On Completion of \textit{C}*-algebra-valued metric spaces

TL;DR

This paper explores the structure of $ C $*-algebra-valued metric spaces, showing their relation to cone metric spaces, and establishes their completion theorem along with connections to Hilbert $ C $*-modules.

Contribution

It introduces the completion theorem for $ C $*-algebra-valued metric and normed spaces, extending the theory and linking it to Hilbert $ C $*-modules.

Findings

$ C $*-algebra-valued metric spaces are cone metric spaces.

Established the completion theorem for these spaces.

Connected $ C $*-algebra-valued normed spaces to Hilbert $ C $*-modules.

Abstract

The concept of a $ C $*-algebra-valued metric space was introduced in 2014. It is a generalization of a metric space by replacing the set of real numbers by a $ C $*-algebra. In this paper, we show that $ C $*-algebra-valued metric spaces are cone metric spaces in some point of view which is useful to extend results of the cone case to $ C $*-algebra-valued metric spaces. Then the completion theorem of $ C $*-algebra-valued metric spaces is obtained. Moreover, the completion theorem of $ C $*-algebra-valued normed spaces is verified and the connection with Hilbert $ C $*-modules, generalized inner product spaces, is also provided.

Equations

$$(a,\alpha )(b,\beta )=(ab+\beta a+\alpha b,\alpha \beta ),$$

$$a\mapsto (a,0).$$

$$(|\gamma |a^{1/2}{)}^2=|\gamma {|}^2(a^{1/2}{)}^2=({\alpha }^2+{\beta }^2)a.$$

$$\Vert x-tI\Vert \le \Vert x_n-x\Vert +\Vert x_n-tI\Vert \le \Vert x_n-x\Vert +t.$$

$$d_X(x,y)=d_Y(Tx,Ty),$$

$$({\xi }_n)+({\eta }_n)$$

$$({\xi }_n)({\eta }_n)$$

$$\lambda ({\xi }_n)$$

$$({\xi }_n{)}^{\ast }$$

$$\Vert ({\xi }_n)\Vert$$

$$\ell^{\infty}_{h}=\big{\{}a\in\ell^{\infty}:a^{*}=a\big{\}}=\big{\{}(\xi_{n})\in\ell^{\infty}:\xi_{n}\in\mathbb{R}\text{ for all }n\in\mathbb{N}\big{\}}$$

$$\ell^{\infty}_{+}=\big{\{}a\in\ell^{\infty}_{h}:\sigma(a)\subseteq\mathbb{R}_{+}\big{\}}=\big{\{}(\xi_{n})\in\ell^{\infty}:\xi_{n}\in\mathbb{R}_{+}\text{ for all }n\in\mathbb{N}\big{\}}.$$

$$d(a,b)=(|a-b|,\alpha |a-b|)$$

$$f(d_X(x,y))=d_Y(T(x),T(y)),$$

$$B(x,\epsilon )=\{y\in X:\Vert d(x,y){\Vert }_{\mathbb{A}}<\epsilon \}$$

$$\mathrm{Cl}(M)=\{x\in X:B(x,\epsilon )\cap M\ne \varnothing \text{ for every }\epsilon >0\}.$$

$$\mathrm{Cl}(M)=\{x\in X:B_1(x,c)\cap M\ne \varnothing \text{ for every }c\gg 0\},$$

$$d^s(x,y)=\underset{n\to \infty }{\lim }{d}^s(x_n,y_n).$$

$$d^s(x_n,y_n)=d(T(x_n),T(y_n))\in \mathbb{A},$$

$$d^s(x,y)=\underset{n\to \infty }{\lim }{d}^s(x_n,y_n)\in \mathbb{A}.$$

$$\Vert x{\Vert }_0=(a^2+b^2{)}^{1/2}.$$

$$\Vert \gamma x{\Vert }_0^2$$

$$={\alpha }^2{a}^2+{\beta }^2{b}^2+{\beta }^2{a}^2+{\alpha }^2{b}^2$$

$$=({\alpha }^2+{\beta }^2)(a^2+b^2).$$

$$\Vert x+y{\Vert }_0^2$$

$$=(a+c{)}^2+(b+d{)}^2$$

$$=(a^2+2ac+c^2)+(b^2+2bd+b^2)$$

$$=(a^2+b^2+c^2+d^2)+2(ac+bd),$$

$$(\Vert x{\Vert }_0+\Vert y{\Vert }_0{)}^2$$

$$=(a^2+b^2)+2(a^2+b^2{)}^{1/2}(c^2+d^2{)}^{1/2}+(c^2+d^2)$$

$$=(a^2+b^2+c^2+d^2)+2(a^2+b^2{)}^{1/2}(c^2+d^2{)}^{1/2}.$$

$$(ac+bd{)}^2$$

$$\le (ac{)}^2+(ad{)}^2+(bc{)}^2+(bd{)}^2$$

$$=(a^2+b^2)(c^2+d^2).$$

$$f(\Vert x{\Vert }_X)=\Vert T(x){\Vert }_Y,$$

$$u\oplus v:=T^{-1}(Tu+Tv)\text{ and }\alpha \odot u:=T^{-1}(\alpha Tu),$$

$$T^{-1}(0_X)\oplus u$$

Full text

On Completion of C-algebra-valued metric spaces*

Wanchai Tapanyo1, Wachiraphong Ratiphaphongthon2,

Areerat Arunchai3 1,3Division of Mathematics and Statistics, Nakhon Sawan Rajabhat University, Nakhon Sawan, Thailand,

2Department of Mathematics, Naresuan University, Phisanulok, Thailand

Email: 1wanchai.t@nsru.ac.th, 2wachirapongr58@nu.ac.th, 3areerat.a@nsru.ac.th

Abstract

The concept of a $C$-algebra-valued metric space was introduced in 2014. It is a generalization of a metric space by replacing the set of real numbers by a $C$-algebra. In this paper, we show that $C$-algebra-valued metric spaces are cone metric spaces in some point of view which is useful to extend results of the cone case to $C$-algebra-valued metric spaces. Then the completion theorem of $C$-algebra-valued metric spaces is obtained. Moreover, the completion theorem of $C$-algebra-valued normed spaces is verified and the connection with Hilbert $C$*-modules, generalized inner product spaces, is also provided.

Keywords: $C$*-algebra-valued metric space, Cone metric space, Normed space, Inner product space, Completion

1 Introduction

A metric space is one of attractive objects in mathematics which plays an important role in various branches of mathematics. It is a nonempty set $X$ together with a distance function $d:X\times X\to\mathbb{R}$, which is often called a metric on $X$. Plenty of research papers study various kinds of spaces generalized from the definition of a metric space in different directions. Some authors remove or change initial properties of a metric space while others change the values of the distance function to be in generalized sets of real or complex numbers, such as, a Banach space or a $C$*-algebra which can be seen in [1] and [2], respectively.

The concept of a $C$-algebra-valued metric space was first introduced in 2014 by Z. Ma and others. For this space the distance function was replaced by a function valued in a $C$-algebra $\mathbb{A}$. If we consider the set of all positive elements $\mathbb{A}_{+}$ of $\mathbb{A}$ as a cone of $\mathbb{A}$. A $C$*-algebra-valued metric space is, in fact, a cone metric space which was introduced in 2004 by L.-G. Huang and others, see more details about a cone metric space in [1].

The main purpose of this research is to study the completion for $C$-algebra-valued metric spaces and a $C$-algebra-valued normed spaces. We verify some facts and use them to extend the results from others in [3]. Then we discuss relationships between $C$-algebra-valued metric spaces and Hilbert $C$-modules, generalized inner product spaces whose scalar fields are replaced by some $C$*-algebras.

The rest of the paper is organized as follows. In section 2 we derive the important definitions and theorems used to obtain our results. In section 3 We discuss on $C$-algebra-valued metric and normed spaces and the relation to cone metric spaces. In section 4 the connection to Hilbert $C$-modules is provided.

2 Preliminaries

This section provides a brief review of basic knowledge used in this research which can be found in [2, 4, 5, 3, 1]. We start with the definition of $C$-algebras and some necessary related properties. Then we mention the definition of $C$-algebra-valued and cone metric spaces and some previous results provided in other research papers.

Definition 2.1**.**

An algebra is a vector space $\mathbb{A}$ together with a bilinear map $\mathbb{A}^{2}\to\mathbb{A},$ $(a,b)\mapsto ab$, such that $a(bc)=(ab)c$ for all $a,b,c\in\mathbb{A}$.

Definition 2.2**.**

An involution on an algebra $\mathbb{A}$ is a conjugate-linear map $a\mapsto a^{*}$ on $\mathbb{A}$, such that $a^{**}=a$ and $(ab)^{*}=b^{*}a^{*}$ for all $a,b\in\mathbb{A}$. The pair $(\mathbb{A},*)$ is called an involutive algebra or $*$-algebra.

Definition 2.3**.**

A Banach $\ast$-algebra is a $\ast$-algebra $\mathbb{A}$ together with a complete norm such that $\|ab\|\leq\|a\|\|b\|$ and $\lVert a^{*}\rVert=\lVert a\rVert$ for every $a,b\in\mathbb{A}$.

Definition 2.4**.**

A $C$-algebra* is a Banach $\ast$-algebra such that $\lVert a^{*}a\rVert=\lVert a\rVert^{2}$ for every $a\in\mathbb{A}$. If $\mathbb{A}$ admits a unit $I$ ($aI=Ia=a$ for every $a\in\mathbb{A}$) such that $\|I\|=1$, we call $\mathbb{A}$ a unital $C$-algebra*.

Remark 2.5**.**

The word “unital” is also used for other terminologies, for example, unital algebra, unital $*$-algebra, unital Banach $*$-algebra. If a norm is not defined, the condition $\|I\|=1$ will be omitted.

Definition 2.6**.**

A homomorphism is a linear map $f$ from an algebra $\mathbb{A}$ to an algebra $\mathbb{B}$ such that $f(ab)=f(a)f(b)$. An isomorphism is a bijective homomorphism. In the case that $\mathbb{A}$ and $\mathbb{B}$ are $*$-algebra, a $*$-homomorphism (resp. $*$-isomorphism) is a homomorphism (resp. isomorphism) $f:\mathbb{A}\to\mathbb{B}$ preserving adjoints, that is, $f(a^{*})=f(a)^{*}$ for every $a\in\mathbb{A}$. In addition, if norms are defined on $\mathbb{A}$ and $\mathbb{B}$, the word “isometric” will be added before all of the terminologies to indicate that $\|f(a)\|=\|a\|$.

Consider the Cartesian $(\mathbb{A},\mathbb{C})$ of $\mathbb{A}$ and the complex plane $\mathbb{C}$, it is $*$-algebra together with a unit $(0,1)$ under componentwise operations for addition and involution, and the multiplication defined by

[TABLE]

for every $a,b\in\mathbb{A}$ and every $\alpha,\beta\in\mathbb{C}$. By the norm obtained in [4, Theorem 2.1.6] we obtain that $(\mathbb{A},\mathbb{C})$ becomes a unital $C$-algebra. The $C$-algebra $\mathbb{A}$ can be embedded in a unital $C$*-algebra $(\mathbb{A},\mathbb{C})$ by the injective isometric $*$-homomorphism defined by

[TABLE]

The Cartesian $(\mathbb{A},\mathbb{C})$ is called the unitization of $\mathbb{A}$ and denoted by $\widetilde{\mathbb{A}}$. Therefore, we may consider $\mathbb{A}$ as a $C$*-subalgebra of $\widetilde{\mathbb{A}}$.

For an element $a$ of a unital algebra $\mathbb{A}$, we say that $a$ is invertible if there is an element $b\in\mathbb{A}$ such that $ab=I=ba$. We denote by $\mathrm{Inv}(\mathbb{A})$ the set of all invertible elements of $\mathbb{A}$. We define the spectrum of $a$ to be the set $\sigma(a)=\sigma_{\mathbb{A}}(a)\{\lambda\in\mathbb{C}:\lambda I-a\notin\mathrm{Inv}(A)\}$. If $\mathbb{A}$ is nonunital, we define $\sigma_{\mathbb{A}}(a)=\sigma_{\widetilde{\mathbb{A}}}(a)$. The following definitions are about characterization of elements in $C$*-algebra.

Definition 2.7**.**

An element $a$ of a $*$-algebra $\mathbb{A}$ is called self-adjoint or hermitian if $a^{*}=a$. The set of all hermitian elements of $\mathbb{A}$ is denoted by $\mathbb{A}_{h}$. If $\mathbb{A}$ is a $C$*-algebra, a self-adjoint element $a\in\mathbb{A}$ with $\sigma(a)\subseteq[0,+\infty)$ is called positive and the set of all positive elements of $\mathbb{A}$ is denoted by $\mathbb{A}_{+}$.

If $\mathbb{A}$ is a $C$*-algebra, $\mathbb{A}_{h}$ becomes a partially ordered set by defining $a\leq b$ to mean $b-a\in\mathbb{A}_{+}$. It is obvious that $0_{\mathbb{A}}\leq a$ precisely for $a\in\mathbb{A}_{+}$ where $0_{\mathbb{A}}$ is a zero in $\mathbb{A}$. Thus, we may write $0_{\mathbb{A}}\leq a$ to indicate that $a$ is positive.

Proposition 2.8** (Murphy).**

Let $\mathbb{A}$ be a $C$**-algebra. Then for each $x\in\mathbb{A}$ there is a unique pair of hermitian elements $a,b\in\mathbb{A}$ such that $x=a+bi$. More precisely, $a=\frac{1}{2}(x+x^{*})$ and $b=\frac{1}{2i}(x-x^{*})$.*

Theorem 2.9**.**

[4, Theorem 2.2.1]** Let $a$ be a positive element of a $C$**-algebra $\mathbb{A}$. Then there is a unique $b\in\mathbb{A}_{+}$ such that $b^{2}=a$.*

By the previous theorem we can define the square root of the positive element $a$ to be the element $b$, we denote it by $a^{1/2}$. The theorem below is a brief review of some necessary properties for positive elements of a $C$*-algebra, see more details in [4].

Lemma 2.10**.**

The sum of two positive elements in a $C$**-algebra are positive.*

Theorem 2.11**.**

Let $\mathbb{A}$ be a $C$**-algebra. The the following properties are satisfied.*

Suppose that $\mathbb{A}$ is unital and $a\in\mathbb{A}$ is hermitian. If $\|a-tI\|\leq t$ for some $t\in\mathbb{R}$, then $a$ in positive. In the reverse direction, for every $t\in\mathbb{R}$, if $\|a\|\leq t$ and $a$ is positive, then $\|a-tI\|\leq t$ 2. 2.

For every $a,b,c\in\mathbb{A}_{h}$, $a\leq b$ implies $a+c\leq b+c$, 3. 3.

For every real numbers $\alpha,\beta\geq 0$ and every $a,b\in\mathbb{A}_{+}$, $\alpha a+\beta b\in\mathbb{A}_{+}$, 4. 4.

$A_{+}=\{a^{*}a:a\in\mathbb{A}\}$, 5. 5.

If $a,b\in A_{h}$ and $c\in A$, then $a\leq b$ implies $c^{*}ac\leq c^{*}bc$, 6. 6.

If $0_{\mathbb{A}}\leq a\leq b$, then $\|a\|\leq\|b\|$.

Lemma 2.12**.**

Let $\gamma=\alpha+\beta i\in\mathbb{C}$ and $a\in\mathbb{A}_{+}$. Then $((\alpha^{2}+\beta^{2})a)^{1/2}=|\gamma|a^{1/2}$.

Proof.

It is obvious that $|\gamma|a^{1/2}$ is positive. Consider

[TABLE]

By Theorem 2.9 , we have $((\alpha^{2}+\beta^{2})a)^{1/2}=|\gamma|a^{1/2}$. ∎

Theorem 2.13**.**

Let $a,b\in\mathbb{A}_{+}$. Then $a\leq b$ implies $a^{1/2}\leq b^{1/2}$.

Next, we provide the definitions of a $C$*-algebra-valued metric space, convergent sequences and Cauchy sequences in the space which are our main study.

Definition 2.14**.**

Let $X$ be a nonempty set and $d:X\times X\to\mathbb{A}$ be a function satisfying the following properties:

1. (C1)

$d(x,y)\geq 0$, 2. (C2)

$d(x,y)=0$ if and only if $x=y$, 3. (C3)

$d(x,y)=d(y,x)$, 4. (C4)

$d(x,y)\leq d(x,z)+d(z,y)$,

for every $x,y,z\in X$. We call the function $d$ a $C$-algebra-valued metric and call the triple $(X,\mathbb{A},d)$ a $C$-algebra-valued metric space.

We know that every $C$-algebra $\mathbb{A}$ can be embedded in $\widetilde{\mathbb{A}}$ which is a unital $C$-algebra. This means that we can work on $\widetilde{\mathbb{A}}$ instead. In other words, an $\mathbb{A}$-valued metric $d$ of the space $(X,\mathbb{A},d)$ is an $\widetilde{\mathbb{A}}$-valued metric as concluded in the remark below.

Remark 2.15**.**

A $C$-algebra-valued metric space $(X,\mathbb{A},d)$ is a $C$-algebra-valued metric space $(X,\widetilde{\mathbb{A}},d)$.

Therefore, we will assume $\mathbb{A}$ to be unital in our research. The following statements are definitions of convergent and Cauchy sequences in a $C$*-algebra-valued metric space which are defined in [2, Definition 2.2]. We change some inequality in the definitions to correspond them to other similar definitions we use frequently.

Definition 2.16**.**

Let $(X,\mathbb{A},d)$ be a $C$*-algebra-valued metric space. A sequence $\{x_{n}\}$ in $X$ is said to converge to an element $x\in X$ (with respect to $\mathbb{A}$) if and only if for every $\varepsilon>0$ there is a positive integer $N$ such that for every integer $n\geq N$ we have $\|d(x_{n},x)\|<\varepsilon$. In this case we write $\lim_{n\to\infty}x_{n}=x$, and say that the sequence $\{x_{n}\}$ is convergent.

A sequence $\{x_{n}\}$ in $X$ is said to be Cauchy (with respect to $\mathbb{A}$) If and only if for every $\varepsilon>0$ there is a positive integer $N$ such that for every integer $n,m\geq N$ we have $\|d(x_{n},x_{m})\|<\varepsilon$.

We say that a $C^{\ast}$-algebra-valued metric space $(X,\mathbb{A},d)$ is complete if every Cauchy sequence (with respect to $\mathbb{A}$) is convergent.

Next, we discuss cone metric spaces. We start with a cone of a real Banach Space which was introduced in [1]. The definition is different one from [5] which allows a cone to be trivial.

Definition 2.17**.**

Let $\mathbb{E}$ be ba real Banach space. A nonempty closed subset $P$ of $\mathbb{E}$ is called a cone if and only if it satisfies the following properties:

1. (P1)

$P\neq\{0\}$, 2. (P2)

For every real numbers $\alpha,\beta\geq 0$ and every $a,b\in P$, $\alpha a+\beta b\in P$, 3. (P3)

If $x\in P$ and $-x\in P$, then $x=0$.

Now we can define a partial order $\leq$ on $\mathbb{E}$ with respect to $P$ by $x\leq y$ to mean $y-x\in P$. We write $x<y$ to indicate that $x\leq y$ and $x\neq y$, and write $x\ll y$ if $y-x\in\mathrm{Int}(P)$.

A cone $P$ is said to be normal if and only if there exists a positive real number $K$ such that for every $x,y\in\mathbb{E}$, $0\leq x\leq y$ implies $\|x\|\leq K\|y\|$. The following proposition is a consequence of Theorem 2.11. $\mathbb{A}_{+}$ is a cone in the sense of the preceding definition.

Proposition 2.18**.**

$\mathbb{A}_{+}$* is a cone of a unital $C$**-algebra $\mathbb{A}$.

Proof.

The proof is immediate from Lemma 2.10 and Theorem 2.11. Let $\{x_{n}\}$ be a sequence in $\mathbb{A}_{+}$ converging to $x\in\mathbb{A}$. Since $\mathbb{A}_{h}$ is closed in $\mathbb{A}$ and $\mathbb{A}_{+}\subseteq\mathbb{A_{h}}$, we have $x\in\mathbb{A}_{h}$. To show that $\mathbb{A}_{+}$ is closed we need to show that $x\in\mathbb{A}_{+}$.

Since $\{x_{n}\}$ is convergent, it is certainly bounded. Then there is a positive real number $t$ such that $\|x_{n}\|\leq t$ for every $n\in\mathbb{N}$. We know that $x_{n}$ is positive for every $n\in\mathbb{N}$. Thus, Theorem 2.11 implies that $\|x_{n}-tI\|\leq t$ for every $n\in\mathbb{N}$. Consider

[TABLE]

This implies that $\|x-tI\|\leq t$. Since $x$ is hermitian, again by Theorem 2.11 we have $x\in\mathbb{A}_{+}$. Therefore, $\mathbb{A}_{+}$ is closed in $\mathbb{A}$. ∎

Definition 2.19**.**

Let $X$ be a nonempty set and $d:X\times X\to\mathbb{E}$ be a function satisfying the following properties:

1. (M1)

$d(x,y)\geq 0$, 2. (M2)

$d(x,y)=0$ if and only if $x=y$, 3. (M3)

$d(x,y)=d(y,x)$, 4. (M4)

$d(x,y)\leq d(x,z)+d(z,y)$,

for every $x,y,z\in X$. We call the function $d$ a cone metric and call the pair $(X,d)$ a cone metric space.

Consider a $C$-algebra $\mathbb{A}$. If the scalar filed is restricted to the set of real numbers, $\mathbb{A}$ becomes a real Banach space. Thus, a $C$-algebra-valued metric space becomes a cone metric space.

Definition 2.20**.**

Let $(X,d)$ be a cone metric space. A sequence $\{x_{n}\}$ in $X$ is said to converge to $x\in X$ (with respect to $\mathbb{E}$) if and only if for every $c\in\mathbb{E}$ with $c\gg 0$ there is a positive integer $N$ such that for every integer $n\geq N$ we have $d(x_{n},x)\ll c$. In this case we write $\lim_{n\to\infty}x_{n}=x$, and say that the sequence $\{x_{n}\}$ is convergent.

A sequence $\{x_{n}\}$ in $X$ is said to be Cauchy (with respect to $\mathbb{E}$) If and only if for every $c\in\mathbb{E}$ with $c\gg 0$ there is a positive integer $N$ such that for every integer $n,m\geq N$ we have $d(x_{n},x_{m})\ll c$.

We say that a cone metric space $(X,d)$ is complete if every Cauchy sequence (with respect to $\mathbb{E}$) is convergent.

Lemma 2.21**.**

Let $(X,d)$ be a cone metric space together with a normal cone. A sequence $\{x_{n}\}$ converses to $x\in X$ if and only if $\lim_{n\to+\infty}d(x_{n},x)=0$. A sequence $\{x_{n}\}$ is Cauchy if and only if $\lim_{n,m\to+\infty}d(x_{n},x_{m})=0$.

Definition 2.22**.**

Let $X$ be a vector space over the real field and $\|\cdot\|:X\to\mathbb{E}$ be a function. A pair $(X,\|\cdot\|)$ is called a cone normed space if $\|\cdot\|$ satisfies the following properties:

$\|x\|=0_{\mathbb{E}}$ if and only if $x=0_{X}$, 2. 2.

$\|\alpha x\|=|\alpha|\|x\|$, 3. 3.

$\|x+y\|\leq\|x\|+\|y\|$,

for every $x,y\in X$ and every scalar $\alpha$.

Theorem 2.23**.**

Let $(X,d)$ be a cone metric space over a normal cone. Then there is a complete cone metric space $(X^{s},d^{s})$ which has a dense subspace $W$ isometric to $X$. The space $X^{s}$ is unique except for isometries.

Theorem 2.24**.**

Let $(X,\|\cdot\|)$ be a cone normed space over a normal cone. Then there is a cone Banach space $(X^{s},\|\cdot\|^{s})$ which has a dense subspace $W$ isometrically isomorphic to $X$. The space $X^{s}$ is unique except for isometric isomorphisms.

The two results above are completion theorems obtained in [3]. We apply the first one to obtain our results. The isometry mentioned in the first theorem is a bijective mapping $T:X\to Y$ between cone metric spaces preserving distances, that is,

[TABLE]

for every $x,y\in X$, where $d_{X}$ and $d_{Y}$ are metrics on $X$ and $Y$, respectively. Properties of the mapping $T$ are different from those of the ordinary version only the values of $d$ and $d^{s}$ which are not real numbers. The second theorem is rewritten from the original version of cone normed spaces. The word “isomorphism” refers to a bijective linear operator between cone normed spaces and the word “isometric” indicates that the isomorphism is a cone-norm-preserving. In [3], an isomorphism between cone normed spaces is always cone-norm-preserving.

Concepts of isometries and of $C$-algebra-values metric spaces and isometric isomorphisms of $C$-algebra-values normed spaces will be provided in the next section with more general than those of the cone version.

3 Completion of C*-algebra-valued metric

and normed spaces

In this section we verify that a $C$-algebra-valued metric space can be embedded in a complete $C$-algebra-valued metric space as a dense subspace. The theorem in a version of a $C$-algebra-valued normed space is also provided. We apply the fact that the $C$-algebra-valued metric (resp. normed) spaces are cone metric (resp. normed) spaces to extend the results from the previous results in [3]. To work with a cone metric space, we need to assume that the interior of a cone is nonempty. However, this property does not generally occur for a $C$*-algebra as we show in the series of examples below.

Example 3.1**.**

Let $\mathbb{A}$ be a complex plane $\mathbb{C}$. Then $\mathbb{A}_{+}=[0,\infty)$, so $\mathrm{Int}(\mathbb{A}_{+})$ is empty in $\mathbb{C}$. Observe that $\mathrm{Int}(\mathbb{A}_{+})$ is not empty in $\mathbb{R}$, the set of self-adjoint elements of $\mathbb{C}$.∎

Example 3.2**.**

In this example we consider $\mathbb{A}$ as a $C$*-algebra of all bounded complex sequences $\ell^{\infty}$ with the operators defined as follows:

[TABLE]

for every $(\xi_{n}),(\eta_{n})\in\ell^{\infty}$ and every $\lambda\in\mathbb{C}$. We will show that $\mathrm{Int}(\ell^{\infty}_{+})=\emptyset$

By the definitions of self-adjoint and positive elements of a $C$*-algebra, we have

[TABLE]

and

[TABLE]

To show that $\mathrm{Int}(\ell^{\infty}_{+})=\emptyset$, we let $a=(\xi_{n})\in\ell^{\infty}_{+}$ and $\varepsilon>0$. Then choose $b=(\xi_{1}-i\frac{\varepsilon}{2},\xi_{2},\xi_{3},\ldots)$. Clearly, $b$ is in $\ell^{\infty}\setminus\ell^{\infty}_{+}$ such that $\|a-b\|=\frac{\varepsilon}{2}<\varepsilon$. This implies that $b\in B(a,\varepsilon)$, the open ball in $\ell^{\infty}$ of radius $\varepsilon$ centered at $a$. Since $\varepsilon$ is arbitrary, the element $a$ is not an interior point of $\ell^{\infty}_{+}$. This situation occurs for every element of $\ell^{\infty}_{+}$, so we have $\mathrm{Int}(\ell^{\infty}_{+})=\emptyset$.∎

Example 3.3** (A $C$*-algebra-valued metric space with the empty interior of $\mathbb{A}_{+}$).**

In this example we replace $X$ and $\mathbb{A}$ by $\mathbb{C}$ and $\mathbb{C}^{2}$, respectively. By the same operators in the previous example, the space $\mathbb{C}^{2}$ can be considered as a $C$*-subalgebra of $\ell^{\infty}$ with $\mathrm{Int}(\mathbb{C}^{2}_{+})=\emptyset$. Let $d:\mathbb{C}\times\mathbb{C}\to\mathbb{C}^{2}$ be a function defined by

[TABLE]

such that $\alpha>0$ for every $a,b\in\mathbb{C}$. Therefore, $(\mathbb{C},\mathbb{C}^{2},d)$ is a $C$*-algebra-valued metric space.∎

Although the situation in the previous example can occur, we does not assume that the interior of $\mathbb{A}_{+}$ is not empty. This is the result of that there exists a suitable real Banach subspace of $\mathbb{A}$ containing $\mathbb{A}_{+}$ with a nonempty interior under the topology on the Banach subspace restricted from $\mathbb{A}$, and so, we will work on the subspace instead. We conclude this useful fact in the two following propositions.

Proposition 3.4**.**

$\mathbb{A}_{h}$* is a real Banach subspace of a $C$**-algebra $\mathbb{A}$.

Proof.

Since $\mathbb{A}_{h}\subseteq\mathbb{A}$, $0_{\mathbb{A}}\in\mathbb{A}_{h}$ and $(\alpha a+b)^{*}=\alpha a+b$ for all $\alpha\in{\mathbb{R}}$ and $a,b\in\mathbb{A}_{h}$, we obtain that $\mathbb{A}_{h}$ is a real normed space. The completeness of $\mathbb{A}_{h}$ can be obtained by verifying that $\mathbb{A}_{h}$ is closed in $\mathbb{A}$. Let $\{a_{n}\}$ be a sequence in $\mathbb{A}_{h}$ converging to $a\in\mathbb{A}$. Since $\lVert a_{n}-a\rVert=\lVert(a_{n}-a)^{*}\rVert=\lVert a_{n}^{*}-a^{*}\rVert=\lVert a_{n}-a^{*}\rVert$, we obtain that $a_{n}$ converges to $a^{*}$. By the uniqueness of a limit of a convergent sequence, we have $a=a^{*}$, i.e. $a\in\mathbb{A}_{h}$. Therefore, $\mathbb{A}_{h}$ is closed in $\mathbb{A}$, and so $\mathbb{A}_{h}$ is a real Banach subspace of $\mathbb{A}$. ∎

Proposition 3.5**.**

If $\mathbb{A}$ is a unital $C$**-algebra, then $\mathrm{Int}_{A_{h}}(\mathbb{A}_{+})\neq\emptyset$.*

Proof.

Let $I$ be a unit of $\mathbb{A}$ and $B(I,1)=\{a\in\mathbb{A}_{h}:\|a-I\|<1\}$. Then Theorem 2.11 implies that $B(I,1)\subseteq\mathbb{A}_{+}$. Hence, $I\in\mathrm{Int}_{A_{h}}(\mathbb{A}_{+})$, so $\mathrm{Int}_{A_{h}}(\mathbb{A}_{+})\neq\emptyset$. ∎

Corollary 3.6**.**

If $\mathbb{A}$ is a unital $C$**-algebra and $\mathbb{A}=\mathbb{A}_{h}$, then $\mathrm{Int}(\mathbb{A})\neq\emptyset$.*

Since $\mathbb{A_{+}}$ is a cone of $\mathbb{A}$, it is also a cone of $\mathbb{A}_{h}$. We have known by Theorem 2.11 that $\|a\|\leq\|b\|$ for every positive elements $a$ and $b$ satisfying $a\leq b$. Hence, $\mathbb{A_{+}}$ is a normal cone of $\mathbb{A}_{h}$. This is a fact occurring in every $C^{\ast}$-algebra. Therefore, a $C^{\ast}$-algebra-valued metric space $(X,\mathbb{A},d)$ is a cone metric space $(X,\widetilde{\mathbb{A}}_{h},d)$ with a normal cone $\widetilde{\mathbb{A}}_{+}$ such that $\mathrm{Int}_{\widetilde{\mathbb{A}}_{h}}(\widetilde{\mathbb{A}}_{+})\neq\emptyset$. Finally, we obtain Lemma 2.21 in a version of a $C^{\ast}$-algebra-valued metric space $(X,\mathbb{A},d)$, equivalent definitions of convergent and Cauchy sequences, stated in the following theorem.

Theorem 3.7**.**

Let $(x_{n})$ be a sequence in a $C$**-algebra-valued metric space $(X,\mathbb{A},d)$. Then the following statements are satisfied.*

$(x_{n})$* converges to $x\in\mathbb{X}$ (in the sense of Definition 2.16) if and only if for every $c\in\widetilde{\mathbb{A}}_{h}$ with $c\gg 0$ there is a positive integer $N$ such that for every integer $n\geq N$ we have $d(x_{n},x)\ll c$.* 2. 2.

$(X_{n})$* is Cauchy (in the sense of Definition 2.16) if and only if for every $c\in\widetilde{\mathbb{A}}_{h}$ with $c\gg 0$ there is a positive integer $N$ such that for every integer $n,m\geq N$ we have $d(x_{n},x_{m})\ll c$.*

Proof.

We prove only the convergent case, the other can be proved similarly. Suppose that $(x_{n})$ converges to an element $x$ of $(X,\mathbb{A},d)$. Then $(x_{n})$ converges to an element $x$ of $(X,\widetilde{\mathbb{A}},d)$, and so, converges in $(X,\widetilde{\mathbb{A}}_{h},d)$. Then the forward implication is obtained after applying Lemma 2.21. For the converse implication, we suppose that the condition holds. Then Lemma 2.21 implies that $\lim_{n\to+\infty}\|d(x_{n},x)\|_{\widetilde{\mathbb{A}}_{h}}=0$. Since $d(x_{n},x)$ belongs to $\mathbb{A}$, we have $\lim_{n\to+\infty}\|d(x_{n},x)\|_{{\mathbb{A}}}=0$. Therefore, $(x_{n})$ converges to an element $x$ of $(X,\mathbb{A},d)$. ∎

Before verifying the completion theorem, we need to introduce some necessary definitions first. Let $(X,\mathbb{A},d_{X})$ and $(Y,\mathbb{B},d_{Y})$ be $C$*-algebra-valued metric spaces. A mapping $T:X\to Y$ is call an isometry if there exists an isometric $*$-isomorphism $f:\mathbb{A}\to\mathbb{B}$ such that

[TABLE]

for every $x,y\in X$. The space $(X,\mathbb{A},d_{X})$ and $(Y,\mathbb{B},d_{Y})$ are said to be isometric if there exists a bijective isometry from $(X,\mathbb{A},d_{X})$ to $(Y,\mathbb{B},d_{Y})$.

Proposition 3.8**.**

An isometry between $C$**-algebra-valued metric spaces is always injective.*

Proof.

Suppose that $(X,\mathbb{A},d_{X})$ and $(Y,\mathbb{B},d_{Y})$ are $C$*-algebra-valued metric spaces and $T$ is an isometry from $X$ to $Y$. Without loss of generality, we may assume that $\mathbb{B}=\mathbb{A}$. Let $x,y\in X$ such that $T(x)=T(y)$. Then $d_{X}(x,y)=d_{Y}(T(x),T(y))=0_{\mathbb{A}}$, so $x=y$. Therefore, $T$ is injective. ∎

The definition of denseness of a subset of a topological space is determined using neighborhoods. In the case of a $C$*-algebra-valued metric space, we provide an equivalent definition using only open balls in the space.

Definition 3.9**.**

Let $(X,\mathbb{A},d)$ be a C*-algebra-valued metric space and $M$ be a subset of $X$. For any $\varepsilon>0$, we define

[TABLE]

Let $M$ be a subset of $X$, the set of all limit points or closure of $M$ is determined by

[TABLE]

If $\mathrm{Cl}(M)=X$, we say that $M$ is dense in $X$.

Because of Theorem 3.7, an equivalent definition of closure of the set $M$ is obtained, that is,

[TABLE]

where $B_{1}(x,c)=\{y\in X:d(x,y)<c\}$ with $c\in\mathbb{A}$ such that $c\gg 0$.

We have shown that any $C$-algebra-valued metric space $(X,\mathbb{A},d)$ can be considered as the cone metric space $(X,\widetilde{\mathbb{A}}_{h},d)$ with the normal cone $\widetilde{\mathbb{A}}_{+}$ such that $\mathrm{Int}_{\widetilde{\mathbb{A}}_{h}}(\widetilde{\mathbb{A}}_{+})\neq\emptyset$. Thus, we can work on the cone metric space instead, and obtain the completion of $(X,\mathbb{A}_{h},d)$ after applying Theorem 2.23. Since the values of $d$ belong to $\mathbb{A}$, the $C$-algebra-valued metric space $(X,\mathbb{A},d)$ is actuary contained in the acquired space as a dense subspace. We conclude this result in the following theorem.

Theorem 3.10** (Completion of $C$*-algebra-valued metric spaces).**

For any $C$-algebra-valued metric space $(X,\mathbb{A},d)$, there exists a complete $C$-algebra-valued metric space $(X^{s},\mathbb{A},d^{s})$ which contains a dense subspace $W$ isometric with $X$. The space $X^{s}$ is unique except for isometries.

Proof.

We know that $(X,\mathbb{A},d)$ is a cone metric space $(X,\widetilde{\mathbb{A}}_{h},d)$ containing the normal cone $\widetilde{\mathbb{A}}_{+}$ such that $\mathrm{Int}_{\widetilde{\mathbb{A}}_{h}}(\widetilde{\mathbb{A}}_{+})\neq\emptyset$. Then Theorem 2.23 implies that there is a complete cone metric space $(X^{s},\widetilde{\mathbb{A}}_{h},{d}^{s})$ which contains a dense subspace $W$ isometric with $X$. We see that $(X^{s},\widetilde{\mathbb{A}},{d}^{s})$ is also a $C$*-algebra-valued metric space. We will verify that $d^{s}$ is an $\mathbb{A}$-valued metric for $X^{s}$, in fact, after taking the composition with the inverse of the mapping $a\mapsto(a,0)$ from $\mathbb{A}$ to $\widetilde{\mathbb{A}}$.

Let $x,y\in X^{s}$. Since $W$ is dense in $X^{s}$, there exist sequences $\{x_{n}\}$ and $\{y_{n}\}$ in $W$ converging to $x$ and $y$, respectively. By continuity of ${d}^{s}$, we have

[TABLE]

Let $T$ be a bijective isometry of a cone metric space from $W$ to $(X,\widetilde{\mathbb{A}}_{h},d)$. Then

[TABLE]

for every $n\in\mathbb{N}$. Since $\mathbb{A}$ is closed in $\widetilde{\mathbb{A}}$, we have

[TABLE]

This implies that $d^{s}$ is an $\mathbb{A}$-valued metric for $X^{s}$. Let $(\widehat{X},\mathbb{B},\widehat{d})$ be another $C$*-algebra-valued metric space which contains a dense subspace $\widehat{W}$ isometric with $X$. Then there is a bijective isometry $\widehat{T}$ from $X$ to $\widehat{W}$. Thus, $\widehat{T}\circ T$ is a bijective isometry from $W$ to $\widehat{W}$. Therefore, $\widehat{T}\circ T$ can be extended to be a bijective isometry from $X^{s}$ to $\widehat{X}$ after applying denseness of $W$ and $\widehat{W}$ in $X^{s}$ and $\widehat{X}$, respectively. This means that the space $X^{s}$ exists uniquely except for isometries. ∎

Next, we focus on a $C$-algebra-valued normed space. We know that any incomplete normed space is embeddable in another complete normed space. In [3], the concept of completion is also extended to the case of a cone normed space. The author defined a cone normed space and verified the existence of its completion. Therefore, the result in the case of $C$-algebra-valued normed spaces is obtained directly from the sake of Proposition 3.4 and Proposition 3.5 similar to the case of a $C$*-algebra-valued metric space.

Let $X$ be a vector space over the real or complex fields and $\mathbb{A}$ be a $C$-algebra. A triple $(X,\mathbb{A},\|\cdot\|)$ is called a $C$-algebra-valued normed space if $\|\cdot\|$ is a function from $X$ to $A_{+}$ satisfying the following properties:

$\|x\|=0_{\mathbb{A}}$ if and only if $x=0_{X}$, 2. 2.

$\|\alpha x\|=|\alpha|\|x\|$, 3. 3.

$\|x+y\|\leq\|x\|+\|y\|$,

for every $x,y\in X$ and every scalar $\alpha$. Notice that $0_{\mathbb{A}}$ and $0_{X}$ are zeros in $\mathbb{A}$ and $X$ respectively.

By the definition of a $C$-algebra-valued norm, we can investigate that the function $d:X\times X\to\mathbb{A}$ determined by $d(x,y)=\|x-y\|$ is a $C$-algebra-valued metric. We call it the $C$-algebra-valued metric* induced by the norm $\|\cdot\|$ We conclude this fact in the proposition below

Proposition 3.11**.**

A $C$-algebra-valued normed space $(X,\mathbb{A},\|\cdot\|)$ is a $C$-algebra-valued metric space with a metric $d:X\times X\to\mathbb{A}$ given by $d(x,y)=\|x-y\|$.

A complete $C$-algebra-valued normed space under the metric defined above is called a $C$-algebra-valued Banach Space. In the next example, we show that every commutative $C$-algebra is a $C$-algebra-valued normed space. We provide a lemma before verify this claim.

Lemma 3.12**.**

Let $A$ be commutative $C$**-algebra. Then $\mathbb{A}_{h}$ is a closed $*$-subalgebra of $\mathbb{A}$ over the real field. Moreover, if $a,b\in\mathbb{A}_{+}$, then $ab\in\mathbb{A}_{+}$ and $(ab)^{1/2}=a^{1/2}b^{1/2}$.*

Proof.

Since $\mathbb{A}$ is commutative, $(ab)^{*}=a^{*}b^{*}=ab$ for every $a,b\in\mathbb{A}_{h}$. Combine with Proposition 3.4, $\mathbb{A}_{h}$ is a real $*$-subalgebra of $\mathbb{A}$. Now, suppose that $a,b\in\mathbb{A}_{+}$ Theorem 2.11 implies that $a=c^{*}c$ for some $c\in\mathbb{A}$. Thus, we have $0_{\mathbb{A}}=c^{*}0_{\mathbb{A}}c\leq c^{*}bc=c^{*}cb=ab$, so $ab$ is positive. By the same way, $a^{1/2}b^{1/2}$ is also positive. Since $(a^{1/2}b^{1/2})^{2}=ab$, Theorem 2.9 implies that $a^{1/2}b^{1/2}=(ab)^{1/2}$. ∎

Example 3.13**.**

Let $\mathbb{A}$ be a commutative $C$*-algebra and $X=\mathbb{A}$. We know from Proposition 2.8 that every element $x\in\mathbb{A}$ can be uniquely decomposed as $x=a+bi$ for some $a,b\in\mathbb{A}_{h}$. Then we define $\|\cdot\|_{0}:X\to\mathbb{A}_{+}$ by

[TABLE]

We will show that $(X,\|\cdot\|_{0},\mathbb{A})$ is a $C$*-algebra-valued normed space.

Since $a$ and $b$ are hermitian, Theorem 2.11 implies that $a^{2}$ and $b^{2}$ are positive. Thus, $(a^{2}+b^{2})^{1/2}$ is also positive after applying Lemma 2.10 and Theorem 2.9, respectively. This shows that $\|\cdot\|_{0}$ is a well-defined function with its values in $\mathbb{A}_{+}$. Since $x=0_{X}$ if and only if $a=b=0_{X}$, we obtain that $\|x\|_{0}=0_{\mathbb{A}}$ if and only if $x=0_{X}$. Next, let $\gamma\in\mathbb{C}$. Then $\gamma=\alpha+\beta i$ where $\alpha,\beta\in\mathbb{R}$. Hence, $\gamma x=(\alpha+\beta i)(a+bi)=(\alpha a-\beta b)+(\beta a+\alpha b)i$, so

[TABLE]

Theorem 2.9 and Lemma 2.12 imply that $\|\gamma x\|_{0}=((\alpha^{2}+\beta^{2})(a^{2}+b^{2}))^{1/2}=|\alpha|\|x\|_{0}$.

Finally, we prove the triangle inequality. Let $y\in X$ be uniquely represented by $c+di$ where $c,d\in\mathbb{A}_{h}$. Consider

[TABLE]

and

[TABLE]

We obtain by Theorem 2.13 that

$\|x+y\|_{0}\leq\|x\|_{0}+\|y\|_{0}$ whenever $\|x+y\|^{2}_{0}\leq(\|x\|_{0}+\|y\|_{0})^{2}.$

Thus, to complete this proof, we need to show that $ac+bd\leq(a^{2}+b^{2})^{1/2}(c^{2}+d^{2})^{1/2}$.

We may see that $0_{\mathbb{A}}\leq(ad-bc)^{2}=(ad)^{2}-2abcd+(bc)^{2}$, so $2abcd\leq(ad)^{2}+(bc)^{2}$. Therefore,

[TABLE]

Theorem 2.13 implies $((ac+bd)^{2})^{1/2}\leq((a^{2}+b^{2})(c^{2}+d^{2}))^{1/2}$. Then apply Theorem 2.9 and Lemma 3.12 to the left and right sides of the inequality, respectively, so we obtain $ac+bd\leq(a^{2}+b^{2})^{1/2}(c^{2}+d^{2})^{1/2}$. Now, the proof of triangle inequality is completed. Consequently, $\|\cdot\|_{0}$ is an $\mathbb{A}$-valued norm for $\mathbb{A}$. ∎

Consider a linear operator between normed spaces. It is an isometry if and only if it is norm-preserving. Thus, the isometric property may be replaced by the norm-preserving property to define isometric isomorphisms of normed spaces. We do the same for $C$-algebra-valued normed spaces. Let $(X,\mathbb{A},\|\cdot\|_{X})$ and $(Y,\mathbb{B},\|\cdot\|_{Y})$ be $C$-algebra-valued normed spaces. A linear operator $T:X\to Y$ is called an isometric if there exists a $*$-isomorphism $f:\mathbb{A}\to\mathbb{B}$ such that

[TABLE]

for every $x\in X$. A bijective isometric linear operator is called an isometric isomorphism. We say that the spaces $(X,\mathbb{A},\|\cdot\|_{X})$ and $(Y,\mathbb{B},\|\cdot\|_{Y})$ are isometrically isomorphic if there exists an isometric isomorphism from $(X,\mathbb{A},\|\cdot\|_{X})$ to $(Y,\mathbb{B},\|\cdot\|_{Y})$.

Lemma 3.14**.**

Let $(W,\mathbb{A},d)$ and $(X,\mathbb{A},\|\cdot\|_{X})$ be $C$**-algebra-valued metric and normed spaces, respectively. Assume that $T$ is a bijective isometry from $W$ and $X$. Then the following statements are satisfied.*

$(W,\mathbb{A},\|\cdot\|_{W})$* is a $C$**-algebra-valued normed space such that $\|u\|_{W}=\|Tu\|_{X}$ for every $u\in W$, so $T$ is a norm-preserving operator. Moreover, $d(u,v)=\|u-v\|_{W}$ for every $u,v\in W$. 2. 2.

$T$* becomes a linear operator from $W$ to $X$.*

Proof.

We define the additive operation $\oplus$ and the scalar multiplication $\odot$ on $W$ by

[TABLE]

for every $u,v\in W$ and every scalar $\alpha\in\mathbb{C}$. We will show that the operations satisfy all axioms of a vector space. Let $u,v,w\in W$ and $\alpha,\beta\in\mathbb{C}$ be scalars.

1. V1)

Closure property of $\oplus$ and $\odot$: Clearly, $u\oplus v,\alpha\odot u\in W$. 2. V2)

Associativity of $\oplus$: \begin{aligned} (u\oplus v)\oplus w&=T^{-1}\Big{(}T\big{(}T^{-1}(Tu+Tv)\big{)}+Tw\Big{)}\\ &=T^{-1}(Tu+Tv+Tw)\\ &=T^{-1}\Big{(}Tu+T\big{(}T^{-1}(Tu+Tv)\big{)}\Big{)}\\ &=u\oplus(v\oplus w)\end{aligned} 3. V3)

Commutativity of $\oplus$: $\begin{aligned} u\oplus v&=T^{-1}(Tu+Tv)\\ &=T^{-1}(Tv+Tu)\\ &=v\oplus u\end{aligned}$ 4. V4)

Let $0_{X}$ be the identity of $X$. We show that $T^{-1}(0_{X})$ is the identity of $W$ under $\oplus$. For any element $u\in W$, we have

[TABLE] 5. V5)

We show that $T^{-1}(-Tu)$ is the inverse of $u$ in $W$ under $\oplus$,

[TABLE] 6. V6)

Compatibility of $\odot$ with multiplication of the filed $\mathbb{C}$:

[TABLE] 7. V7)

Distributivity of $\odot$ with respect to $\oplus$:

[TABLE] 8. V8)

Distributivity of $\odot$ with respect to addition of the field $\mathbb{C}$:

[TABLE] 9. V9)

Identity under $\odot$: $1\odot u=T^{-1}(1Tu)=u$.

Now, $(W,\oplus,\odot)$ is a vector space over the field $\mathbb{C}$. Next, we verify that $\|\cdot\|_{W}$ is an $\mathbb{A}$-valued norm on $W$. Let $0_{W}=T^{-1}(0_{X})$, the identity of $W$. Clearly, $\|0_{W}\|=\|0_{X}\|_{X}=0_{\mathbb{A}}$. In the reverse direction, we assume that $\|u\|=0_{\mathbb{A}}$. Then $\|Tu\|_{X}=0_{\mathbb{A}}$. Thus, $Tu=0_{X}$, so $u=T^{-1}(0_{X})=0_{W}$. Moreover, we have

[TABLE]

and

[TABLE]

Therefore, $(W,\mathbb{A},\|\cdot\|_{W})$ is a $C$*-algebra-valued normed space such that

[TABLE]

In addition, for every $u,v\in W$ and every $\alpha,\beta\in\mathbb{C}$, we have

[TABLE]

Therefore, $T$ is a linear operator. The proof of the lemma is now complete. ∎

Lemma 3.15**.**

Let $(W,\mathbb{A},\|\cdot\|_{W})$ be a $C$-algebra-valued normed space contained as a subspace of a complete $C$-algebra-valued metric space $(X^{s},\mathbb{A},d^{s})$. Assume that $W$ is dense in $X^{s}$. Then $X^{s}$ becomes $C$**-algebra-valued Banach space.*

Proof.

Let $x^{s},y^{s}\in X^{s}$ and $\alpha\in\mathbb{C}$ be a scalar. Then there are sequences $\{x_{n}\}$ and $\{y_{n}\}$ in $W$ converging to $x^{s}$ and $y^{s}$, respectively. Consider

[TABLE]

and

[TABLE]

These imply that $\{x_{n}+y_{n}\}$ and $\{\alpha x_{n}\}$ are Cauchy, so they converge in $X^{s}$. We extend the addition and the scalar multiplication of $W$ to $X^{s}$ by $x^{s}+y^{s}=\lim_{n\to\infty}(x_{n}+y_{n})$ and $\alpha x^{s}=\lim_{n\to\infty}\alpha x_{n}$. Next, we show that the extended operations are well-defined.

Assume that $x^{\prime}_{n}$ and $y^{\prime}_{n}$ are other sequences in $W$ converging to $x^{s}$ and $y^{s}$, respectively. We define a sequence $\{z_{n}\}$ by $z_{2n-1}=x_{n}+y_{n}$ and $z_{2n}=x^{\prime}_{n}+y^{\prime}_{n}$ for every $n\in\mathbb{N}$. We see that

[TABLE]

Thus, $\{z_{n}\}$ is Cauchy in $X^{s}$, so it converges in $X^{s}$. This implies that its subsequences $\{z_{2n}\}$ and $\{z_{2n-1}\}$ converge to the same limit. Therefore, the addition is well-defined. The proof for scalar multiplication can be done similarly. Compatibility of the operations with the axiom of a vector space can be obtained by considering sequences in $W$. Finally, $X^{s}$ becomes a vector space with the identity $0_{W}$ and the limit of a sequence $\{-x_{n}\}$ as the inverse of $x^{s}$. We see that $\{\|x_{n}\|\}$ is a convergent sequence in $\mathbb{A}$, so we put $\|x^{s}\|=\lim_{n\to\infty}\|x_{n}\|$. Consequently, $X^{s}$ is an $\mathbb{A}$-valued normed space. ∎

Theorem 3.16** (Completion of $C$*-algebra-valued normed spaces).**

For any $C$-algebra-valued normed space $(X,\mathbb{A},\|\cdot\|)$, there exists a $C$-algebra-valued Banach space $(X^{s},\mathbb{A},\|\cdot\|^{s})$ which contains a dense subspace $W$ isometric with $X$. The space $X^{s}$ is unique except for isometries.

Proof.

Let $d$ be the metric induced by the norm as in Proposition 3.11. We obtain that the $C$-algebra-valued normed space $(X,\mathbb{A},\|\cdot\|)$ becomes an $\mathbb{A}$-valued metric space. Then apply Theorem 3.10 to obtain a complete $C$-algebra-valued metric space $(X^{s},\mathbb{A},d^{s})$ containing a dense metric subspace $W$ isometric with $X$. By using the previous two lemmas $X^{s}$ becomes a $C$*-algebra-valued Banach space with an $\mathbb{A}$-valued norm $\|\cdot\|^{s}$ such that $d^{s}(x^{s},y^{s})=\|x^{s}-y^{s}\|^{s}$ for every $x^{s},y^{s}\in X^{s}$. ∎

In Lemma 3.14, we show that the bijective isometry $T$ from the space $(W,\mathbb{A},d)$ to the space $(X,\mathbb{A},\|\cdot\|_{X})$ finally becomes a linear operator. Suppose that this situation occurs for other spaces $\widehat{W}$ and $\widehat{X}^{s}$ together with a bijective isometry $\widehat{T}:\widehat{W}\to X$. Then the composition $\widehat{T}^{-1}\circ T$ is a bijective linear operator from $W$ to $\widehat{W}$. By applying denseness of the spaces $W$ and $\widehat{W}$, we can extend the bijective linear operator to be an isometric isomorphism from $X^{s}$ to $\widehat{X}^{s}$. Now, we obtain another version of the preceding theorem stated in the corollary below by using isometric isomorphisms instead of isometries. The similar result studied in [3] is concluded in Theorem 2.24 for cone normed spaces.

Corollary 3.17**.**

For any $C$-algebra-valued normed space $(X,\mathbb{A},\|\cdot\|)$, there exists a $C$-algebra-valued Banach space $(X^{s},\mathbb{A},\|\cdot\|^{s})$ which contains a dense subspace $W$ isometrically isomorphic with $X$. The space $X^{s}$ is unique except for isometric isomorphism.

4 Connection with Hilbert C*-modules.

This section provides certain relationships between concepts of a $C$-algebra-valued metric space and an inner-product $C$-module which is a generalization of an inner product space. The concept of inner-product $C$-module was first introduced in [6], the study of I. Kaplansky in 1953, to develop the theory for commutative unital algebras. In the 1970s, the definition was extended to the case of noncommutative $C$-algebra, see more details in [7, 8]. Let $\mathbb{A}$ be a $C$*-algebra and $X$ be a complex vector space which is a right $\mathbb{A}$-module with compatible scalar multiplication:

[TABLE]

for every $\alpha\in\mathbb{C},x\in X$ and $a\in\mathbb{A}$. The triple $(X,\mathbb{A},\langle\cdot,\cdot\rangle)$ is called an inner product $\mathbb{A}$-module if the mapping $\langle\cdot,\cdot\rangle:X\times X\to\mathbb{A}$ satisfies the following conditions;

$\langle x,\alpha y+\beta z\rangle=\alpha\langle x,y\rangle+\beta\langle x,z\rangle$, 2. 2.

$\langle x,ya\rangle=\langle x,y\rangle a$, 3. 3.

$\langle y,x\rangle=\langle x,y\rangle^{*}$, 4. 4.

$\langle x,x\rangle\geq 0_{\mathbb{A}}$, 5. 5.

if $\langle x,x\rangle=0_{\mathbb{A}}$, then $x=0_{X}$,

for every $\alpha\in\mathbb{C}$ and every $x,y\in X$. It is known that any inner product $C$*-module $(X,\mathbb{A},\langle\cdot,\cdot\rangle)$ is a norm space with a scalar-valued norm $\|\cdot\|_{m}$ given by

[TABLE]

for every $x\in X$ where $\|\cdot\|_{\mathbb{A}}$ is a norm on $\mathbb{A}$. It is called a Hilbert $C$-module* if the induced norm is complete.

Let $\mathbb{A}$ be a commutative unital $C$-algebra such that every nonzero element is invertible. We have $\mathbb{A}=\mathbb{C}I$ where $I$ is a unit of $\mathbb{A}$. In this case an inner product $C$-module is almost like a traditional inner product space that we can see easily. So the Cauchy-Schwarz inequality is also satisfied in context for a $C$-algebra-valued inner product. In fact, we require only the values of the $C$-algebra-valued inner product are invertible elements in the commutative unital $C$*-algebra. The inequality is proved in the following lemma.

Lemma 4.1** (Cauchy-Schwarz inequality).**

Let $(X,\mathbb{A},\langle\cdot,\cdot\rangle)$ be an inner product $C$-module with a commutative unital $C$-algebra $\mathbb{A}$ such that every nonzero value of $\langle\cdot,\cdot\rangle$ is invertible. Then

[TABLE]

for every $x,y\in X$. Moreover,

[TABLE]

where $\|\cdot\|_{0}$ is the norm defined in Example 3.13.

Proof.

Let $x,y\in X$, $a=\langle x,y\rangle$, $b=\alpha I$ where $\alpha\in\mathbb{R}_{+}$ and $I$ is a unit of $\mathbb{A}$. Then

[TABLE]

This implies that $2a^{*}ab\leq\langle x,x\rangle a^{*}a+\langle y,y\rangle b^{2}$.

If $\langle x,x\rangle=0_{\mathbb{A}}$, then $2a^{*}a\leq\langle y,y\rangle b=\alpha\langle y,y\rangle$. This is true for every $\alpha\in\mathbb{R}_{+}$, so we have $2a^{*}a=0$. Thus, $\langle x,y\rangle\langle y,x\rangle\leq\langle x,x\rangle\langle y,y\rangle$. Now we assume that $\langle x,x\rangle\neq 0_{\mathbb{A}}$. In this case, we let $b=\langle x,x\rangle$. Hence, $2a^{*}ab\leq ba^{*}a+\langle y,y\rangle b^{2}$, so $a^{*}a\leq\langle y,y\rangle b$. Therefore, $\langle x,y\rangle\langle y,x\rangle\leq\langle x,x\rangle\langle y,y\rangle$.

Next, we show that $\|\langle x,y\rangle\|_{0}\leq\langle x,x\rangle^{1/2}\langle y,y\rangle^{1/2}$. By commutativity of $\mathbb{A}$ and the representation

[TABLE]

for some $a,b\in\mathbb{A}_{h},$ we have $\|\langle x,y\rangle\|^{2}_{0}=\langle x,y\rangle\langle y,x\rangle\leq\langle x,x\rangle\langle y,y\rangle$. Then Theorem 2.13 implies that

[TABLE]

Consider $(\langle x,x\rangle^{1/2}\langle y,y\rangle^{1/2})^{2}=(\langle x,x\rangle^{1/2})^{2}(\langle y,y\rangle^{1/2})^{2}=\langle x,x\rangle\langle y,y\rangle.$ Thus,

[TABLE]

Therefore,

[TABLE]

Lemma 4.2**.**

Let $a$ be a positive element of a $C$**-algebra $\mathbb{A}$. Then $\|a\|_{0}=a$ where $\|\cdot\|_{0}$ is the norm defined in Example 3.13*

Proof.

The definition of $\|\cdot\|_{0}$ implies that $\|a\|_{0}=(a^{2})^{1/2}$. Since $a$ is the unique positive element such that $a^{2}=a^{2}$, we have $(a^{2})^{1/2}=a$. The proof is now completed. ∎

Theorem 4.3**.**

Let $(X,\mathbb{A},\langle\cdot,\cdot\rangle)$ is an inner product $C$-module. If $\mathbb{A}$ is a commutative unital $C$-algebra such that every nonzero value of $\langle\cdot,\cdot\rangle$ is invertible, then $X$ becomes a $C$**-algebra-valued normed space with a $\mathbb{A}$-valued norm $\|\cdot\|$ given by $\|x\|=\langle x,x\rangle^{1/2}$, for every $x\in X$.*

Proof.

Let $x,y\in X$ and $\alpha$ be a scalar. Since $\langle x,x\rangle\in\mathbb{A}_{+}$, $\|x\|=\langle x,x\rangle^{1/2}\in\mathbb{A}_{+}$.

(1) Assume that $\langle x,x\rangle^{1/2}=\|x\|=0_{\mathbb{A}}$. Then $\langle x,x\rangle=0_{\mathbb{A}}$, so $x=0_{X}$. In reverse direction we assume that $x=0_{X}$. Then apply the second property in the definition of inner product $C$*-module and obtain $\langle x,x\rangle=0_{\mathbb{A}}=0_{\mathbb{A}}^{2}$, so $\|x\|=\langle x,x\rangle^{1/2}=0_{\mathbb{A}}$.

(2) Consider $\|\alpha x\|^{2}=\langle\alpha x,\alpha x\rangle=\alpha\bar{\alpha}\langle x,x\rangle=|\alpha|^{2}\langle x,x\rangle$, so $\langle x,x\rangle=(\frac{1}{|\alpha|}\|\alpha x\|)^{2}$. This implies that $\|x\|=\langle x,x\rangle^{1/2}=\frac{1}{|\alpha|}\|\alpha x\|$, so $|\alpha|\|x\|=\|\alpha x\|$.

(3) Let us consider $\|x-y\|^{2}$. Clearly,

[TABLE]

Then take the norm $\|\cdot\|_{0}$ determined in Example 3.13 to both sides of the equation and apply the two preceding lemmas, so we have

[TABLE]

This means that $\|x-y\|=(\|x-y\|^{2})^{1/2}\leq((\|x\|+\|y\|)^{2})^{1/2}=\|x\|+\|y\|$.

From (1) to (3), we obtain that $(X,\mathbb{A},\|\cdot\|)$ is a $C$*-algebra-valued normed space. ∎

Corollary 4.4**.**

Let $(X,\mathbb{A},\langle\cdot,\cdot\rangle)$ is an inner product $C$-module with a commutative unital $C$-algebra $\mathbb{A}$ such that every nonzero element is invertible. Then $X$ becomes a $C$**-algebra-valued normed space with the same norm determined in the preceding theorem.*

In case the inner product $C$-module $(X,\mathbb{A},\langle\cdot,\cdot\rangle)$ is a $C$-algebra-valued normed space, and so a $C$-algebra-valued metric space, we can consider whether the space is complete by using a $C$-algebra-valued metric. The following theorem shows that these two definitions of completeness are identical in this situation.

Theorem 4.5**.**

Assume that an inner product $C$-module $(X,\mathbb{A},\langle\cdot,\cdot\rangle)$ is a $C$-algebra-valued norm space with an $\mathbb{A}$-valued norm $\|\cdot\|_{X}$ induced by $\langle\cdot,\cdot\rangle$. Then it is a Hilbert $C$-module if and only if it is a $C$-algebra-valued Banach space.

Proof.

Let $x$ be any element of $X$ and $\|\cdot\|_{X}$ be an $\mathbb{A}$-valued norm on $X$ induced by $\langle\cdot,\cdot\rangle$. Since $\|x\|_{X}^{2}=\langle x,x\rangle$, we have

[TABLE]

Thus,

[TABLE]

Then by Definition 2.16 we obtain that the two concepts of convergence of any sequence $\{x_{n}\}$ in $X$ by $\|\cdot\|_{X}$ and $\|\cdot\|$ are equivalence. Therefore, $X$ is a Hilbert $C$-module if and only if it is a $C$-algebra-valued Banach space. ∎

The concept of completion is also extended to inner product $C$-module. It is mentioned in [9] that for any inner product $C$-module $X$ over a $C$*-algebra $\mathbb{A}$, one can form its completion $X^{s}$, a Hilbert $\mathbb{A}$-module, using a similar way to the case of the scalar-valued inner product space. An $\mathbb{A}$-valued inner product on $X^{s}$ is constructed from one of $X$ using the completeness of $\mathbb{A}$.

By using the scalar-valued norm $\|\cdot\|_{m}$, Corollary 3.17 implies that there is a Banach space $X^{s}$ which contains a dense subspace $W$ isometrically isomorphic with $(X,\|\cdot\|_{m})$. Let $T$ be an isometric isomorphism from $W$ to $X$. By the same argument of Lemma 3.14, the algebraic operation of a right $\mathbb{A}$-module compatible with scalar multiplication (1) on $W$ can be induced by that on $X$ through the mapping $T$, that is,

[TABLE]

for every $u\in W$ and every $a\in\mathbb{A}$. An $\mathbb{A}$-valued inner product for $W$ can be induced in a similar way, that is,

[TABLE]

for every $u,v\in W$. Now $W$ becomes an inner product $\mathbb{A}$-module and $T$ is an isomorphism between inner product $\mathbb{A}$-modules. Finally, we extend all the induced operations on $W$ to $X^{s}$ by the similar argument used in Lemma 3.15. Let $x^{s},y^{s}\in X^{s}$. Then there exist a sequence $\{x_{n}\}$ and $\{y_{n}\}$ in $W$ converging by the norm $\|\cdot\|_{m}$ to $x^{s}$ and $y^{s}$, respectively. For every $a\in\mathbb{A}$, we define an $\mathbb{A}$-module operation and an $\mathbb{A}$-valued inner product on $X^{s}$ by

[TABLE]

It is the fact that norm $\|\cdot\|_{m}$ makes $W$ be a right normed $\mathbb{A}$-module, that is,

[TABLE]

for every $u\in W$ and every $a\in\mathbb{A}$. Moreover, It follows from [9, Proposition 1.1] that

[TABLE]

for every $u,v\in W$. Therefore, the limits above exist and the operations for $X^{s}$ is well-defined. Thus, $X^{s}$ becomes a Hilbert $\mathbb{A}$-module. We conclude this result in the remark below.

Remark 4.6**.**

The completion for any inner product $C$-module $(X,\mathbb{A},\langle\cdot,\cdot\rangle)$ exists. That is, there is a Hilbert $C$-module $(X^{s},\mathbb{A},\langle\cdot,\cdot\rangle^{s})$ containing $W$ as an inner product $C$*-submodule such that $W$ is isomorphic to $X$ as $\mathbb{A}$-valued inner product spaces.

Assume that an inner product $C$-module $(X,\mathbb{A},\langle\cdot,\cdot\rangle)$ is a $C$-algebra-valued norm space with an $\mathbb{A}$-valued norm $\|\cdot\|_{X}$ induced by $\langle\cdot,\cdot\rangle$. Then $W$ is also a $C$*-algebra-valued norm space. In this case the completion of $X$ can be constructed by using the norm $\|\cdot\|_{W}$ instead of $\|\cdot\|_{m}$ on $W$. This is a result of the following identity,

[TABLE]

for every $u\in W$. Consequently, we also obtain the completion of an inner product $C$*-module $X$ by applying the completion theorem for $\mathbb{A}$-valued normed spaces if the $\mathbb{A}$-valued norm induced by $\langle\cdot,\cdot\rangle$ exists. An $\mathbb{A}$-valued Inner product for the complete space can be induced from $\langle\cdot,\cdot\rangle$ by using its continuity together with the Cauchy-Schwarz inequality. The continuity of $\langle\cdot,\cdot\rangle$ is proved in the following theorem by applying the original version of the Cauchy-Schwarz inequality for $X$, that is,

[TABLE]

for every $x,y\in X$. Since $\langle y,x\rangle\langle x,y\rangle$ is positive, we have

[TABLE]

Therefore, the inequality below holds,

[TABLE]

Theorem 4.7**.**

Let $(X,\mathbb{A},\langle\cdot,\cdot\rangle)$ be an inner product $C$**-module. Assume that the $\mathbb{A}$-valued norm $\|\cdot\|$ induced by $\langle\cdot,\cdot\rangle$ exists. If $x_{n}\to x$ and $y_{n}\to y$ by the norm, then $\langle x_{n},y_{n}\rangle\to\langle x,y\rangle$ in $\mathbb{A}$.*

Proof.

We apply the triangle inequality for $\|\cdot\|_{\mathbb{A}}$ and then the inequality (2), finally, rewrite the inner product in the form of the norm $\|\cdot\|$, so we obtain

[TABLE]

It not difficult to see that \big{\|}\|x_{n}\|\big{\|}_{\mathbb{A}} is bounded. Since $x_{n}\to x$ and $y_{n}\to y$ by the $\mathbb{A}$-valued norm $\|\cdot\|$, $\|\langle x_{n},y_{n}\rangle-\langle x,y\rangle\|_{\mathbb{A}}\to 0$. Therefore, $\langle x_{n},y_{n}\rangle\to\langle x,y\rangle$ in $\mathbb{A}$. ∎

Corollary 4.8**.**

Let $(X,\mathbb{A},\langle\cdot,\cdot\rangle)$ be an inner product $C$-module with a commutative unital $C$-algebra $\mathbb{A}$ such that every nonzero value of $\langle\cdot,\cdot\rangle$ is invertible. If $x_{n}\to x$ and $y_{n}\to y$ by the norm, then $\langle x_{n},y_{n}\rangle\to\langle x,y\rangle$ in $\mathbb{A}$.