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{C}^{\ast}algebravalued metric spaces and related fixed point theorems
Fixed Point Theory and Applications volume 2014, Article number: 206 (2014)
Abstract
Based on the concept and properties of {C}^{\ast}algebras, the paper introduces a concept of {C}^{\ast}algebravalued metric spaces and gives some fixed point theorems for selfmaps 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.
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 conjugatelinear 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 infinitedimensional 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 operatorvalued 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 operatorvalued metric spaces, and give some related fixed point theorems for selfmaps 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}algebravalued metric spaces. Moreover, some fixed point theorems for mappings satisfying the contractive or expansive conditions on {C}^{\ast}algebravalued 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 yx\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}algebravalued 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}algebravalued metric on X and (X,\mathbb{A},d) is called a {C}^{\ast}algebravalued metric space.
It is obvious that {C}^{\ast}algebravalued 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}algebravalued 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}algebravalued metric space if every Cauchy sequence with respect to (X,\mathbb{A},d) is convergent.
It is obvious that if X is a Banach space, then (X,\mathbb{A},d) is a complete {C}^{\ast}algebravalued metric space if we set
The following are nontrivial examples of complete {C}^{\ast}algebravalued 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.
Define d:X\times X\to L(H) by
where {\pi}_{h}:H\to H is the multiplication operator defined by
for \phi \in H. Then d is a {C}^{\ast}algebravalued metric and (X,L(H),d) is a complete {C}^{\ast}algebravalued metric space.
Indeed, it suffices to verity the completeness. Let {\{{f}_{n}\}}_{n=1}^{\mathrm{\infty}} in 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}.
It follows that
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}algebravalued metric and (X,{M}_{2}(\mathbb{R}),d) is a complete {C}^{\ast}algebravalued metric space by the completeness of ℝ.
Now we give the definition of a {C}^{\ast}algebravalued contractive mapping on X.
Definition 2.3 Suppose that (X,\mathbb{A},d) is a {C}^{\ast}algebravalued metric space. We call a mapping T:X\to X is a {C}^{\ast}algebravalued 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}algebravalued 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).
Notice that in a {C}^{\ast}algebra, if a,b\in {\mathbb{A}}_{+} and a\u2aafb, then for any x\in \mathbb{A} both {x}^{\ast}ax and {x}^{\ast}bx are positive elements and {x}^{\ast}ax\u2aaf{x}^{\ast}bx [1]. Thus
So for n+1>m,
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}_{n1}=x.
Since
hence, Tx=x, i.e., x is a fixed point of T.
Now suppose that y (≠x) is another fixed point of T, since
we have
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}algebravalued 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}algebravalued 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.
Next, we will show T has a unique fixed point in X. In fact, since T is invertible and for any x,y\in X,
In the above formula, substitute x, y with {T}^{1}x, {T}^{1}y, respectively, and we get
This means
and thus
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 Ia is invertible and \parallel a{(Ia)}^{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 Ia\in {\mathbb{A}}_{+}^{\prime} is a invertible operator, then
{(Ia)}^{1}b\u2ab0{(Ia)}^{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.
Choose {x}_{0}\in X, set {x}_{n+1}=T{x}_{n}={T}^{n+1}{x}_{0}, n=1,2,\dots , by B we denote the element d({x}_{1},{x}_{0}) in (X,\mathbb{A},d). Then
Thus,
Since A\in {\mathbb{A}}_{+}^{\prime} with \parallel A\parallel <\frac{1}{2}, one have {(IA)}^{1}\in {\mathbb{A}}_{+}^{\prime} and furthermore A{(IA)}^{1}\in {\mathbb{A}}_{+}^{\prime} with \parallel A{(IA)}^{1}\parallel <1 by Lemma 2.1. Therefore,
where t=A{(IA)}^{1}.
For n+1>m,
This implies that \{{x}_{n}\} is a Cauchy sequence with respect to (X,\mathbb{A},d). By the completeness of (X,\mathbb{A},d), there exists x\in X such that {lim}_{n\to \mathrm{\infty}}{x}_{n}=x, i.e. {lim}_{n\to \mathrm{\infty}}T{x}_{n1}=x. Since
This is equivalent to
Then
This implies that Tx=x i.e., x is a fixed point of T.
Now if y (≠x) is another fixed point of T, then
i.e.,
Since \parallel A{(IA)}^{1}\parallel <1,
This means that
Therefore the fixed point is unique and the proof is complete. □
3 Applications
As applications of contractive mapping theorem on complete {C}^{\ast}algebravalued 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.
Suppose that

(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)(uv),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}algebravalued metric and (X,L(H),d) is a complete {C}^{\ast}algebravalued metric space with respect to L(H).
Let T:{L}^{\mathrm{\infty}}(E)\to {L}^{\mathrm{\infty}}(E) be
Set A=kI, then A\in L{(H)}_{+} and \parallel A\parallel =k<1. For any h\in 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.
Choose a positive operator T\in L(H). For X,Y\in L(H), set
It is easy to verify that d(X,Y) is a {C}^{\ast}algebravalued metric and (L(H),d) is complete since L(H) is a Banach space.
Consider the map F:L(H)\to L(H) defined by
Then
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. □
As a special case of Example 3.2, one can consider the following matrix equation, which can also be found in [16]:
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.
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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 (YJXTD2014A08).
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Ma, Z., Jiang, L. & Sun, H. {C}^{\ast}algebravalued metric spaces and related fixed point theorems. Fixed Point Theory Appl 2014, 206 (2014). https://doi.org/10.1186/168718122014206
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DOI: https://doi.org/10.1186/168718122014206
Keywords
 {C}^{\ast}algebra
 {C}^{\ast}algebravalued metric
 contractive mapping
 expansive mapping
 fixed point theorem