The Crossed Product of Finite Hopf C*-Algebra and C*-Algebra

Abstract

Let H be a finite Hopf $C^{*}$-algebra and A a $C^{*}$-algebra of finite dimension. In this paper, we focus on the crossed product $A⋊H$ arising from the action of H on A, which is a ∗-algebra. In terms of the faithful positive Haar measure on a finite Hopf $C^{*}$-algebra, one can construct a linear functional on the ∗-algebra $A⋊H$, which is further a faithful positive linear functional. Here, the complete positivity of a positive linear functional plays a vital role in the argument. At last, we conclude that the crossed product $A⋊H$ is a $C^{*}$-algebra of finite dimension according to a faithful ∗- representation.

1. Introduction

Quantum groups are a powerful tool to reveal symmetry in physics. Motivated by the quantum method of the inverse problem in integrable quantum systems, Drinfeld [1] popularized the term of quantum groups based on his and Jimbo’s work. Quantum groups stand for Hopf algebras that are nontrivial deformations of the enveloping algebras of the semi-simple Lie algebras, or of those algebras of continuous functions on compact groups. In a word, the quantum group can be considered as the spectrum of a non-commutative Hopf algebra, and braided Hopf algebras provide solutions of the Young–Baxter equation in a systematic way. However, the extension of the Pontryagin duality for non-abelian groups leads necessarily to extensive investigations of the dual objects for a locally compact group. Ernest [2] was the first to use Hopf algebras in this context as far as we know. Building on the work of Enock and Schwartz [3], Kustermans and Vaes [4] proposed locally compact quantum groups, which developed locally compact groups and their duality. Locally compact quantum groups carry a very rich structure; generalizing the quantum groups in the framework of analytics, they can be viewed as $C^{*}$-algebras equipped with comultiplications as well as left and right Haar weights.

Let G be a locally compact space. The set $C_0\left(G\right)$ of all complex continuous functions vanishing at infinity on G is a $C^{*}$-algebra. Indeed, any abelian $C^{*}$-algebra has this form via the famous Gelfand–Naimark theorem. That is the reason why a $C^{*}$-algebra could be viewed as a non-commutative locally compact quantum space. Thus a Hopf $C^{*}$-algebra can be considered as a non-commutative locally compact quantum group. Moreover, Woronowicz [5] developed the standard Hopf algebras in the $C^{*}$-algebra context, which led to the research into Hopf $C^{*}$-algebras. Instead of the formulation of Woronowicz, van Daele [6] established elaborate algebraic Hopf $C^{*}$-algebras, possessing the useful Haar functionals, which are powerful tools for our purpose of studying the structure of Hopf $C^{*}$-algebras.

In this paper, we are interested in finite Hopf $C^{*}$-algebras. As it is well known, the information of the direct product $G\times H$ of groups of G and H could be retrieved from the G and H, respectively. However, the situation becomes complicated when we consider the ubiquitous crossed products, like the case of Hopf $C^{*}$-algebra and $C^{*}$-algebra. In the infinite dimensional case, there is a need for a topological approach; this seems to be very difficult, since there are some subtle constraints on the choice of a $C^{*}$-norm, such that the crossed product of Hopf $C^{*}$-algebra and $C^{*}$-algebra is a $C^{*}$-algebra again. Therefore, there is a possible approach to consider the crossed product algebras in the finite dimensional case. In this case, we prove that the crossed product of a finite Hopf $C^{*}$-algebra and a $C^{*}$-algebra is a $C^{*}$-algebra. Notice that, this result without topological hypothesis has been shown by Blattner R. J. et al. [7]. They demonstrated that when H is a finite dimensional, semisimple Hopf algebra, and A is a semisimple Artinian algebra, the crossed product A#_σH with invertible cocycle σ is semisimple Artinian. That implies the result of the trivial case for σ [8], and it coincides with our conclusion. Whereas, our work highlights the method of proof, which is to construct a faithful positive linear functional, to show the crossed product $A⋊H$ is a $C^{*}$-algebra.

Hopf algebra actions are used in certain contexts in order to achieve some important results, i.e., Kemer’s theory for H-module algebras, applying to the polynomial identity exponent [9]. What is more, to reveal the quantum symmetry in quantum spin models, finite Hopf $C^{*}$-algebra actions are used to construct the quantum double. In particular, Liu et al. [10] established a faithful positive linear functional, and then showed that the quantum double of the pairing of two finite Hopf $C^{*}$-algebras is a $C^{*}$-algebra. However, the proof of the main result in their work is insufficient. Our work provides an efficiency method to demonstrate that the functional they constructed is an exactly faithful positive. In addition, the conclusion of this paper is necessary to construct the observable algebra and the field algebra in Hopf spin models [11].

The paper is organized as follows. To fix our conventions and notations, Section 2 reviews some necessary conceptions and notations on finite Hopf $C^{*}$-algebras and their actions. Section 3 proves that if a finite Hopf $C^{*}$-algebra H acts on a finite dimensional $C^{*}$-algebra A, then the crossed product $A⋊H$ is a $C^{*}$-algebra. Here, the complete positivity of a positive linear functional contributes greatly in the announcement. Section 4 concludes our obtained results and further applications.

2. Preliminaries

Definition 1

([12,13]). Let $(H,m,\iota ,\ast )$ be a ∗-algebra with a unit 1, where $m:H\otimes H\to H$ denotes the multiplication of A, $\iota :\mathbb{C}\to A$ denotes unit map. If there exist ∗-homomorphisms $\Delta :H\to H\otimes H$ and $\epsilon :H\to \mathbb{C}$ such that

$$\begin{array}{ccc}\hfill (\Delta \otimes \mathrm{id})\circ \Delta & =& (\mathrm{id}\otimes \Delta )\circ \Delta ,\hfill \\ \hfill (\epsilon \otimes \mathrm{id})\circ \Delta & =& (\mathrm{id}\otimes \epsilon )\circ \Delta =\mathrm{id},\hfill \end{array}$$

and a ∗-preserving anti-multiplicative map $S:H\to H$ so that, for all $a\in H$,

$$\begin{array}{ccc}\hfill S{\left(S{\left(a\right)}^{*}\right)}^{*}& =& a,\hfill \\ \hfill m(S\otimes \mathrm{id})\Delta \left(a\right)& =& m(\mathrm{id}\otimes S)\Delta \left(a\right)=\epsilon \left(a\right)1,\hfill \end{array}$$

then we call $(H,m,\iota ,\Delta ,\epsilon ,S,\ast )$ a Hopf ∗-algebra, and the structure maps $\Delta ,\epsilon$ and S comultiplication, counit and antipode, respectively.

Furthermore, if H is a Hopf ∗-algebra of finite dimension, and also a $C^{*}$-algebra, then H is said to be a finite Hopf $C^{*}$-algebra.

For any element $a\in H$, we shall denote the element $\Delta \left(a\right)$ in $H\otimes H$ by Sweedler’s sigma notation, which is standard in Hopf algebra theory:

$$\Delta \left(a\right)=\sum _{\left(a\right)}{a}_{\left(1\right)}\otimes a_{\left(2\right)}.$$

Moreover, since the coassociative law $(\Delta \otimes \mathrm{id})\Delta \left(a\right)=(\mathrm{id}\otimes \Delta )\Delta \left(a\right)={\displaystyle \sum _{\left(a\right)}}{a}_{\left(1\right)}\otimes a_{\left(2\right)}\otimes a_{\left(3\right)}$ holds, then the map ${\Delta }^{\left(n\right)}:H\to H^{\otimes (n+1)}$ defined inductively on $n\ge 1$ by ${\Delta }^{\left(1\right)}=\Delta$ is given by

$${\Delta }^n\left(a\right)=({\mathrm{id}}^{\otimes (n-1)}\otimes \Delta )\circ {\Delta }^{n-1}\left(a\right)=\sum _{\left(a\right)}{a}_{\left(1\right)}\otimes a_{\left(2\right)}\otimes \cdots \otimes a_{(n+1)}.$$

Throughout this paper, H denotes a finite Hopf $C^{*}$-algebra. For more details about Hopf algebra one can refer to [14,15].

For a finite Hopf $C^{*}$-algebra H, it is semi-simple and involutive, namely, $S^2=\mathrm{id}$. This implies that ${\displaystyle \sum _{\left(a\right)}}{a}_{\left(2\right)}S\left(a_{\left(1\right)}\right)={\displaystyle \sum _{\left(a\right)}}S\left(a_{\left(2\right)}\right)a_{\left(1\right)}=\epsilon \left(a\right)1$ for any $a\in H$. Furthermore, there exists an invariant functional $\phi$ on H satisfying that for every $a\in H$, $\phi =\phi \circ S$, and

$$(\mathrm{id}\otimes \phi )\Delta \left(a\right)=\phi \left(a\right)1=(\phi \otimes \mathrm{id})\Delta \left(a\right).$$

We call this a Haar functional and also a Haar measure [16], according to the correspondence between positive linear functionals and measures.

Definition 2.

Let H be a Hopf ∗-algebra, and A be a ∗-algebra. A is said to be a left H-module algebra if there exists a bilinear map $\gamma :H\otimes A\to A$ such that the following hold for any $x,y\in H,a,b\in A$:

$$\begin{array}{ccc}\hfill {\gamma }_{xy}\left(a\right)& =& {\gamma }_x\circ {\gamma }_y\left(a\right),\hfill \\ \hfill {\gamma }_x\left(ab\right)& =& \sum _{\left(x\right)}{\gamma }_{x_{\left(1\right)}}\left(a\right){\gamma }_{x_{\left(2\right)}}\left(b\right),\hfill \\ \hfill {\gamma }_x{\left(a\right)}^{*}& =& {\gamma }_{S\left(x^{*}\right)}\left(a^{*}\right).\hfill \end{array}$$

Before proceeding let us briefly introduce some examples of finite Hopf $C^{*}$-algebras and their actions on $C^{*}$-algebras.

Example 1

([17]). Let G be a finite group, $\mathbb{C}G$ is the group algebra of G, and $C\left(G\right)$ is the algebra that consists of the $\mathbb{C}$-valued continuous function on G. Define the structure maps as follows:

$$\Delta \left(g\right)=g\otimes g,\epsilon \left(g\right)=1,S\left(g\right)=g^{-1},g^{*}=g^{-1},g\in G.$$

Thus $(\mathbb{C}G,\Delta ,\epsilon ,S,*)$ is a finite Hopf $C^{*}$-algebra and so is its dual algebra $C\left(G\right)$. Moreover, the following results hold:

(1)

There exists an element $z=\frac{1}{\left|G\right|}{\displaystyle \sum _{g\in G}}g\in \mathbb{C}G$ such that for every $t\in \mathbb{C}G,zt=tz=\epsilon \left(t\right)z$, and $z=z^{*}=z^2=S\left(z\right)$, where S is the antipode of $\mathbb{C}G$, we call the element such as z the normalized Haar integral with $\epsilon \left(z\right)=1$. The element ${\delta }_e:\mathbb{C}G\to \mathbb{C},{\delta }_e\left(g\right)={\delta }_{g,e}$ is a normalized Haar integral in Hopf $C^{*}$-algebra $C\left(G\right)$, where e is the unit of G, ${\delta }_{g,e}=\left\{\begin{array}{cc}1,\hfill & g=e\hfill \\ 0,\hfill & otherwise\hfill \end{array}\right.$. One can check that $z=\frac{1}{\left|G\right|}{\displaystyle \sum _{g\in G}}g$ and ${\delta }_e$ are normalized tracial Haar measures on $C\left(G\right)$ and on $\mathbb{C}G$, respectively.

(2)

Suppose that N is a normal group of G. For $g\in G$, the left adjoint action $A{d}_g:\mathbb{C}N\to \mathbb{C}N$, $A{d}_g\left(h\right)=gh{g}^{-1}$ makes $\mathbb{C}N$ a left $\mathbb{C}G$-module algebra.

Example 2

([14]). Let H be a finite Hopf $C^{*}$-algebra and $\widehat{H}$ be the dual of H, which is also a finite Hopf $C^{*}$-algebra with $\langle {\phi }^{*},a\rangle \overline{\langle \phi ,S{\left(a\right)}^{*}\rangle }$. Then $\widehat{H}$ is an H-module algebra under the natural left action of H, denoted by Sweedler’s arrow:

$$a\to \phi =\sum _{\left(\phi \right)}{\phi }_{\left(1\right)}\langle {\phi }_{\left(2\right)},a\rangle ,a\in H,\phi \in \widehat{H}.$$

Indeed, for $a,b\in H,\phi ,\psi \in \widehat{H}$, one has

$$\begin{array}{ccc}\hfill a\to (b\to \phi )& =& \sum _{\left(\phi \right)}{\phi }_{\left(1\right)}\langle {\phi }_{\left(2\right)},a\rangle \langle {\phi }_{\left(3\right)},b\rangle \hfill \\ & =& \sum _{\left(\phi \right)}{\phi }_{\left(1\right)}\langle {\phi }_{\left(2\right)},ab\rangle \hfill \\ & =& \left(ab\right)\to \phi ,\hfill \end{array}$$

$$\begin{array}{ccc}\hfill a\to \left(\phi \psi \right)& =& \sum _{\left(\phi \right)\left(\psi \right)}{\phi }_{\left(1\right)}{\psi }_{\left(1\right)}\langle {\phi }_{\left(2\right)}{\psi }_{\left(2\right)},a\rangle \hfill \\ & =& \sum _{\left(\phi \right)\left(\psi \right)\left(a\right)}{\phi }_{\left(1\right)}{\psi }_{\left(1\right)}\langle {\phi }_{\left(2\right)},a_{\left(1\right)}\rangle \langle {\psi }_{\left(2\right)},a_{\left(2\right)}\rangle \hfill \\ & =& \sum _{\left(a\right)}(a_{\left(1\right)}\to \phi )(a_{\left(2\right)}\to \psi ).\hfill \end{array}$$

Moreover,

$$\begin{array}{ccc}\hfill \left(S\left(a^{*}\right)\right)\to {\phi }^{*}& =& \sum _{\left(\phi \right)}{\phi }_{\left(1\right)}^{*}\langle {\phi }_{\left(2\right)}^{*},S\left(a^{*}\right)\rangle \hfill \\ & =& \sum _{\left(\phi \right)}{\phi }_{\left(1\right)}^{*}\overline{\langle {\phi }_{\left(2\right)},{\left(S\left(S\left(a^{*}\right)\right)\right)}^{*}\rangle }\hfill \\ & =& \sum _{\left(\phi \right)}{\phi }_{\left(1\right)}^{*}\overline{\langle {\phi }_{\left(2\right)},a\rangle },\hfill \\ & =& {(a\to \phi )}^{*}.\hfill \end{array}$$

where the penultimate equality follows from the property ${(S\circ *)}^2=\mathrm{id}$ in H.

3. Conclusions

In this section, we focus on the crossed product that arises from the action of finite Hopf $C^{*}$-algebra on a $C^{*}$-algebra. What is more, we show this crossed product is a $C^{*}$-algebra on a finite dimension.

We start with reviewing some well known facts. Let H be a finite Hopf $C^{*}$-algebra and A a finite dimensional $C^{*}$-algebra. Suppose that A is a left H-module algebra, then we can define the maps on the vector space $A\otimes H$ in following way, for all $a,b\in A,x,y\in H$,

$$\begin{array}{ccc}\hfill (a\otimes x)(b\otimes y) :=& & \sum _{\left(x\right)}a(x_{\left(1\right)}.b)\otimes x_{\left(2\right)}y,\hfill \\ \hfill {(a\otimes x)}^{*} :=& & \sum _{\left(x\right)}{x}_{\left(1\right)}^{*}.a^{*}\otimes x_{\left(2\right)}^{*},\hfill \end{array}$$

where “$x.a$” denotes the action of H on A. Then $A\otimes H$ together with above maps becomes an associative algebra with unit $1_A\otimes 1_H$, which is called the crossed product of A and H, denoted as $A⋊H$. The elements in $A⋊H$ will be presented in form of “$a⋊x$”. One can easily check that $A⋊H$ contains A and H as unital sub-algebras through injective homomorphisms $a\mapsto a⋊1_H$, and $x\mapsto 1_A⋊x$.

Moreover, $A⋊H$ is a unital ∗-algebra. The claim follows from the following short discussion. By virtue of the relation ${\displaystyle \sum _{\left(x\right)}}{x}_{\left(2\right)}S\left(x_{\left(1\right)}\right)=\epsilon \left(x\right)1$, one has ${\left({(a⋊x)}^{*}\right)}^{*}=a⋊x$ for any $a\in A,x\in H$. In addition,

$$\begin{array}{ccc}\hfill {\left((a⋊x)(b⋊y)\right)}^{*}& =& \sum _{\left(x\right)}{(a(x_{\left(1\right)}.b)⋊x_{\left(2\right)}y)}^{*}\hfill \\ & =& \sum _{\left(x\right)\left(y\right)}\left(y_{\left(1\right)}^{*}{x}_{\left(2\right)}^{*}\right).\left((S\left(x_{\left(1\right)}^{*}\right).b^{*})a^{*}\right)⋊y_{\left(2\right)}^{*}{x}_{\left(3\right)}^{*}\hfill \\ & =& \sum _{\left(x\right)\left(y\right)}(\left(y_{\left(1\right)}^{*}{x}_{\left(2\right)}^{*}S\left(x_{\left(1\right)}^{*}\right)\right).b^{*})(\left(y_{\left(2\right)}^{*}{x}_{\left(3\right)}^{*}\right).a^{*})⋊y_{\left(3\right)}^{*}{x}_{\left(4\right)}^{*}\hfill \\ & =& \sum _{\left(x\right)\left(y\right)}(y_{\left(1\right)}^{*}.b^{*})(\left(y_{\left(2\right)}^{*}{x}_{\left(1\right)}^{*}\right).a^{*})⋊y_{\left(3\right)}^{*}{x}_{\left(2\right)}^{*}\hfill \\ & =& \sum _{\left(x\right)\left(y\right)}(y_{\left(1\right)}^{*}.b^{*}⋊y_{\left(2\right)}^{*})(x_{\left(1\right)}^{*}.a^{*}⋊x_{\left(2\right)}^{*})\hfill \\ & =& {(b⋊y)}^{*}{(a⋊x)}^{*}.\hfill \end{array}$$

We now proceed to the main result of this section, showing that the crossed product $A⋊H$ is further a $C^{*}$-algebra. Before moving forward we want to derive the corresponding knowledge from the completely positive maps, which plays a very important role in the argument.

Let A be a $C^{*}$-algebra and $M_n\left(A\right)$ be the set of all $n\times n$-matrices $a=\left[a_{ij}\right]$ with entries $a_{ij}$ in A. Endowed with the matrix multiplication, and ∗-operation, $M_n\left(A\right)$ is an associative ∗-algebra. Let $(\pi ,\mathcal{H})$ be a faithful ∗-representation of A, and ${\mathbb{C}}^n$ a Hilbert space with an orthogonal basis $\{e_1,\dots ,e_n\}$. Then

$$\tilde{\pi }\left(a\right)(\xi \otimes e_j)=\sum _{i=1}^n\pi \left(a_{ij}\right)\xi \otimes e_i,a=\left[a_{ij}\right]\in M_n\left(A\right)$$

defines a representation of $M_n\left(A\right)$, and it is a faithful ∗-representation, which means that $M_n\left(A\right)$ is a $C^{*}$-algebra.

Definition 3

([18]). Let A and B be $C^{*}$-algebras. For each linear map $\omega :A\to B$, define the linear map ${\omega }_n:M_n\left(A\right)\to M_n\left(B\right)$ by

$${\omega }_n\left(\left[a_{ij}\right]\right)=\left[\omega \left(a_{ij}\right)\right].$$

If ${\omega }_n$ is positive, then ω is said to be n-positive. Moreover, if ω is n-positive for all n, then ω is said to be completely positive.

Example 3.

Set $A=M_n\left(\mathbb{C}\right)$. Then the map $a=\left[a_{ij}\right]\mapsto \left[a_{ji}\right]$ is completely positive.

Now we recall the definition of a conditional expectation.

Definition 4.

A linear map $E:A\to B$ from a unital $C^{*}$-algebra A onto its $C^{*}$-subalgebra B with a common unit, satisfying the following conditions:

(a)

(unit preserving) $E\left(1\right)=1$;

(b)

(bimodular property) $E\left(b_{1}a{b}_2\right)=b_{1}E\left(a\right)b_2,\forall b_1,b_2\in B,a\in A$;

(c)

(positivity) $E\left(a^{*}a\right)\ge 0$;

is called a conditional expectation.

Example 4.

The map E given in Definition 4 is completely positive.

It suffices to show that for any n, the matrix $\left[E\left(a_{ij}\right)\right]\in M_n\left(B\right)$ is positive, where $a=\left[a_{ij}\right]\in M_n\left(A\right)$. For a positive element a in $C^{*}$-algebra $M_n\left(A\right)$, it is necessarily a sum of matrices of the form $\left[a_i^{*}{a}_j\right]$ with $a_1,\dots ,a_n\in A$ [18]. Therefore, we need only to verify that ${\displaystyle \sum _{i,j=1}^n}{b}_i^{*}E\left(a_i^{*}{a}_j\right)b_j\ge 0$ for every $b_1,\dots ,b_n\in B,a_1,\dots ,a_n\in A$.

In fact, by the bimodular property and positivity of the map E, one has

$$\begin{array}{ccc}\hfill \sum _{i,j=1}^n{b}_i^{*}E\left(a_i^{*}{a}_j\right)b_j& =& \sum _{i,j=1}^{n}E\left(b_i^{*}{a}_i^{*}{a}_j{b}_j\right)\hfill \\ & =& E((\sum _{j=1}^n{a}_j{b}_j){}^{*}(\sum _{j=1}^n{a}_j{b}_j))\ge 0,\hfill \end{array}$$

which forms the desired result.

Lemma 1

([18]). Let A and B be $C^{*}$-algebras. If B is commutative, then any positive linear map $\omega :A\to B$ is completely positive.

As an immediate consequence from Lemma 1, any positive linear functional $\phi$ on a $C^{*}$-algebra A is always a completely positive map.

Now it is time to arrive at the main result of this paper, which is given by the following theorem.

Theorem 1.

Let H be a finite Hopf $C^{*}$-algebra and A be a finite dimensional $C^{*}$-algebra. Suppose that A is a left H-module algebra, then the crossed product $A⋊H$ is a $C^{*}$-algebra of finite dimension.

Proof. 

The strategy is to construct a faithful positive linear functional $\theta$ on $A⋊H$ first, then using the GNS representation associated to $\theta$ [19], one can obtain that the ∗-algebra $A⋊H$ is a ∗-sub-algebra of $B\left(\mathcal{H}\right)$ for some Hilbert space $\mathcal{H}$, and hence it is a $C^{*}$-algebra with $C^{*}$-norm $\parallel a⋊x\parallel =\theta {\left((a⋊x){(a⋊x)}^{*}\right)}^{1/2}$, which forms the desired result.

Recall that there exists a faithful positive linear functional $\varphi$ on A, and a faithful positive Haar measure $\phi$ on H, with $(\mathrm{id}\otimes \phi )\Delta \left(x\right)=(\phi \otimes \mathrm{id})\Delta \left(x\right)=\phi \left(x\right)1,x\in H$ [16]. For $a\in A,x\in H$, define a linear map $\theta$ on $A⋊H$ as follows:

$$\theta (a⋊x)=\varphi \left(a\right)\phi \left(x\right).$$

For a general element of $A⋊H$, which is of the form ${\displaystyle \sum _{k=1}^n}{a}_k⋊x_k,a_k\in A,x_k\in H$, without loss of generality, assume that vectors ${\left\{a_k\right\}}_{k=1}^n$ are linearly independent in A. One has

$$\begin{array}{ccc}\hfill \theta \left((\sum _{k=1}^n{a}_k⋊x_k)(\sum _{k=1}^n{a}_k⋊x_k){}^{*}\right)& =& \theta \left((\sum _{k=1}^n{a}_k⋊x_k)(\sum _{k=1}^n(\sum _{\left(x_k^{*}\right)}(x_{k_{\left(1\right)}}^{*}.a_k^{*})⋊x_{k_{\left(2\right)}}^{*}))\right)\hfill \\ & =& \theta \left(\sum _{i,j=1}^n\sum _{\left(x_i{x}_j^{*}\right)}{a}_i(\left(x_{i_{\left(1\right)}}{x}_{j_{\left(1\right)}}^{*}\right).a_j^{*})⋊x_{i_{\left(2\right)}}{x}_{j_{\left(2\right)}}^{*}\right)\hfill \\ & =& \sum _{i,j=1}^n\sum _{\left(x_i{x}_j^{*}\right)}\varphi \left(a_i(\left(x_{i_{\left(1\right)}}{x}_{j_{\left(1\right)}}^{*}\right).a_j^{*})\right)\phi \left(x_{i_{\left(2\right)}}{x}_{j_{\left(2\right)}}^{*}\right)\hfill \\ & =& \sum _{i,j=1}^n\varphi \left(a_i(\left(\phi \left(x_i{x}_j^{*}\right)1\right).a_j^{*})\right)\hfill \\ & =& \varphi (\sum _{i,j=1}^n{a}_i\phi \left(x_i{x}_j^{*}\right)a_j^{*}).\hfill \end{array}$$

The next step is to show that the element ${\displaystyle \sum _{i,j=1}^n}{a}_i\phi \left(x_i{x}_j^{*}\right)a_j^{*}\in A$ is positive. Setting $x=[x_i{x}_j^{*}]$, then $x\in M_n\left(H\right)$ is positive. Letting ${\phi }_n:M_n\left(H\right)\to M_n\left(\mathbb{C}\right)$ be given by ${\phi }_n\left(x\right)=\left[\phi \left(x_i{x}_j^{*}\right)\right]$, one can obtain that ${\phi }_n\left(x\right)$ is a positive element in $M_n\left(\mathbb{C}\right)$; since a positive linear functional on a $C^{*}$-algebra is completely positive in terms of Lemma 1, there is an element $C=\left[C_{ij}\right]\in M_n\left(\mathbb{C}\right)$ such that ${\phi }_n\left(x\right)=C{C}^{*}$. Hence ${\left({\phi }_n\left(x\right)\right)}_{ij}={\displaystyle \sum _{k=1}^n}{C}_{ik}\overline{C_{jk}}.$ It leads to the following:

$$\begin{array}{ccc}\hfill \sum _{i,j=1}^n{a}_i\phi \left(x_i{x}_j^{*}\right)a_j^{*}& =& \sum _{i,j=1}^n{a}_i\left(\sum _{k=1}^n{C}_{ik}\overline{C_{jk}}\right)a_j^{*}\hfill \\ & =& \sum _{k=1}^n((\sum _{i=1}^n{C}_{ik}{a}_i)(\sum _{i=1}^n{C}_{ik}{a}_i){}^{*})\ge 0,\hfill \end{array}$$

the claim following from the map $\varphi$ is a positive linear functional.

The last step of completing the proof is to verify that the map $\theta$ is faithful. Indeed, $\theta \left(({\displaystyle \sum _{k=1}^n}{a}_k⋊x_k){({\displaystyle \sum _{k=1}^n}{a}_k⋊x_k)}^{*}\right)=0$ suggests that for every $k=1,\dots ,n$, one has ${\displaystyle \sum _{i=1}^n}{C}_{ik}{a}_i=0.$ All $a_i$ are linearly independent in A and this implies that $\forall k,C_{ik}=0$. As an immediate consequence of the fact that $\phi$ is faithful, one has $x_i=0,i=1,\dots ,n$, which indicates that ${\displaystyle \sum _{k=1}^n}{a}_k⋊x_k=0$. Combining these results, $A⋊H$ is a finite dimensional $C^{*}$-algebra. This completes the proof. □

An immediate consequence is established in the following.

Corollary 1.

With assumptions and notations as above in Theorem 1, if A is a left H-module algebra, then for $x\in H$, the map $x:A\to A,a\mapsto x.a$ is continuous with respect to the $C^{*}$-norm on A.

Remark 1.

In [20], there is an abstract description about the crossed product between $C^{*}$-algebras as follows. If $A,B,C$ are unital $C^{*}$-algebras and $\alpha ,\beta$ are the ∗-homomorphisms from $A,B$ to C, respectively, such that $\alpha \left(A\right)\beta \left(B\right)=C$, then the triple $(C,\alpha ,\beta )$ is said to be a crossed product of A and B. Note that the $C^{*}$-algebra $A⋊H$ given in Theorem 1 is consistent with the above crossed product. Indeed, in the case of Theorem 1, the maps α and β are given by $\alpha :a\in A\mapsto a⋊1\in A⋊H$, and $x\in H\mapsto 1⋊x\in A⋊H$.

4. Applications

In Section 3, we demonstrated that the crossed product of a Hopf $C^{*}$-algebra and a $C^{*}$-algebra is a $C^{*}$-algebra on a finite dimension. This section is devoted to the discussion of the obtained results in Theorem 1, and further applications.

Notice that the crossed product of any two finite Hopf $C^{*}$-algebras makes it possible to determine the observable algebra, and the field algebra in general Hopf spin models [11].

Moreover, the quantum symmetry in Hopf spin models is implemented by the quantum double of finite Hopf $C^{*}$-algebras. In particular, Liu et al. [10] investigated the relations between the pairing of two finite Hopf $C^{*}$-algebras A and B, and then proved that non-degenerate pairing leads to the quantum double $D(A,B)$, which is a new finite Hopf $C^{*}$-algebra. This result is illustrated by their Lemma 3.7.

We first collect some basic facts about $D(A,B)$ that will be needed in the sequel. The quantum double $D(A,B)$ is an algebra $(A\otimes B,m)$ with the multiplication defined by

$$(a,b)(a^{\prime },b^{\prime })\sum _{\left(a^{\prime }\right)\left(b\right)}(a{a}_{\left(2\right)}^{{}^{\prime }},b_{\left(2\right)}{b}^{\prime })\langle a_{\left(1\right)}^{{}^{\prime }},S_B^{-1}\left(b_{\left(3\right)}\right)\rangle \langle a_{\left(3\right)}^{{}^{\prime }},b_{\left(1\right)}\rangle ,$$

where $a,a^{\prime }\in A,b,b^{\prime }\in B$, “$\langle ·,·\rangle$” means non-degenerate bilinear form between A and B. Moreover, $D(A,B)$ is a ∗-algebra with respect to the involution

$${(a,b)}^{*}:= \sum _{\left(a\right)\left(b\right)}(a_{\left(2\right)}^{*},b_{\left(2\right)}^{*})\langle a_{\left(3\right)}^{*},b_{\left(1\right)}^{*}\rangle \langle a_{\left(1\right)}^{*},S_B{\left(b_{\left(3\right)}\right)}^{*}\rangle .$$

Define the map

$$\theta (a,b){\phi }_A\left(a\right){\phi }_B\left(b\right),$$

where ${\phi }_A,{\phi }_B$ are Haar functionals on A and B, respectively. Then $\theta$ is a faithful positive linear functional on $D(A,B)$.

Lemma 3.7 in [10] plays an important role in the above assertion. However, the proof of Lemma 3.7 is incomplete. Notice that in order to establish the positivity of $\theta$, one needs to verify $\theta \left(\left({\displaystyle \sum _{i=1}^n}(a_i,b_i)\right){\left({\displaystyle \sum _{i=1}^n}(a_i,b_i)\right)}^{*}\right)\ge 0$ instead of just $\theta \left((a,b){(a,b)}^{*}\right)\ge 0$. In the same way, $\theta$ is faithful means that the condition $\theta \left(\left({\displaystyle \sum _{i=1}^n}(a_i,b_i)\right){\left({\displaystyle \sum _{i=1}^n}(a_i,b_i)\right)}^{*}\right)=0$ implies ${\displaystyle \sum _{i=1}^n}(a_i,b_i)=0$.

We now give a few words to sketch the proof for Lemma 3.7 in [10] as follows. For ${\displaystyle \sum _{i=1}^n}(a_i,b_i)\in D(A,B)$, one has

$$\begin{array}{cc}\hfill & \theta \left((\sum _{i=1}^n(a_i,b_i))(\sum _{i=1}^n(a_i,b_i)){}^{*}\right)\hfill \\ \hfill =& \sum _{i,j=1}^n\sum _{(a_j^{*})(b_i{b}_j^{*})}(a_i{a}_{j_{(2)}}^{*},{(b_i{b}_j^{*})}_{(2)})\langle a_{j_{(1)}}^{*},S^{-1}{(b_i{b}_j^{*})}_{(3)}\rangle \langle a_{j_{(3)}}^{*},{(b_i{b}_j^{*})}_{(1)}\rangle \hfill \\ \hfill =& \sum _{i,j=1}^n\sum _{(a_j^{*})(b_i{b}_j^{*})}{\phi }_A(a_i{a}_{j_{(2)}}^{*}){\phi }_B({(b_i{b}_j^{*})}_{(2)})\langle a_{j_{(1)}}^{*},S^{-1}{(b_i{b}_j^{*})}_{(3)}\rangle \langle a_{j_{(3)}}^{*},{(b_i{b}_j^{*})}_{(1)}\rangle \hfill \\ \hfill =& \sum _{i,j=1}^n\sum _{(a_j^{*})(b_i{b}_j^{*})}{\phi }_A(a_i{a}_{j_{(2)}}^{*})\langle a_{j_{(1)}}^{*},S^{-1}{(b_i{b}_j^{*})}_{(2)}\rangle \langle a_{j_{(3)}}^{*},{\phi }_B({(b_i{b}_j^{*})}_{(1)})\rangle \hfill \\ \hfill =& \sum _{i,j=1}^n\sum _{(a_j^{*})(b_i{b}_j^{*})}{\phi }_A(a_i{a}_{j_{(2)}}^{*})\langle a_{j_{(1)}}^{*},S^{-1}({\phi }_B({(b_i{b}_j^{*})}_{(1)}){(b_i{b}_j^{*})}_{(2)})\rangle \hfill \\ \hfill =& \sum _{i,j=1}^n{\phi }_A(a_i{\phi }_B(b_i{b}_j^{*})a_j^{*}).\hfill \end{array}$$

Then one can reach the conclusion in terms of the method given in Theorem 1.

The natural subsequent question leads one to consider the case of infinite dimensional Hopf $C^{*}$-algebra acting on $C^{*}$-algebra. However, the choice of $C^{*}$-norm would impose some subtle constraints, which motivates one to work on these in the future.