# ${C}^{\ast}$-algebra-valued metric spaces and related fixed point theorems

- Zhenhua Ma
^{1, 2}, - Lining Jiang
^{1}Email author and - Hongkai Sun
^{2}

**2014**:206

https://doi.org/10.1186/1687-1812-2014-206

© Ma et al.; licensee Springer. 2014

**Received: **20 May 2014

**Accepted: **17 September 2014

**Published: **13 October 2014

## Abstract

Based on the concept and properties of ${C}^{\ast}$-algebras, the paper introduces a concept of ${C}^{\ast}$-algebra-valued metric spaces and gives some fixed point theorems for self-maps with contractive or expansive conditions on such spaces. As applications, existence and uniqueness results for a type of integral equation and operator equation are given.

**MSC:**47H10, 46L07.

## Keywords

## 1 Introduction

We begin with the concept of ${C}^{\ast}$-algebras.

Suppose that $(X,\mathbb{A},d)$ is a unital algebra with the unit *I*. An involution on $(X,\mathbb{A},d)$ is a conjugate-linear map $a\mapsto {a}^{\ast}$ on $(X,\mathbb{A},d)$ such that ${a}^{\ast \ast}=a$ and ${(ab)}^{\ast}={b}^{\ast}{a}^{\ast}$ for all $a,b\in \mathbb{A}$. The pair $(\mathbb{A},\ast )$ is called a ∗-algebra. A Banach ∗-algebra is a ∗-algebra $(X,\mathbb{A},d)$ together with a complete submultiplicative norm such that $\parallel {a}^{\ast}\parallel =\parallel a\parallel $ ($\mathrm{\forall}a\in \mathbb{A}$). A ${C}^{\ast}$-algebra is a Banach ∗-algebra such that $\parallel {a}^{\ast}a\parallel ={\parallel a\parallel}^{2}$ [1, 2].

Notice that the seeming mild requirement on a ${C}^{\ast}$-algebra above is in fact very strong. Moreover, the existence of the involution ${C}^{\ast}$-algebra theory can be thought of as infinite-dimensional real analysis. Clearly that under the norm topology, $L(H)$, the set of all bounded linear operators on a Hilbert space *H*, is a ${C}^{\ast}$-algebra.

As we have known, the Banach contraction principle is a very useful, simple and classical tool in modern analysis. Also it is an important tool for solving existence problems in many branches of mathematics and physics. In general, the theorem has been generalized in two directions. On the one side, the usual contractive (expansive) condition is replaced by weakly contractive (expansive) condition. On the other side, the action spaces are replaced by metric spaces endowed with an ordered or partially ordered structure. In recent years, O’Regan and Petrusel [3] started the investigations concerning a fixed point theory in ordered metric spaces. Later, many authors followed this research by introducing and investigating the different types of contractive mappings. For example in [4] Caballero *et al.* considered contractive like mapping in ordered metric spaces and applied their results in ordinary differential equations. In 2007, Huang and Zhang [5] introduced the concept of cone metric space, replacing the set of real numbers by an ordered Banach space. Later, many authors generalized their fixed point theorems on different type of metric spaces [6–13]. In [14], the authors studied the operator-valued metric spaces and gave some fixed point theorems on the spaces. In this paper, we introduce a new type of metric spaces which generalize the concepts of metric spaces and operator-valued metric spaces, and give some related fixed point theorems for self-maps with contractive or expansive conditions on such spaces.

The paper is organized as follows: Based on the concept and properties of ${C}^{\ast}$-algebras, we first introduce a concept of ${C}^{\ast}$-algebra-valued metric spaces. Moreover, some fixed point theorems for mappings satisfying the contractive or expansive conditions on ${C}^{\ast}$-algebra-valued metric spaces are established. Finally, as applications, existence and uniqueness results for a type of integral equation and operator equation are given.

## 2 Main results

To begin with, let us start from some basic definitions, which will be used later.

Throughout this paper, $(X,\mathbb{A},d)$ will denote an unital ${C}^{\ast}$-algebra with a unit *I*. Set ${\mathbb{A}}_{h}=\{x\in \mathbb{A}:x={x}^{\ast}\}$. We call an element $x\in \mathbb{A}$ a positive element, denote it by $x\u2ab0\theta $, if $x\in {\mathbb{A}}_{h}$ and $\sigma (x)\subset {\mathbb{R}}_{+}=[0,\mathrm{\infty})$, where $\sigma (x)$ is the spectrum of *x*. Using positive elements, one can define a partial ordering ⪯ on ${\mathbb{A}}_{h}$ as follows: $x\u2aafy$ if and only if $y-x\u2ab0\theta $, where *θ* means the zero element in $(X,\mathbb{A},d)$. From now on, by ${\mathbb{A}}_{+}$ we denote the set $\{x\in \mathbb{A}:x\u2ab0\theta \}$ and $|x|={({x}^{\ast}x)}^{\frac{1}{2}}$.

**Remark 2.1** When $(X,\mathbb{A},d)$ is a unital ${C}^{\ast}$-algebra, then for any $x\in {\mathbb{A}}_{+}$ we have $x\u2aafI\iff \parallel x\parallel \le 1$ [1, 2].

With the help of the positive element in $(X,\mathbb{A},d)$, one can give the definition of a ${C}^{\ast}$-algebra-valued metric space.

**Definition 2.1**Let

*X*be a nonempty set. Suppose the mapping $d:X\times X\to \mathbb{A}$ satisfies:

- (1)
$\theta \u2aafd(x,y)$ for all $x,y\in X$ and $d(x,y)=\theta \iff x=y$;

- (2)
$d(x,y)=d(y,x)$ for all $x,y\in X$;

- (3)
$d(x,y)\u2aafd(x,z)+d(z,y)$ for all $x,y,z\in X$.

Then *d* is called a ${C}^{\ast}$-algebra-valued metric on *X* and $(X,\mathbb{A},d)$ is called a ${C}^{\ast}$-algebra-valued metric space.

It is obvious that ${C}^{\ast}$-algebra-valued metric spaces generalize the concept of metric spaces, replacing the set of real numbers by ${\mathbb{A}}_{+}$.

**Definition 2.2** Let $(X,\mathbb{A},d)$ be a ${C}^{\ast}$-algebra-valued metric space. Suppose that $\{{x}_{n}\}\subset X$ and $x\in X$. If for any $\epsilon >0$ there is *N* such that for all $n>N$, $\parallel d({x}_{n},x)\parallel \le \epsilon $, then $\{{x}_{n}\}$ is said to be convergent with respect to $(X,\mathbb{A},d)$ and $\{{x}_{n}\}$ converges to *x* and *x* is the limit of $\{{x}_{n}\}$. We denote it by ${lim}_{n\to \mathrm{\infty}}{x}_{n}=x$.

If for any $\epsilon >0$ there is *N* such that for all $n,m>N$, $\parallel d({x}_{n},{x}_{m})\parallel \le \epsilon $, then $\{{x}_{n}\}$ is called a Cauchy sequence with respect to $(X,\mathbb{A},d)$.

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

*X*is a Banach space, then $(X,\mathbb{A},d)$ is a complete ${C}^{\ast}$-algebra-valued metric space if we set

The following are nontrivial examples of complete ${C}^{\ast}$-algebra-valued metric space.

**Example 2.1** Let $X={L}^{\mathrm{\infty}}(E)$ and $H={L}^{2}(E)$, where *E* is a Lebesgue measurable set. By $L(H)$ we denote the set of bounded linear operators on Hilbert space *H*. Clearly $L(H)$ is a ${C}^{\ast}$-algebra with the usual operator norm.

for $\phi \in H$. Then *d* is a ${C}^{\ast}$-algebra-valued metric and $(X,L(H),d)$ is a complete ${C}^{\ast}$-algebra-valued metric space.

*X*be a Cauchy sequence with respect to $L(H)$. Then for a given $\epsilon >0$, there is a natural number $N(\epsilon )$ such that for all $n,m\ge N(\epsilon )$,

then ${\{{f}_{n}\}}_{n=1}^{\mathrm{\infty}}$ is a Cauchy sequence in the space *X*. Thus, there is a function $f\in X$ and a natural number ${N}_{1}(\epsilon )$ such that ${\parallel {f}_{n}-f\parallel}_{\mathrm{\infty}}\le \epsilon $ if $n\ge {N}_{1}$.

Therefore, the sequence ${\{{f}_{n}\}}_{n=1}^{\mathrm{\infty}}$ converges to the function *f* in *X* with respect to $L(H)$, that is, $(X,L(H),d)$ is complete with respect to $L(H)$.

**Example 2.2**Let $X=\mathbb{R}$ and $\mathbb{A}={M}_{2}(\mathbb{R})$. Define

where $x,y\in \mathbb{R}$ and $\alpha \ge 0$ is a constant. It is easy to verify *d* is a ${C}^{\ast}$-algebra-valued metric and $(X,{M}_{2}(\mathbb{R}),d)$ is a complete ${C}^{\ast}$-algebra-valued metric space by the completeness of ℝ.

Now we give the definition of a ${C}^{\ast}$-algebra-valued contractive mapping on *X*.

**Definition 2.3**Suppose that $(X,\mathbb{A},d)$ is a ${C}^{\ast}$-algebra-valued metric space. We call a mapping $T:X\to X$ is a ${C}^{\ast}$-algebra-valued contractive mapping on

*X*, if there exists an $A\in \mathbb{A}$ with $\parallel A\parallel <1$ such that

**Theorem 2.1** *If* $(X,\mathbb{A},d)$ *is a complete* ${C}^{\ast}$-*algebra*-*valued metric space and* *T* *is a contractive mapping*, *there exists a unique fixed point in* *X*.

*Proof* It is clear that if $A=\theta $, *T* maps the *X* into a single point. Thus without loss of generality, one can suppose that $A\ne \theta $.

Choose ${x}_{0}\in X$ and set ${x}_{n+1}=T{x}_{n}={T}^{n+1}{x}_{0}$, $n=1,2,\dots $ . For convenience, by *B* we denote the element $d({x}_{1},{x}_{0})$ in $(X,\mathbb{A},d)$.

Therefore $\{{x}_{n}\}$ is a Cauchy sequence with respect to $(X,\mathbb{A},d)$. By the completeness of $(X,\mathbb{A},d)$, there exists an $x\in X$ such that ${lim}_{n\to \mathrm{\infty}}{x}_{n}={lim}_{n\to \mathrm{\infty}}T{x}_{n-1}=x$.

hence, $Tx=x$, *i.e.*, *x* is a fixed point of *T*.

*y*(≠

*x*) is another fixed point of

*T*, since

It is impossible. So $d(x,y)=\theta $ and $x=y$, which implies that the fixed point is unique. □

Similar to the concept of contractive mapping, we have the concept of an expansive mapping and furthermore have the related fixed point theorem.

**Definition 2.4**Let

*X*be a nonempty set. We call a mapping

*T*is a ${C}^{\ast}$-algebra-valued expansion mapping on

*X*, if $T:X\to X$ satisfies:

- (1)
$T(X)=X$;

- (2)
$d(Tx,Ty)\u2ab0{A}^{\ast}d(x,y)A$, $\mathrm{\forall}x,y\in X$,

where $A\in \mathbb{A}$ is an invertible element and $\parallel {A}^{-1}\parallel <1$.

**Theorem 2.2** *Let* $(X,\mathbb{A},d)$ *be a complete* ${C}^{\ast}$-*algebra*-*valued metric space*. *Then for the expansion mapping* *T*, *there exists a unique fixed point in* *X*.

*Proof*Firstly,

*T*is injective. Indeed, for any $x,y\in X$ with $x\ne y$, if $Tx=Ty$, we have

Since ${A}^{\ast}d(x,y)A\in {\mathbb{A}}_{+}$, ${A}^{\ast}d(x,y)A=\theta $. Also *A* is invertible, $d(x,y)=\theta $, which is impossible. Thus *T* is injective.

*T*has a unique fixed point in

*X*. In fact, since

*T*is invertible and for any $x,y\in X$,

*x*,

*y*with ${T}^{-1}x$, ${T}^{-1}y$, respectively, and we get

Using Theorem 2.1, there exists a unique *x* such that ${T}^{-1}x=x$, which means there has a unique fixed point $x\in X$ such that $Tx=x$. □

Before introducing another fixed point theorem, we give a lemma first. Such a result can be found in [1, 15].

**Lemma 2.1**

*Suppose that*$(X,\mathbb{A},d)$

*is a unital*${C}^{\ast}$-

*algebra with a unit*

*I*.

- (1)
*If*$a\in {\mathbb{A}}_{+}$*with*$\parallel a\parallel <\frac{1}{2}$,*then*$I-a$*is invertible and*$\parallel a{(I-a)}^{-1}\parallel <1$; - (2)
*suppose that*$a,b\in \mathbb{A}$*with*$a,b\u2ab0\theta $*and*$ab=ba$,*then*$ab\u2ab0\theta $; - (3)
*by*${\mathbb{A}}^{\prime}$*we denote the set*$\{a\in \mathbb{A}:ab=ba,\mathrm{\forall}b\in \mathbb{A}\}$.*Let*$a\in {\mathbb{A}}^{\prime}$,*if*$b,c\in \mathbb{A}$*with*$b\u2ab0c\u2ab0\theta $*and*$I-a\in {\mathbb{A}}_{+}^{\prime}$*is a invertible operator*,*then*${(I-a)}^{-1}b\u2ab0{(I-a)}^{-1}c.$

Notice that in a ${C}^{\ast}$-algebra, if $\theta \u2aafa,b$, one cannot conclude that $\theta \u2aafab$. Indeed, consider the ${C}^{\ast}$-algebra ${\mathbb{M}}_{2}(\mathbb{C})$ and set $a=\left(\begin{array}{cc}3& 2\\ 2& 3\end{array}\right)$, $b=\left(\begin{array}{cc}1& -2\\ -2& 4\end{array}\right)$, then $ab=\left(\begin{array}{cc}-1& 2\\ -4& 8\end{array}\right)$. Clearly $a,b\in {\mathbb{M}}_{2}{(\mathbb{C})}_{+}$, while *ab* is not.

**Theorem 2.3**

*Let*$(X,\mathbb{A},d)$

*be a complete*${C}^{\ast}$-

*valued metric space*.

*Suppose the mapping*$T:X\to X$

*satisfies for all*$x,y\in X$

*where* $A\in {\mathbb{A}}_{+}^{\prime}$ *and* $\parallel A\parallel <\frac{1}{2}$. *Then there exists a unique fixed point in* *X*.

*Proof* Without loss of generality, one can suppose that $A\ne \theta $. Notice that $A\in {\mathbb{A}}_{+}^{\prime}$, $A(d(Tx,y)+d(Ty,x))$ is also a positive element.

*B*we denote the element $d({x}_{1},{x}_{0})$ in $(X,\mathbb{A},d)$. Then

where $t=A{(I-A)}^{-1}$.

*i.e.*${lim}_{n\to \mathrm{\infty}}T{x}_{n-1}=x$. Since

This implies that $Tx=x$ *i.e.*, *x* is a fixed point of *T*.

*y*(≠

*x*) is another fixed point of

*T*, then

*i.e.*,

Therefore the fixed point is unique and the proof is complete. □

## 3 Applications

As applications of contractive mapping theorem on complete ${C}^{\ast}$-algebra-valued metric spaces, existence and uniqueness results for a type of integral equation and operator equation are given.

**Example 3.1**Consider the integral equation

where *E* is a Lebesgue measurable set.

- (1)
$K:E\times E\times \mathbb{R}\to \mathbb{R}$ and $g\in {L}^{\mathrm{\infty}}(E)$;

- (2)there exists a continuous function $\phi :E\times E\to \mathbb{R}$ and $k\in (0,1)$ such that$|K(t,s,u)-K(t,s,v)|\le k|\phi (t,s)(u-v)|,$
for $t,s\in E$ and $u,v\in \mathbb{R}$

- (3)
${sup}_{t\in E}{\int}_{E}|\phi (t,s)|\phantom{\rule{0.2em}{0ex}}\mathrm{d}s\le 1$.

Then the integral equation has a unique solution ${x}^{\ast}$ in ${L}^{\mathrm{\infty}}(E)$.

*Proof* Let $X={L}^{\mathrm{\infty}}(E)$ and $H={L}^{2}(E)$. Set *d* as Example 2.1, then *d* is a ${C}^{\ast}$-algebra-valued metric and $(X,L(H),d)$ is a complete ${C}^{\ast}$-algebra-valued metric space with respect to $L(H)$.

Since $\parallel A\parallel <1$, the integral equation has a unique solution ${x}^{\ast}$ in ${L}^{\mathrm{\infty}}(E)$. □

**Example 3.2**Suppose that

*H*is a Hilbert space, $L(H)$ is the set of linear bounded operators on

*H*. Let ${A}_{1},{A}_{2},\dots ,{A}_{n}\in L(H)$, which satisfy ${\sum}_{n=1}^{\mathrm{\infty}}{\parallel {A}_{n}\parallel}^{2}<1$ and $X\in L(H)$, $Q\in L{(H)}_{+}$. Then the operator equation

has a unique solution in $L(H)$.

*Proof* Set $\alpha ={\sum}_{n=1}^{\mathrm{\infty}}{\parallel {A}_{n}\parallel}^{2}$. Clear that if $\alpha =0$, then the ${A}_{n}=\theta $ ($(X,\mathbb{A},d)$), and the equation has a unique solution in $L(H)$. Without loss of generality, one can suppose that $\alpha >0$.

It is easy to verify that $d(X,Y)$ is a ${C}^{\ast}$-algebra-valued metric and $(L(H),d)$ is complete since $L(H)$ is a Banach space.

Using Theorem 2.1, there exists a unique fixed point *X* in $L(H)$. Furthermore, since ${\sum}_{n=1}^{\mathrm{\infty}}{A}_{n}^{\ast}X{A}_{n}+Q$ is a positive operator, the solution is a Hermitian operator. □

where *Q* is a positive definite matrix and ${A}_{1},\dots ,{A}_{m}$ are arbitrary $n\times n$ matrices. Using Example 3.2, there exists a unique Hermitian matrix solution.

## Declarations

### Acknowledgements

This work is supported financially by the NSFC (11371222) and by the project for the Construction of the Graduate Teaching Team of Beijing Institute of Technology (YJXTD-2014-A08).

## Authors’ Affiliations

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