot\mathcal{J},g\in G\}) $.
\end{lemma}
\begin{proof}
The integrated form $\Phi\times L$ of the covariant representation $(\Phi,L)$ is a faithful representation of $C_c(G,D_\delta^{(\infty)}(S))$ on $L^2(G)$ and can be extended to a covariant representation $(\Phi\times L)\otimes 1$ on $L^2(G)\otimes L^2(G)=L^2(G\times G)$. Consider a unitary operator $W\in B(L^2(G)\otimes L^2(G))$, given by $W\xi(x,y)=\xi(x,xy)$ for any $\xi\in L^2(G)\otimes L^2(G)$. We claim that $W$ provides unitary equivalence between the representations $(\Phi\times L)\otimes 1$ and $(\Phi\times L)\otimes L$. To show this it is sufficient to check the equation for the components of the representations on the generators of the algebra $D_\delta^{(\infty)}(S)$ and of the group $G$. Denote $\Phi^{-1}(E_X)=\widetilde{E_X}$ for any $X\in S^{-1}\cdot\mathcal{J}$ and take $g,x,y\in G$.

$$W(L\otimes L)(g)W^*\xi(x,y)=(L\otimes L)(g)W^*\xi(x,xy)=W^*\xi(g^{-1}x,g^{-1}xy)=$$ $$\quad =\xi(g^{-1}x,y)=(L\otimes 1)(g) \xi(x,y),$$
$$W(\Phi\otimes 1)(\widetilde{E_X})W^*\xi(x,y)= (\Phi\otimes 1)(\widetilde{E_X})W^*\xi(x,xy)=I_X(x)W^*\xi(x,xy)=$$ $$\quad =I_X(x)\xi(x,y)=(\Phi\otimes 1)(\widetilde{E_X})$$

Using Lemma A.18 in \cite{Echt} one can see that $(\Phi\times L)\otimes L$ is unitarily equivalent to the integrated form of the regular representation $\mathrm{Ind}\Phi$, which gives the reduced crossed product $D_\delta^{(\infty)}(S)\rtimes_{r,\tau}G $. Finally, combining this fact with unitary equivalence by operator $W$  we deduce that $C^*(\{E_X,L_g\colon X\in S^{-1}\cdot\mathcal{J},g\in G\}) $ is isomorphic to $D_\delta^{(\infty)}(S)\rtimes_{r,\tau}G $.
\end{proof}


\begin{theorem}\label{crcross}
The algebra $C^*_\delta(S)$ is isomorphic to $E_S(\pi(D_\delta^{(\infty)}(S)\rtimes_{r,\tau}G))E_S$, where $\pi$ is the isomorphism defined in Lemma \ref{cros}.
\end{theorem}
\begin{proof}
In fact, the algebra $\mathcal A$ in question is the same as in Lemma \ref{l2}, what we will now show.
By Lemma \ref{cros}, $E_S(\pi(D_\delta^{(\infty)}(S)\rtimes_{r,\tau}G))E_S$ is generated as a linear space by the operators $E_SE_{q^{-1}\cdot_G X} L_{a^{-1}}L_b E_S$ with $q,a,b\in S$, $X\in\mathcal{J}$.
Such an operator can be written in another form, using the fact that $L_gE_X=E_{g\cdot_G X}L_g= E_{g\cdot_G X}L_g E_S$ for all $g\in G$, $X\subset S$, and $r^{-1}\cdot_GX\cap S=r^{-1}X$ for $r\in S$, $X\in\mathcal{J}$. Using these identities we obtain:
\begin{align*}
E_SE_{q^{-1}\cdot_G X} L_{a^{-1}}L_b E_S&=E_{q^{-1}X} L_{a^{-1}}E_{bS}L_b =\\
E_{q^{-1}X} E_{a^{-1}\cdot_G bS}L_{a^{-1}}L_b &=E_{q^{-1}X} E_{a^{-1}bS}L_{a^{-1}b} E_S.
\end{align*}
Consider the isomorphism in Lemma \ref{l2} and denote it by $\Psi$. Then by Lemma \ref{l2}
 $$E_{a^{-1}bS}L_{a^{-1}b} E_S=\Psi(T_{a^{-1}b});$$
By Lemma \ref{pro} $E_{q^{-1}X}=\Psi(T_{ww^{-1}})$ for some $w\in\mathcal{F}$ which depends on $qX$.
Thus, $E_{q^{-1}X} E_{a^{-1}bS}L_{a^{-1}b} E_S\in \Psi\big(C^*_\delta(S)\big)$. From the other side, $\Psi\big(C^*_\delta(S)\big)$ is generated as a $C^*$-algebra by the operators $L_{a^{-1}b}E_S=\Psi(T_{a^{-1}b})$, $a,b\in S$ which are contained in $\mathcal A$; this shows that $\mathcal A=\Psi(C^*_\delta(S))$, what proves the theorem.
\end{proof}

\begin{theorem}\label{crq}
There exists a comultiplication $\Delta\colon C^*_\delta(S)\to C^*_\delta(S)\mathop{\otimes}\limits_{min}C^*_\delta(S)$, with which $\mathbb{Q}(S) = \big(C^*_\delta(S),\Delta\big)$ is a compact quantum semigroup.
\end{theorem}
\begin{proof}
The map $\Delta_u$ defined at the beginning of the Section 4 is a comultiplication on $C^*(S)$ and admits a restriction $D(S)\to D(S)\otimes_{max} D(S)$. By Lemma \ref{D} $D_\delta(S)$ is isomorphic to $D(S)$. Hence $D_\delta(S)$ is endowed with a comultiplication, denote it by $\Delta$. To see that $\Delta$ induces a limit map on $D_\delta^{(\infty)}(S)$ let us note that $\Delta$ respects the maps $\tau_{qp^{-1}}$. Indeed, for a monomial $V\in \mathcal{A}_p$ and $q\geq p$ we have

$$\Delta\tau_{qp^{-1}}(V)=\Delta(T_{qp^{-1}}VT_{qp^{-1}}^*)=T_{qp^{-1}}VT_{qp^{-1}}^*\otimes T_{qp^{-1}}VT_{qp^{-1}}^*=$$
$$\tau_{qp^{-1}}\otimes \tau_{qp^{-1}}(\Delta(V)) $$
On the generators of $\pi(D_\delta^{(\infty)}(S))$, obviously $\Delta$ acts as
$$ \Delta(E_{q^{-1}\cdot_GX})= E_{q^{-1}\cdot_GX}\otimes E_{q^{-1}\cdot_GX}$$
for $q\in S, X\in\mathcal{J}$.
It follows that $\Delta$ commutes with the action of $G$ on $D_\delta^{(\infty)}(S) $ defined in (\ref{taug}). Consequently, $\Delta$ gives rise to a comultiplication on $\pi(D_\delta^{(\infty)}(S)\rtimes_{r,\tau}G)$, which we also denote by $\Delta$. Due to the fact that $E_S\in \pi(D_\delta^{(\infty)}(S))$, using Lemma \ref{crcross} we obtain the required comultiplication $\Delta$ on $C^*_\delta(S)$.
\end{proof}

\begin{remark}
The bialgebras $C^*(S)$ and $C^*_\delta(S)$ are co-commutative, as for example the group $C^*$-algebra $C^*(G)$ of $G$. But their dual algebras, unlike the Fourier-Stieltjes algebra $B(G)=C^*(G)^*$, cannot be viewed as function algebras on $S$ or even on $G$.
It is possible that $\phi,\psi\in (C^*_\delta(S))^*$ are nonequal but have the same values on $T_a$ and $T_a^*$ for all $a\in S$.

More specifically, consider $G=\Z$, $S=\Z_+$ and $\phi_k(T) = \langle T\delta_k,\delta_k\rangle$, $k\in\Z$. Then $\phi_k(T_a)=\phi_k(T_a^*)=\delta_0(a)$ for all $k\in\Z$, $a\in\Z_+$, but $\phi_k(T_aT_a^*) = I_{\Z_+}(k-a)$ while $\delta_0(T_aT_a^*) = \delta_0(a)$.
\end{remark}

\begin{remark}\label{comm}
If $S$ is abelian and has a non-empty interior in $G=S^{-1}S$ then there exists a natural short exact sequence connecting $C^*_\delta(S)$ with $C^*_\delta(G)$.

Consider the commutator ideal $K$ in $C^*_\delta(S)$, i.~e.\ the ideal generated by $\{[A,B]=AB-BA\colon A,B\in C^*_\delta(S)\}$.
Due to Lemma \ref{l2}, $C^*_\delta(S)=\overline{\lin}\{ E_X L_g: X\in\mathcal{J}, g\in G, X\subset gS\}$. Therefore, 
any generating commutator can be written as:
\begin{equation}\label{EXEY-intersection}
E_XL_{g_1}E_YL_{g_2}-E_YL_{g_2}E_XL_{g_1}=(E_{X\cap (g_1Y)}-E_{Y\cap (g_2X)})L_{g_1g_2}.
\end{equation}
In particular, the commutator ideal contains the following operators:
\begin{align*}
T_aT_a^*-T_a^*T_a&=E_{aS}-E_S,\\
T_a^*(E_X-E_S)T_a&=E_{a^{-1}X}-E_S,\\
T_a(E_X-E_S)T_a^*&=E_{aX}-E_{aS}
\end{align*}
for all $a\in S$, $X\in\mathcal J$.
Consequently, in the quotient $C^*_\delta(S)/K$ we have the equivalence classes $[E_X] = [E_S]$ for all $X\in \mathcal{J}$ and $[E_X L_g] = [E_{gS}L_g]$ for all $X\mathcal J$, $g\in G$ such that $X\subset gS$. Denote $\hat L_g = [E_{gS} L_g]$. For all $g_1,g_2\in G$
$$
\hat L_{g_1} \hat L_{g_2} = [ E_{g_1 \!S\,\cap\, g_1g_2 S} L_{g_1g_2}] = \hat L_{g_1g_2},
$$
and every $\hat L_g$ is unitary since $\hat L_g^*=\hat L_{g^{-1}}$.

Let us show that $\|\sum_{k=1}^n c_k \hat L_{g_k}\|_{C^*_\delta(S)/K} = \|\sum c_k L_{g_k}\|_{B(L^2(G))}$ for all $c_k\in \mathbb{C}$, $g_k\in G$. Since $E_{g_kS}L_{g_k} = L_{g_k}E_S$, we have
$$
\|\sum_{k=1}^n c_k \hat L_{g_k}\|_{C^*_\delta(S)/K} \le \|\sum c_k L_{g_k} E_S\|_{B(L^2(G))}
  \le \|\sum c_k L_{g_k}\|_{B(L^2(G))}.
$$
From the other side, the fact that $S$ has non-empty interior implies that the norm of $T=\sum_{k=1}^n c_k L_{g_k}$ is attained on $H_S$. Indeed, for every $f\in L^2(G)$ and every $\e>0$ there is a compact set $K\subset G$ such that $\|f-E_K f\|<\e$; there exists \cite{mukhin} $g\in G$ such that $gK\subset S$, so that $E_SE_{gK}=E_{gK}$. Set $h=L_gf$. We have $E_{gK}h = L_g(E_Kf)$, and
$$
\|E_Sh-h\| \le \|E_S(h-E_{gK}h)\|+\|E_SE_{gK}h-h\| \le 2\|h-E_{gK}h\| = 2\|f-E_K f\| <2\e.
$$
If $f$ is such that $\|f\|=1$ and $\|Tf\|>\|T\|-\e$, then $\|h\|=1$,
$$
\|T h\| = \|L_g T f\| = \|Tf\|>\|T\|-\e,
$$
and at the same time $\|T E_Sh-Th\| \le 2\|T\|\e$, what implies $\|TE_Sh\|>\|T\|-(1+2\|T\|)\e$. This proves the statement.

Thus, we have an isomorphism $C^*_\delta(S)/K\simeq C^*_\delta(G)$ and the short exact sequence:
\begin{equation}\label{ses}
0\rightarrow K \rightarrow C^*_\delta(S)\rightarrow C^*_\delta(G)\rightarrow 0.\end{equation}
\end{remark}

\begin{example}
Let us calculate the algebra $C^*_\delta(\R_+)$; we specify that in our notation $\R_+=[0,+\infty)$.
\end{example}
By our assumptions, $G=\R$. Below, denote $E_{[t,+\infty)}=E_t$, $t\in\R$.
We have
\begin{align*}
\cal J  & = \{ [t,+\infty): t\in\R_+\} \cup\{\emptyset\},
\end{align*}
and according to Lemma \ref{l2},
$$
C^*_\delta(\R_+) = \overline{\lin}\{ E_t L_g: t\in\R_+, g\in\R, t\geq g\}.
$$
It is worth noting that
$$
D(\R_+) = C^*( E_t: t\in\R_+)\subset B(L^2(\R_+)).
$$
This algebra can be described as the space of functions supported in $\R_+$ and such that $\lim_{t\to t_0-0} f(t)$ exists for every $t_0\in(0,+\infty]$ and $f(t_0)=\lim_{t\to t_0+0} f(t)$ for every $t_0\in[0,+\infty)$. This is the uniform closure of the algebra of piecewise continuous functions, and is sometimes called by the same name.

The  short exact sequence (\ref{ses}) in this case is written as
$$0\rightarrow K \rightarrow C^*_\delta(\R_+)\rightarrow C^*_\delta(\R)\rightarrow 0.$$
By \eqref{EXEY-intersection}, %Using Remark (\ref{comm}) one can see that
the commutator ideal $K$ in $C^*_\delta(\R_+)$ has the following form:
$$
K=\overline{\lin}\{ E_{[a;b)} L_g: a,b\in\R_+, g\in\R, b \geq a\geq g\}.
$$



\section{The semigroup von Neumann algebra}

In the case when $S$ has a non-empty interior (this does not hold automatically, see \cite{Ganesan}), the bialgebra structure can be extended to the weak closure of $C^*_\delta(S)$ in $B(H_S)$. This closure will be denoted $VN(S)$ and termed the {\it semigroup von Neumann algebra}. Moreover, $\Delta$ admits a family of ``multiplicative partial isometries'', which play a role analogous to that of a multiplicative unitary of a locally compact quantum group.

For $d\in S$, denote $S_d = \{(a,b)\in S\times S: da^{-1}b\in S\}$, $I_d =  I_{S_d}$, $I'_d = I_{(a,b): ad^{-1}b\in S}$,
and $H_d = I_d\cdot L^2(S\times S)$.

\begin{lemma}\label{K-subset-Sd}
Let $K\subset S\times S$ be a compact set. There is $d\in S$ such that $K\subset S_d$.
\end{lemma}
\begin{proof}
Set $L=\{ a^{-1}b: a,b\in K\} \subset G$. This is a compact set. Let $U$ be an open neighbourhood of identity and $c\in S$ such that $cU\subset S$. There are $g_k\in G$, $k=1,\dots,n$ such that $L \subset \cup_{k=1}^n g_k U$.

There is $d\in S$ such that $dg_kc^{-1}\in S$ for all $k$: to see this, represent $g_k$ as $g_k = s_kt_k^{-1}$ with $s_k,t_k\in S$; since $S$ is a reversible semigroup, $Ss_k\cap Sct_k\ne\emptyset$ for every $k$, so we can pick $d_k\in S$ such that $d_ks_kt_k^{-1}c^{-1}=d_kg_kc^{-1}\in S$. Next pick $d\in S$ such that $d_k\le d$ for all $k$. Then we have $dd_k^{-1}\in S$, and  $dg_kc^{-1} = dd_k^{-1} d_kg_kc^{-1}\in S$.

It follows that
$$dL \subset d\,\mathop{\cup}_k g_kU = \mathop{\cup}_k dg_kc^{-1}cU \subset \mathop{\cup}_k dg_kc^{-1} S \subset S,
$$ what implies $K\subset S_d$.
\end{proof}

\begin{lemma}\label{cupHd}
$L^2(S\times S) = [\cup_{d\in S} H_d]$.
\end{lemma}
\begin{proof}
It follows from Lemma \ref{K-subset-Sd} that $I_K\in \cup_{d\in S} H_d$ for every compact set $K\subset S\times S$. This implies the statement immediately.
\end{proof}

Define now the following family of operators $W_d$, $d\in S$, on $L^2(S\times S)$: set $(W_d\xi)(a,b) = \xi(a, da^{-1}b)$ for every $\xi\in L^2(S\times S)$. Then $(W_d^*\xi)(a,b) = \xi(a,ad^{-1}b)$,
and $ W_d W_d^* = I_d$, $ W_d^* W_d = I'_d$.

We claim that
\[\label{DeltaWd}
\Delta(T) W_d = W_d (T\otimes 1)
\]
for all $T\in C^*_\delta(S)$.
For $T=T_c$ one verifies directly: for every $a,b\in S$, $\xi\in L^2(S\times S)$
\begin{align*}
\big(\Delta(T_c) W_d \xi\big)(a,b) &= \big((T_c\otimes T_c) W_d \xi\big)(a,b)
 = \big(W_d \xi\big)(c^{-1}a,c^{-1}b)\\
  &= \xi(c^{-1}a, da^{-1}b),
\\\big(W_d (T_c\otimes1) \xi\big)(a,b) &= \big((T_c\otimes 1) \xi\big)(a,da^{-1}b)
  = \xi(c^{-1}a, da^{-1}b).
\end{align*}
A similar calculation shows that the formula \eqref{DeltaWd} is valid also for $T=T_c^*$ with $c\in S$. Since both sides of \eqref{DeltaWd} are norm continuous homomorphisms, it follows that \eqref{DeltaWd} holds for all $T\in C^*_\delta(S)$.

As a consequence, $H_d$ (being the range of $W_d$) is invariant under $\Delta(T)$ for every $T\in C^*_\delta(S)$, $d\in S$.
The identity $\Delta(T)I_d = W_d(T\otimes 1)W_d^*$ shows that $\Delta(\cdot)I_d$ is ultraweakly continuous for every $d$, and as such admits an extension by continuity from $C^*_\delta(S)$ to its weak closure in $B(H_S)$, which we denote by $VN(S)$ and call the {\it semigroup von Neumann algebra} of $S$. By Lemma \ref{cupHd}, we have $\Delta(T)=\lim_{d\in S} \Delta(T)I_d$ (strongly) for every $T$. It follows (immediately by definition, since the net $(I_d)$ is increasing) that $\Delta$ is normal.

The reasoning above results in the following
\begin{theorem}
$VN(S)$ is a quantum semigroup with the comultiplication extended from $C^*_\delta(S)$.
\end{theorem}


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\end{document}


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{AGL M.~A.~Aukhadiev, S.~A.~Grigoryan, E.~V.~Lipacheva, {An operator approach to quantization of semigroups, Sbornik: Mathematics 205:3 1�24, 2014.
1504.00407.extracted_bib
{barnes B.~A.~Barnes, {Representations of the $l^1$-algebra of an inverse semigroup, Trans. Amer. Math. Soc. 218 (1976), 361--396.
1504.00407.extracted_bib
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