On Graded Semigroup $C^*$-Algebras and Hilbert Modules

Abstract

Reduced semigroup $C^*$-algebras for arbitrary cancellative semigroups are studied. It is proved that if there exists a semigroup epimorphism from a semigroup to an arbitrary group $G$, then the corresponding semigroup $C^*$-algebra is topologically $G$-graded. It is also demonstrated that if the group is finite, then the graded semigroup $C^*$-algebra has the structure of a projective Hilbert $C^*$-module.

Introduction

In this paper, we consider reduced semigroup $C^*$-algebras, namely, the algebras generated by left regular representations of cancellative semigroups. The study of such $C^*$-algebras was initiated by Coburn [2, 3], Douglas [4], and Murphy [21, 22]. The further development of the theory of semigroup $C^*$-algebras is due to many authors (see, for example, the references cited in [16]).

The present work is a continuation of the studies of reduced semigroup $C^*$-algebras started in [1, 8–13, 17–19]. Here we study the following problems: the construction of a grading for reduced semigroup $C^*$-algebras, and the existence of a Hilbert $C^*$-module structure on a graded semigroup $C^*$-algebra.

The grading of an object of a category makes it possible to better understand the structure of this object. In the category of $C^*$-algebras, one uses $C^*$-bundles, or Fell bundles, to construct a grading. These bundles were introduced by Fell [7], who employed them to extend the notions of harmonic analysis to the noncommutative case. Exel [5] introduced the notion of a topologically graded $C^*$-algebra. A refinement of the definition of such an algebra is contained in [24]. It is important that the topological grading of a $C^*$-algebra guarantees the existence of special operators which are analogs of the Fourier coefficients. For a detailed account of the theory of graded $C^*$-algebras, we refer the reader to the monograph [6]. In [1, 8, 9, 11, 17], the authors studied questions related to the construction of gradings for different semigroup $C^*$-algebras. In [11], the notion of the $\sigma$-index of a monomial was introduced and used to construct a $G$-grading of a semigroup $C^*$-algebra for a finite cyclic group $G$. In the present study, we employ this notion to construct a $G$-grading of a semigroup $C^*$-algebra for an arbitrary group $G$, including a nonabelian one.

It is well known that the theory of Hilbert $C^*$-modules is a convenient tool for studying $C^*$-algebras. A detailed introduction to the theory of Hilbert $C^*$-modules, as well as many of its applications, is contained in the book [20]. In [23], the authors used the theory of Hilbert $C^*$-modules to define a noncommutative analog of branched coverings. To define the structure of a Hilbert $C^*$-module on a semigroup $C^*$-algebra, we use the conditional expectation of algebraically finite index (see [25]) and thus define a noncommutative covering of a semigroup $C^*$-algebra [23, Definition 1.4].

The paper consists of the introduction and four sections. In Section 1, we give necessary information on semigroup $C^*$-algebras and the definitions of graded and topologically graded $C^*$-algebras. We present all necessary definitions related to Banach and Hilbert modules over $C^*$-algebras.

In Sections 2 and 3, we construct a grading and show that this grading is topological. The main result of these two sections is formulated as follows (Theorem 2). If there exists a semigroup epimorphism $\sigma \colon\, S\to G$ from an arbitrary cancellative semigroup $S$ to an arbitrary group $G$, then the semigroup $C^*$-algebra $C_{\textrm{r}} ^*(S)$ is a topologically $G$-graded $C^*$-algebra. This theorem generalizes the result obtained in [11], where a similar statement was proved for an abelian semigroup and a finite cyclic group.

In Section 4, we study the existence of a Hilbert $C^*$-module structure on the $G$-graded $C^*$-algebra $C_{\textrm{r}} ^*(S)$ under consideration. Namely, we prove that if $G$ is a finite group, then the $C^*$-algebra $C_{\textrm{r}} ^*(S)$ has the structure of a projective Hilbert $C^*$-module.

1. Preliminaries

Throughout the paper, we denote by $S$ an arbitrary cancellative semigroup with identity. Denote the identity in $S$ by $e$.

The object of the present study is the reduced semigroup $C^*$-algebra $C_{\textrm{r}} ^*(S)$. In this connection, we recall the definition of this $C^*$-algebra.

Consider the Hilbert space $l^2(S)$ of square summable complex-valued functions defined on $S$:

$$l^2(S):=\Biggl\{f \colon\, \, S\to {\mathbb C} \biggm| \sum_{a\in S} |f(a)|^2 <+\infty\Biggr\}.$$

Denote the canonical orthonormal basis of $l^2(S)$ by $\{e_a \mid a\in S\}$, where

$$e_a(b):= \begin{cases} \displaystyle 1 & \text{ if }\, \displaystyle a=b,\\ \displaystyle 0 & \text{ if }\, \displaystyle a\neq b. \end{cases}$$

In the algebra of all bounded operators on $l^2(S)$, consider the $C^*$-subalgebra generated by the set of isometries $\{T_a \mid a\in S\}$, where the operator $T_a$ is defined by the formula

$$T_a(e_b)=e_{ab}, \qquad a,b\in S.$$

It is this $C^*$-subalgebra that is called the reduced semigroup $C^*$-algebra $C_{\textrm{r}} ^*(S)$.

Next, for the convenience of the reader, we recall the definition of a $G$-graded $C^*$-algebra from [6, Definition 16.2].

Let ${\mathfrak A}$ be an arbitrary $C^*$-algebra and $G$ be a group. Then ${\mathfrak A}$ is said to be a $G$-graded $C^*$-algebra if there exists a family of linearly independent closed subspaces $\{ {\mathfrak A} _g\subset {\mathfrak A} \}_{g\in G}$ such that the following properties hold for any $g,h\in G$:

• (1) ${\mathfrak A} _g {\mathfrak A} _h\subset {\mathfrak A} _{gh}$,

• (2) ${\mathfrak A} _g^*= {\mathfrak A} _{g^{-1}}$, and

• (3) ${\mathfrak A} = \overline{\bigoplus_{g\in G} {\mathfrak A} _g}$.

In this case, the family of Banach spaces $\{ {\mathfrak A} _g\}_{g\in G}$ is called a $C^*$-algebraic bundle or a Fell bundle over the group $G$.

In [6, Definition 19.2], Exel also considered the notion of grading in a stronger sense. Namely, a $G$-graded $C^*$-algebra ${\mathfrak A}$ is said to be topologically graded if there exists a contractive linear map

$$F \colon\, \, {\mathfrak A} \to {\mathfrak A} _e$$

that coincides with the identity map on the subspace ${\mathfrak A} _e$, where $e$ is the identity element of the group, and vanishes on every subspace ${\mathfrak A} _g$, where $g\in G$, $g\neq e$.

An important property of a topologically graded $C^*$-algebra is the existence of Fourier coefficients (see [6, Corollary 19.6]). This means that for every $g\in G$ there exists a contractive linear map

$$F_g \colon\, \, {\mathfrak A} \to {\mathfrak A} _g$$

such that the equality $F_g(A)=A_g$ holds for any finite sum $A=\sum_{h\in G}A_h$ with $A_h\in {\mathfrak A} _h$. Moreover, the maps $F_g$, $g\in G$, have the following property: the equalities

$$F_g(BA)=BF_{h^{-1}g}(A)\qquad\text{and}\qquad F_g(AB)=F_{gh^{-1}}(A)B$$

hold for any $B\in {\mathfrak A} _h$, $h\in G$, and $A\in {\mathfrak A}$.

The notions related to Banach and $C^*$-Hilbert modules are contained in the books [14, 20]. Recall the necessary definitions. By a module we mean a left module.

A module ${\mathfrak M}$ over a $C^*$-algebra ${\mathfrak A}$ is called a Banach ${\mathfrak A}$-module if it is a Banach space with norm satisfying the inequality $\|A\cdot M\|\leq\|A\|\cdot\|M\|$, where $A\in {\mathfrak A}$ and $M\in {\mathfrak M}$. A subset $X$ in a Banach ${\mathfrak A}$-module ${\mathfrak M}$ is called a set of generators if the finite ${\mathfrak A}$-linear combinations of elements of $X$ are dense in ${\mathfrak M}$. If $X$ is a finite set, then the module ${\mathfrak M}$ is said to be finitely generated.

An element $M$ in an ${\mathfrak A}$-module ${\mathfrak M}$ is said to be cyclic if the following equality holds:

$${\mathfrak M} = {\mathfrak A} \cdot M:=\{A\cdot M \mid A\in {\mathfrak A} \}.$$

A module with a cyclic element is called a cyclic module.

A module ${\mathfrak M}$ over a $C^*$-algebra ${\mathfrak A}$ is called a pre-Hilbert ${\mathfrak A}$-module if it is equipped with a sesquilinear form $\langle \kern1pt \cdot \kern1pt ,\cdot \kern1pt \rangle \colon\, {\mathfrak M} \times {\mathfrak M} \to {\mathfrak A}$, called an ${\mathfrak A}$-valued scalar (or inner) product, that has the following properties for any $M,N\in {\mathfrak M}$ and $A\in {\mathfrak A}$:

• (1) $\langle M,M\rangle\geq 0$,

• (2) $\langle M,M\rangle=0\,$ if and only if $\,M=0$,

• (3) $\langle M,N\rangle=\langle N,M\rangle^*,\,$ and

• (4) $\langle A\cdot M,N\rangle=A\langle M,N\rangle$.

If ${\mathfrak M}$ is a pre-Hilbert ${\mathfrak A}$-module, then one can define a norm $\| \kern1pt {\cdot}\,\|_{ {\mathfrak M} }$ on it as $\|M\|_{ {\mathfrak M} }=\|\langle M,M\rangle\|^{1/2}$ for any $M\in {\mathfrak M}$ (see [20, Proposition 1.2.4]).

A pre-Hilbert ${\mathfrak A}$-module ${\mathfrak M}$ that is complete in the norm $\| \kern1pt {\cdot} \kern1pt \|_{ {\mathfrak M} }$ is called a Hilbert $C^*$-module.

For a $C^*$-algebra ${\mathfrak A}$ and a $C^*$-subalgebra ${\mathfrak B}$ in ${\mathfrak A}$, a linear map $E \colon\, {\mathfrak A} \to {\mathfrak B}$ such that $E(B)=B$ for any $B\in {\mathfrak B}$ and $\|E(A)\|\leq\|A\|$ for any $A\in {\mathfrak A}$ is called a conditional expectation (see [20, 25]). A conditional expectation is said to be faithful if for any positive $A\in {\mathfrak A}$ the equality $E(A)=0$ implies $A=0$. It is well known (see [20, Example 1.3.6]) that in this case one can introduce the structure of a left pre-Hilbert ${\mathfrak B}$-module on the $C^*$-algebra ${\mathfrak A}$ by defining the inner product

$$\langle A,B\rangle=E(AB^*)\qquad\text{for any}\quad A,B\in {\mathfrak A} .$$

2. Grading of the $C^*$-algebra $C_{\textrm{r}} ^*(S)$

Let $G$ be an arbitrary group. Denote the identity element of the group by $e$.

Suppose that there exists a surjective semigroup homomorphism

$$ \sigma \colon\, \,S\to G.$$

(2.1)

Then the semigroup $S$ can be represented as a disjoint union of subsets $S_g$,

$$ S=\bigsqcup_{g\in G}S_g,$$

(2.2)

such that every $S_g$ is the complete preimage of the element $g\in G$, i.e., $\sigma^{-1}(g)=S_g$.

To construct a grading of the semigroup $C^*$-algebra $C_{\textrm{r}} ^*(S)$, we introduce the notion of a monomial as well as the notion of a $\sigma$-index for a monomial and for the corresponding operator in this $C^*$-algebra, where $\sigma$ is the surjective homomorphism (2.1). We use the construction that was first introduced by the author in the joint paper [11].

Consider the free semigroup generated by the set $\{T_a^{-1},T_a^1 \mid a\in S\}$. The elements of this semigroup are words of the form

$$ V=T_{a_k}^{i_k}T_{a_{k-1}}^{i_{k-1}}\dots T_{a_1}^{i_1},$$

(2.3)

where $a_1,\dots,a_k\in S$ and $i_1,\dots,i_k\in\{-1,1\}$. We will call these words monomials. The number $k$ in (2.3) is called the length of the monomial. The semigroup itself is called the monomial semigroup and is denoted by $\mathrm{Mon}$.

The monomial semigroup is an involutive semigroup. The involution is defined on a monomial of the form (2.3) by the formula

$$V^*=T_{a_1}^{-i_1}T_{a_2}^{-i_2}\dots T_{a_k}^{-i_k}.$$

Now we define a map of semigroups $\operatorname{ind} \colon\, \mathrm{Mon} \to G$. For a monomial of the form (2.3), we set by definition

$$\operatorname{ind} V=\sigma(a_k)^{i_k}\sigma(a_{k-1})^{i_{k-1}}\dots\sigma(a_1)^{i_1}.$$

It is easy to see that the equalities

$$ \operatorname{ind} (V\cdot W) = \operatorname{ind} V\cdot \operatorname{ind} W \qquad\text{and}\qquad \operatorname{ind} (V^*) =( \operatorname{ind} V)^{-1}$$

(2.4)

hold for any $V,W\in \mathrm{Mon}$. Hence, the map $\operatorname{ind}$ is an involutive surjective semigroup homomorphism.

Every monomial $V$ defines an operator $\widehat{V}{}$ on the Hilbert space $l^2(S)$ as follows:

$$\widehat{T}{} _a^1=T_a, \qquad \widehat{T}{} _a^{-1}=T_a^*,$$

and if $V$ is a monomial of the form (2.3), then

$$ \widehat{V}{} = \widehat{T}{} _{a_k}^{\,i_k} \widehat{T}{} _{a_{k-1}}^{\,i_{k-1}}\dots \widehat{T}{} _{a_1}^{\,i_1}.$$

(2.5)

We will call operators of the form (2.5) operator monomials.

Lemma 1.

Let $V\in \mathrm{Mon}$ and $\widehat{V}{} e_a\neq 0$ for some basis vector $e_a\in l^2(S)$. Then there exists an element $b\in S$ such that the following equalities hold :

$$\widehat{V}{} e_a=e_b \qquad\textit{and}\qquad \sigma(b)= \operatorname{ind} V\cdot\sigma(a).$$

Proof.

Let $V$ be a monomial of the form (2.3). We will prove the lemma by induction on the length $k$ of the monomial.

Let $k=1$. Then two cases are possible: either $V=T_{a_1}^1$ or $V=T_{a_1}^{-1}$. Let $\widehat{V}{} e_a\neq 0$. Then, in the first case, we obtain $T_{a_1}e_a=e_{a_1a}$. Hence, $b=a_1a$ and

$$\sigma(b)=\sigma(a_1)\sigma(a)= \operatorname{ind} V\cdot\sigma(a).$$

In the second case, we have $T_{a_1}^*e_a=e_b$, where $a=a_1b$. Therefore, $\sigma(a)=\sigma(a_1)\sigma(b)$, and so

$$\sigma(b)=(\sigma(a_1))^{-1}\sigma(a)= \operatorname{ind} V\cdot\sigma(a).$$

Now, consider a monomial $V$ of arbitrary length $k$. Obviously, we can write it as $V=T_{a_k}^{i_k}V'$, where $V'$ satisfies the equalities $\widehat{V}{} 'e_a=e_{b'}$ and $\sigma(b')= \operatorname{ind} V'\cdot\sigma(a)$ by the induction hypothesis. Then we have $\widehat{V}{} e_a= \widehat{T}{} _{a_k}^{i_k}e_{b'}=e_b$. In the same way as in the case of $k=1$, we obtain the equality $\sigma(b)= \operatorname{ind} T_{a_k}^{i_k}\cdot\sigma(b')$. This implies the required assertion

$$\sigma(b)= \operatorname{ind} T_{a_k}^{i_k}\cdot \operatorname{ind} V'\cdot\sigma(a)= \operatorname{ind} V\cdot\sigma(a).\ \quad\Box$$

It follows from Lemma 1 that if $\widehat{V}{} _1= \widehat{V}{} _2$, then $\operatorname{ind} V_1= \operatorname{ind} V_2$.

For an arbitrary monomial $V\in \mathrm{Mon}$, we refer to the value $\operatorname{ind} V$ of the map $\operatorname{ind}$ on $V$ both as the $\sigma$-index of the monomial $V$ and the $\sigma$-index of the operator monomial $\widehat{V}{}$.

Finite linear combinations of operator monomials form an involutive subalgebra, which is dense in the $C^*$-algebra $C_{\textrm{r}} ^*(S)$. Denote this subalgebra by $P(S)$.

It is easy to see that the monomials of $\sigma$-index $e$ form an involutive subsemigroup in the monomial semigroup $\mathrm{Mon}$. Denote by ${\mathfrak A} _e$ the $C^*$-subalgebra in $C_{\textrm{r}} ^*(S)$ generated by the operator monomials of $\sigma$-index $e$.

For any $g\in G$, the closure of the linear hull of the set of all operator monomials of $\sigma$-index $g$ is a Banach subspace in the $C^*$-algebra $C_{\textrm{r}} ^*(S)$, which we denote by ${\mathfrak A} _g$.

Next, using the expansion (2.2), we represent the Hilbert space $l^2(S)$ as an orthogonal sum of subspaces:

$$ l^2(S)=\bigoplus_{g\in G}H_g,$$

(2.6)

where every subspace $H_g$, $g\in G$, has a Hilbert basis given by the family of functions $\{e_a \mid a\in S_g\}$.

The following lemma shows how the spaces $H_g$ behave under the action of the elements of the space ${\mathfrak A} _h$, for $g,h\in G$.

Lemma 2.

For any $g,h\in G$ and any operator $A\in {\mathfrak A} _h,$ the following inclusion holds :

$$A(H_g)\subset H_{hg}.$$

In particular, for any $g\in G,$ the subspace $H_g$ is invariant under the action of any element of the $C^*$-algebra ${\mathfrak A} _e$.

Proof.

Since finite linear combinations of operator monomials of $\sigma$-index $h$ form a dense subspace in the Banach space ${\mathfrak A} _h$, it suffices to prove the lemma for such operator monomials.

Fix elements $h,g\in G$. Let $V$ be an arbitrary monomial of $\sigma$-index $h$. Take any basis vector $e_a\in H_g$ with $\widehat{V}{} e_a\neq 0$. If there is no such vector, then $\widehat{V}{} (H_g)=\{0\}\subset H_{hg}$. By Lemma 1, we obtain $\widehat{V}{} e_a=e_b$, where $b\in S$ is an element such that $\sigma(b)= \operatorname{ind} V\cdot\sigma(a)$. Since $\sigma(a)=g$, we have $\sigma(b)=hg$. Thus, $b\in S_{hg}$ and $e_b\in H_{hg}$. $\quad\Box$

Next, we apply Lemma 2 in order to prove that the family of subspaces $\{ {\mathfrak A} _g \mid g\in G\}$ is a $C^*$-algebraic bundle over the group $G$. This statement is a generalization of Lemma 3 from [11].

Lemma 3.

The following assertions hold for the system of subspaces $\{ {\mathfrak A} _g\mid g\in G\}$ :

• (1) ${\mathfrak A} _g {\mathfrak A} _h\subset {\mathfrak A} _{gh},$

• (2) ${\mathfrak A} _g^*= {\mathfrak A} _{g^{-1}},$

• (3) the family $\{ {\mathfrak A} _g \mid g\in G\}$ is a linearly independent system of closed subspaces in the $C^*$-algebra $C_{\textrm{r}} ^*(S),$ and

• (4) $C_{\textrm{r}} ^*(S)=\overline{\bigoplus_{g\in G} {\mathfrak A} _g}$.

Proof.

For operator monomials, assertions (1) and (2) follow from equalities (2.4). In the general case, the assertions hold since the finite linear combinations of operator monomials of $\sigma$-index $g$ are dense in the Banach space ${\mathfrak A} _g$.

Let us prove assertion (3). Let $A=\sum_{g\in G}A_g=0$, where $A_g\in {\mathfrak A} _g$. Note that this sum contains a finite number of nonzero terms. We will show that then $A_g=0$ for any $g$. Let $g_0\in G$ be an element such that $A_{g_0}\neq 0$. Then we have the representation

$$A_{g_0}=-\sum_{g\in G,\; g\neq g_0}A_g.$$

By Lemma 2, the inclusion $A_{g_0}(H_h)\subset H_{g_0h}$ holds for any $h\in G$. This implies the relation

$$-\sum_{g\in G,\;g\neq g_0}A_g(H_h) \subset H_{g_0h}.$$

On the other hand, by the same Lemma 2, we have the inclusion

$$-\sum_{g\in G,\;g\neq g_0}A_g(H_h)\subset \bigoplus_{g\in G,\;g\neq g_0}H_{gh}.$$

Since $g_0h\neq gh$ and the subspaces $H_g$, $g\in G$, are orthogonal, we obtain the equality

$$H_{g_0h}\cap\bigoplus_{g\in G,\;g\neq g_0} H_{gh}=\{0\}.$$

This implies the relation

$$A_{g_0}=-\sum_{g\in G,\;g\neq g_0}A_g=0.$$

Let us prove assertion (4). Note that the finite linear combinations of operator monomials are dense in the $C^*$-algebra $C_{\textrm{r}} ^*(S)$. On the other hand, they are dense in $\bigoplus_{g\in G} {\mathfrak A} _g$, since every such finite linear combination can be represented as $\sum_{g\in G}A_g$, where $A_g$ is a finite linear combination of operator monomials of $\sigma$-index $g$. From the inclusion $\bigoplus_{g\in G} {\mathfrak A} _g\subset C_{\textrm{r}} ^*(S)$ we find that the subspace $\bigoplus_{g\in G} {\mathfrak A} _g$ is dense in $C_{\textrm{r}} ^*(S)$. $\quad\Box$

Thus, Lemma 3 implies the following result on the $G$-grading of the semigroup $C^*$-algebra $C_{\textrm{r}} ^*(S)$.

Theorem 1.

Let $\sigma \colon\, S\to G$ be a surjective semigroup homomorphism. Let ${\mathfrak A} _g$ be a closed subspace in $C_{\textrm{r}} ^*(S)$ generated by the operator monomials of $\sigma$-index $g,$ where $g\in G$. Then the family of subspaces $\{ {\mathfrak A} _g \mid g\in G\}$ forms a Fell bundle of the semigroup $C^*$-algebra $C_{\textrm{r}} ^*(S)$ over the group $G$.

3. Topological grading of the $C^*$-algebra $C_{\textrm{r}} ^*(S)$

Here we prove that the semigroup $C^*$-algebra $C_{\textrm{r}} ^*(S)$ is topologically graded (for the grading constructed in Section 2).

In the next lemma, we construct a linear bounded operator that allows us to talk about the topological $G$-grading of the semigroup $C^*$-algebra $C_{\textrm{r}} ^*(S)$.

Lemma 4.

There exists a contractive linear operator

$$F \colon\, \, C_{\textrm{r}} ^*(S)\to {\mathfrak A} _e$$

that coincides with the identity operator on ${\mathfrak A} _e$ and vanishes on every subspace ${\mathfrak A} _g,$ $g\in G,$ $g\neq e$.

Proof.

Recall that the involutive subalgebra $P(S)$, which consists of all finite linear combinations of operator monomials, is dense in the $C^*$-algebra $C_{\textrm{r}} ^*(S)$. Therefore, to prove the lemma, it suffices to construct a linear bounded operator

$$F \colon\, \, P(S) \to {\mathfrak A} _e$$

that leaves invariant the linear combinations of operator monomials of $\sigma$-index $e$ and vanishes on the linear combinations of operator monomials of $\sigma$-index $g$ for every $g\neq e$.

Since every element $A\in P(S)$ can be uniquely represented as a finite sum of nonzero terms of the form

$$ A=\sum_{g\in G}A_g,$$

(3.1)

where $A_g\in {\mathfrak A} _g$, the formula

$$F(A)=A_e$$

obviously defines a linear operator on the normed space $P(S)$ that satisfies the required conditions.

Let us prove that $F$ is a contractive operator. To this end, we fix an arbitrary element $A\in P(S)$ and consider its representation (3.1).

Let us show that the following estimate for the norms holds:

$$ \|F(A)\|=\|A_e\|\leq\|A\|.$$

(3.2)

Recall that the Hilbert space $l^2(S)$ decomposes into the orthogonal sum (2.6). By Lemma 2, the subspaces $H_g$ are invariant with respect to ${\mathfrak A} _e$. Therefore, for every element $A_e\in {\mathfrak A} _e$, we have the decomposition

$$A_e=\bigoplus_{g\in G} A_e^g,$$

where $A_e^g$ denotes the restriction of the operator $A_e$ to the subspace $H_g$. This implies the following equality for the operator norms:

$$\|A_e\|=\sup_{g\in G} \|A_e^g\|.$$

Fix an arbitrary number $\varepsilon>0$. Let $g_0\in G$ be an element such that the inequality

$$ \|A_e^{g_0}\|\geq\|A_e\|-\varepsilon$$

(3.3)

is satisfied. Note that the following inequality holds:

$$\|A\|=\sup_{\|x\|=1,\; x\in l^2(S)}(Ax,Ax)^{1/2}\geq\sup_{\|x\|=1,\; x\in H_{g_0}}(Ax,Ax)^{1/2}.$$

For $x\in H_{g_0}$, consider the inner product

$$(Ax,Ax)=\Biggl(\;\sum_{g\in G}A_gx,\sum_{h\in G}A_hx\Biggr)=\sum_{g,h\in G}(A_gx,A_hx).$$

Let us show that $(A_gx,A_hx)=0$ for $g\neq h$. Indeed, since $x\in H_{g_0}$, by Lemma 2 we have $A_gx\in H_{gg_0}$ and $A_hx\in H_{hg_0}$. If $g\neq h$, then $gg_0\neq hg_0$, and since the subspaces $H_{gg_0}$ and $H_{hg_0}$ are orthogonal, we obtain $(A_gx,A_hx)=0$. Thus, we have the estimate

$$(Ax,Ax)=\sum_{g\in G}(A_gx,A_gx)\geq(A_ex,A_ex),$$

which implies the inequality

$$ \|A\|\geq\sup_{\|x\|=1,\; x\in H_{g_0}}(Ax,Ax)^{1/2}\geq\sup_{\|x\|=1,\; x\in H_{g_0}}(A_ex,A_ex)^{1/2}=\|A_e^{g_0}\|.$$

(3.4)

Since $\varepsilon$ is arbitrary, from inequalities (3.3) and (3.4) we obtain the required inequality (3.2). $\quad\Box$

Note that the constructed linear operator

$$F \colon\, \, C_{\textrm{r}} ^*(S)\to {\mathfrak A} _e$$

is a conditional expectation. We will call it the conditional expectation associated with the grading $\{ {\mathfrak A} _g \mid g\in G\}$.

Lemmas 3 and 4 allow us to claim that the $C^*$-algebra $C_{\textrm{r}} ^*(S)$ is topologically graded.

Theorem 2.

Let $\sigma \colon\, S\to G$ be a surjective semigroup homomorphism. Let ${\mathfrak A} _g$ be a closed subspace in $C_{\textrm{r}} ^*(S)$ generated by the operator monomials of $\sigma$-index $g,$ where $g\in G$. Then the system of subspaces $\{ {\mathfrak A} _g \mid g\in G\}$ forms a topological $G$-grading of the semigroup $C^*$-algebra $C_{\textrm{r}} ^*(S)$.

4. Finitely generated projective Hilbert ${\mathfrak A} _e$-module

In this section, we show that the $C^*$-algebra $C_{\textrm{r}} ^*(S)$ is a left Banach ${\mathfrak A} _e$-module. Moreover, if $G$ is a finite group, then $C_{\textrm{r}} ^*(S)$ is a finitely generated projective Hilbert ${\mathfrak A} _e$-module.

Recall that the semigroup $S$ can be represented as the disjoint union (2.2). Let us fix an arbitrary element $x_g$ in each set $S_g$. Denote the set of all such $x_g$ by $X$. Thus, $X\subset S$ and $X\cap S_g=\{x_g\}$ for any $g\in G$.

By a set of representatives of the classes $\{S_g \mid g\in G\}=\{\sigma^{-1}(g) \mid g\in G\}$ we will mean an arbitrary subset $X\subset S$ with the following property: for any $g\in G$, there exists a unique $x\in X$ such that $X\cap S_g=\{x\}$.

Lemma 5.

For every $g\in G,$ the equality

$${\mathfrak A} _g= {\mathfrak A} _e\cdot T_{x_g}$$

holds ; i.e., the space ${\mathfrak A} _g$ is a cyclic Banach ${\mathfrak A} _e$-module, and the element $T_{x_g}$ is a cyclic element of the module ${\mathfrak A} _g$.

Proof.

First, let us show that the inclusion ${\mathfrak A} _e\cdot T_{x_g}\subset {\mathfrak A} _g$ holds. If $A^{(e)}=\sum_i\alpha_i \widehat{V}{} ^{(e)}_i$ is a finite linear combination of operator monomials of $\sigma$-index $e$, where $\alpha_i\in{\mathbb C}$, then $A^{(e)}T_{x_g}=\sum_i\alpha_i \widehat{V}{} ^{(e)}_iT_{x_g}$ is obviously a finite linear combination of operator monomials of $\sigma$-index $g$. Let $\{A^{(e)}_n\}$ be a sequence of such linear combinations, and let $\lim_{n\to\infty}A^{(e)}_n=B_e\in {\mathfrak A} _e$. Then, in view of the inequality

$$\|A_n^{(e)}T_{x_g}-B_eT_{x_g}\|\leq\|A_n^{(e)}-B_e\|\cdot\|T_{x_g}\|=\|A_n^{(e)}-B_e\|,$$

we obtain $B_eT_{x_g}=\lim_{n\to\infty}A^{(e)}_nT_{x_g}\in {\mathfrak A} _g$.

Let us establish the reverse inclusion ${\mathfrak A} _g\subset {\mathfrak A} _e\cdot T_{x_g}$. Let $B_g\in {\mathfrak A} _g$. Set $B_e:=B_g T_{x_g}^*$. Then we obviously have $B_g=B_eT_{x_g}$. Just as above, taking into account that the $\sigma$-index of $T_{x_g}^*$ is equal to $g^{-1}$, we can show that $B_e\in {\mathfrak A} _e$. This completes the proof of the lemma. $\quad\Box$

Assertion (4) of Lemma 3 and Lemma 5 imply the following theorem.

Theorem 3.

Let $G$ be an arbitrary group with identity $e,$ $\sigma \colon\, S\to G$ be a surjective semigroup homomorphism, and $X$ be a set of representatives of the classes $\{\sigma^{-1}(g) \mid g\in G\}$. Let ${\mathfrak A} _e$ be the $C^*$-subalgebra in $C_{\textrm{r}} ^*(S)$ generated by the operator monomials of $\sigma$-index $e$. Then the $C^*$-algebra $C_{\textrm{r}} ^*(S)$ is a Banach ${\mathfrak A} _e$-module with generating set $\{T_x \mid x\in X\}$.

As pointed out in Section 1, since the system of subspaces $\{ {\mathfrak A} _g \mid g\in G\}$ forms a topological $G$-grading of the $C^*$-algebra $C_{\textrm{r}} ^*(S)$, for every $g$ there exists a contractive linear map

$$F_g \colon\, \, C_{\textrm{r}} ^*(S)\to {\mathfrak A} _g$$

such that for any finite sum $A=\sum_{g\in G}A_g$ with $A_g\in {\mathfrak A} _g$ we have the equality $F_g(A)=A_g$. Moreover, the maps $F_g$ satisfy the equalities

$$ F_g(AB)=F_{gh^{-1}}(A)B \qquad\text{and}\qquad F_g(BA)=BF_{h^{-1}g}(A)$$

(4.1)

for any $A\in C_{\textrm{r}} ^*(S)$ and $B\in {\mathfrak A} _h$.

Note that the construction of the operator $F$ in Lemma 4 and the principle of extension by continuity (see, for example, [15, Ch. 2, §1, Theorem 2]) immediately imply the equality $F_e=F$.

Next, let $G$ be a finite group.

First, we examine the geometry of the underlying Banach space of the reduced semigroup $C^*$-algebra $C_{\textrm{r}} ^*(S)$.

Theorem 4.

Let $G$ be a finite group with identity $e,$ $\sigma \colon\, S\to G$ be a surjective semigroup homomorphism, and $X$ be a set of representatives of the classes $\{\sigma^{-1}(g) \mid g\in G\}$. Let ${\mathfrak A} _e$ be the $C^*$-subalgebra in $C_{\textrm{r}} ^*(S)$ generated by the operator monomials of $\sigma$-index $e$. Then the $C^*$-algebra $C_{\textrm{r}} ^*(S)$ is a finitely generated Banach ${\mathfrak A} _e$-module with a set of generators $\{T_x \mid x\in X\}$. Moreover, it can be represented as a direct sum of a finite number of cyclic ${\mathfrak A} _e$-modules :

$$C_{\textrm{r}} ^*(S)=\bigoplus_{x\in X} {\mathfrak A} _e\cdot T_x.$$

Proof.

First, we prove the equality of spaces

$$ C_{\textrm{r}} ^*(S)=\bigoplus_{g\in G} {\mathfrak A} _g.$$

(4.2)

To this end, we show that the equality

$$ A=\sum_{g\in G}F_g(A)$$

(4.3)

holds for any $A\in C_{\textrm{r}} ^*(S)$. Let $\{A_n\}_{n\in\mathbb{N}}$ be a sequence in $P(S)$ that converges to $A$. Note that equality (4.3) holds for any finite sum of the form $A=\sum_{g\in G}A_g$; i.e., for any $A_n\in P(S)$, we have

$$A_n=\sum_{g\in G}(A_n)_g=\sum_{g\in G}F_g(A_n).$$

Then we obtain the chain of inequalities

$$\begin{aligned} \, \Biggl\|A-\sum_{g\in G}F_g(A)\Biggr\| &\leq \|A-A_n\|+\Biggl\|\sum_{g\in G}F_g(A_n)-\sum_{g\in G}F_g(A)\Biggr\| \\[4pt] &\leq\|A-A_n\|+\sum_{g\in G}\|F_g\|\cdot\|A_n-A\|=(k+1)\cdot\|A-A_n\|, \end{aligned}$$

where $k$ is the order of the group $G$. This estimate implies equality (4.3).

Now, employing Lemma 5, we obtain the assertion of the theorem. $\quad\Box$

In fact, as we will see below, if the group $G$ is finite, then the $C^*$-algebra $C_{\textrm{r}} ^*(S)$ is a finitely generated projective $C^*$-Hilbert ${\mathfrak A} _e$-module.

Recall the definition of a finitely generated projective Hilbert module [20]. Let ${\mathfrak M}$ be a Hilbert ${\mathfrak A}$-module such that there exists a Hilbert ${\mathfrak A}$-module ${\mathfrak N}$ for which the direct sum ${\mathfrak M} \oplus {\mathfrak N}$ is isomorphic as a module to the direct sum of a finite number of copies of the Hilbert ${\mathfrak A}$-module ${\mathfrak A}$. Then ${\mathfrak M}$ is called a finitely generated projective ${\mathfrak A}$-module.

In [25, Corollary 3.1.4], conditions on the conditional expectation $E \colon\, {\mathfrak A} \to {\mathfrak B} \subset {\mathfrak A}$ are presented under which ${\mathfrak A}$ is a $C^*$-Hilbert module over the $C^*$-algebra ${\mathfrak B}$.

The conditional expectation $E \colon\, {\mathfrak A} \to {\mathfrak B} \subset {\mathfrak A}$ is called a conditional expectation of algebraically finite index (see [25, Definition 1.2.2] and [20, Sect. 4.5]) if there exists a finite set of elements $X_1,\dots,X_n\in {\mathfrak A}$ such that every element $A\in {\mathfrak A}$ can be represented as

$$A=\sum_{k=1}^nE(AX_k^*)X_k.$$

If the conditional expectation is a conditional expectation of algebraically finite index, then it is faithful [25]. In [25, Corollary 3.1.4], it is demonstrated that the algebraic finiteness of the index is equivalent to the fact that ${\mathfrak A}$ is a finitely generated projective Hilbert $C^*$-module over the $C^*$-algebra ${\mathfrak B}$ (see also [23, Theorem 5.7]).

In the following lemma and theorem, we prove that in the case of a finite group $G$ the conditional expectation $F \colon\, C_{\textrm{r}} ^*(S)\to {\mathfrak A} _e$ constructed in Lemma 4 is a conditional expectation of algebraically finite index and that the $C^*$-algebra $C_{\textrm{r}} ^*(S)$ is a finitely generated projective Hilbert ${\mathfrak A} _e$-module.

Lemma 6.

The conditional expectation $F \colon\, C_{\textrm{r}} ^*(S) \to {\mathfrak A} _e$ associated with the grading $\{ {\mathfrak A} _g \mid g \in G\}$ is a conditional expectation of algebraically finite index.

Proof.

By Theorem 4, any element $A\in C_{\textrm{r}} ^*(S)$ can be represented as the finite sum (4.3). Then, using the properties (4.1) and taking into account that $F_e=F$, we obtain for any $A\in C_{\textrm{r}} ^*(G)$ the equalities

$$A=\sum_{g\in G}F_g(A)=\sum_{g\in G}F_g(AT^*_{x_g}T_{x_g})= \sum_{g\in G}F(AT_{x_g}^*)T_{x_g}.$$

This means that the conditional expectation $F$ constructed in Lemma 4 is a conditional expectation of algebraically finite index. $\quad\Box$

Theorem 5.

Let $G$ be a finite group with identity $e$ and $\sigma \colon\, S\to G$ be a surjective semigroup homomorphism. Let ${\mathfrak A} _e$ be the $C^*$-subalgebra in $C_{\textrm{r}} ^*(S)$ generated by the operator monomials of $\sigma$-index $e$. Then the $C^*$-algebra $C_{\textrm{r}} ^*(S)$ is a finitely generated projective Hilbert ${\mathfrak A} _e$-module.

Proof.

It follows from Lemma 6 that the conditional expectation $F \colon\, C_{\textrm{r}} ^*(S)\to {\mathfrak A} _e$ associated with the grading $\{ {\mathfrak A} _g \mid g\in G\}$ is faithful. Then we can define the following ${\mathfrak A} _e$-valued inner product on the $C^*$-algebra $C_{\textrm{r}} ^*(S)$:

$$\langle A,B\rangle=F(AB^*),$$

where $A,B\in C_{\textrm{r}} ^*(S)$. On the other hand, as already mentioned above, the algebraic finiteness of the index of the conditional expectation $F$ associated with the grading $\{ {\mathfrak A} _g \mid g\in G\}$ implies that the $C^*$-algebra $C_{\textrm{r}} ^*(S)$ is a finitely generated projective Hilbert $C^*$-module over the $C^*$-algebra ${\mathfrak A} _e$. $\quad\Box$