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A short note on \(C^{*}\)-valued contraction mappings

  • Hamed H Alsulami1,
  • Ravi P Agarwal2,
  • Erdal Karapınar1, 3Email author and
  • Farshid Khojasteh4
Journal of Inequalities and Applications20162016:50

https://doi.org/10.1186/s13660-016-0992-5

Received: 6 November 2015

Accepted: 27 January 2016

Published: 9 February 2016

Abstract

In this short note we point out that the recently announced notion, the \(C^{*}\)-valued metric, does not bring about a real extension in metric fixed point theory. Besides, fixed point results in the \(C^{*}\)-valued metric can be derived from the desired Banach mapping principle and its famous consecutive theorems.

Keywords

Unit BallFixed Point TheoremBanach AlgebraContraction MappingShort Note

1 Introduction and preliminaries

Very recently, Ma et al. [1] reported a generalization of the Banach contraction principle for self mappings on \(C^{*}\)-valued metric spaces by defining the notion of a \(C^{*}\)-valued metric space. Following this initial article, some further extension of the Banach contraction principle has been reported (see e.g. [2, 3]). In this note, we shall show that the announced fixed point results in [15] in the context of \(C^{*}\)-valued metric spaces can be derived from the corresponding existing fixed point results in the literature.

First of all, we recall some basic definitions, which will be used later.

Suppose that A is a unital algebra with the unit e. An involution on A is a conjugate linear map a \(*: A \to A\) such that \(a^{**} = a\) and \((ab)^{*} = b^{*}a^{*}\) for all \(a, b \in A\). The pair \((A,*)\) is called a -algebra. A Banach -algebra is a -algebra A together with a complete sub-multiplicative norm such that \(\Vert a\Vert = \Vert a^{*}\Vert \) for all \(a\in A\). A \(C^{*}\)-algebra is a Banach -algebra such that \(\Vert a\Vert =\Vert aa^{*}\Vert \).

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

A character on an abelian algebra A is a non-zero homomorphism \(\tau: A \to\Bbb{C}\). We denote by \(\Omega(A)\) the set of characters on A.

Suppose that A is an abelian Banach algebra for which the space \(\Omega(A)\) is nonempty. If \(a\in A\), we define the function â by
$$\textstyle\begin{cases} \hat{a}: \Omega(A)\to\Bbb{C},\\ \tau\mapsto\tau(a). \end{cases} $$
Clearly, the topology on \(\Omega(A)\) is the smallest one making all of the functions a continuous.

The set \(\{\tau\in\Omega(A): |\tau(a)| \geq\epsilon\}\) is weak closed in the closed unit ball of \(A^{*}\) for each \(\epsilon> 0\), and weak compact by the Banach-Alaoglu theorem. Hence, we deduce that \(a \in C (\Omega(A))\).

We call â the Gelfand transform of a.

Theorem 1.1

([6], Gelfand representation)

Suppose that A is an abelian Banach algebra and that \(\Omega(A)\) is nonempty. Then the map
$$\textstyle\begin{cases} \hat{a}: A\to C (\Omega(A)),\\ a\mapsto\hat{a,} \end{cases} $$
is a norm-decreasing homomorphism, and
$$r(a) = \Vert \hat{a}\Vert _{\infty}\quad(a \in A). $$
If A is unital, \(\sigma(a) = \sigma(\hat{a}(\Omega(A)))\), and if A is non-unital, \(\sigma(a) = \sigma(\hat{a}(\Omega(A)))\cup\{0\}\), for each \(a\in A\).

Theorem 1.2

([6])

Let A be a unital Banach algebra generated by 1 and an element a. Then A is abelian and the map
$$\textstyle\begin{cases} \hat{a}: \Omega(A)\to\sigma(a),\\ \tau\mapsto\tau(a), \end{cases} $$
is a homeomorphism.

Theorem 1.3

([6], Theorem 2.2.5)

Let A be a \(C^{*}\)-algebra and \(a\in A^{+}\). Then
  1. (1)

    There exists a unique element \(b \in A^{+}\) such that \(b^{2} = a\).

     
  2. (2)

    The set \(A^{+}\) is equal to \(\{a^{*} a : a \in A\}\).

     
  3. (3)

    If \(a,b\in A\) and \(0\leq a\leq b\), then \(\Vert a\Vert \leq \Vert b\Vert \).

     

We recall the definition of \(C^{*}\)-algebra-valued metric.

Definition 1.1

Let X be a nonempty set. Suppose that the mapping \(d : X \times X \to\mathbb{A} \) satisfies:
  1. (d1)

    \(\theta\leq d(x, y)\) for all \(x, y \in X \) and \(d(x, y) = \theta\iff x = y\);

     
  2. (d2)

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

     
  3. (d3)

    \(d(x, y) \leq d(x, z) + d(z, y) \) for all \(x, y, z \in X\).

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

2 Main result

Theorem 2.1

Let \((X,\mathbb{A},d)\) be a \(C^{*}\)-algebra-valued complete metric space and \(T:X\to X\) be a mapping such that there exists \(a\in A\) with \(\Vert a\Vert <1\) such that
$$d(Tx,Ty)\preceq a^{*}d(x,y) a\quad \textit{for all } x,y\in X. $$
Then T has a unique fixed point in X.

Proof

Since \(d(x,y)\) and \(d(Tx,Ty)\) are positive and we have
$$0_{A}\leq d(Tx,Ty)\preceq a^{*}d(x,y) a. $$
Also, by (2) of Theorem 1.3 there exists \(u_{x,y}\in A\) such that \(d(x,y)=u_{x,y}^{*}u_{x,y}\). Thus \(\Vert d(x,y)\Vert =\Vert u_{x,y}^{*}u_{x,y}\Vert =\Vert u_{x,y}\Vert ^{2}\) and
$$\begin{aligned} 0_{A} \leq& d(Tx,Ty)\leq a^{*}d(x,y) a \\ =&a^{*}u_{x,y}^{*}u_{x,y}a \\ =&(u_{x,y}a)^{*}(u_{x,y}a). \end{aligned}$$
Applying (3) of Theorem 1.3 we have
$$\begin{aligned} \bigl\Vert d(Tx,Ty)\bigr\Vert \leq&\bigl\Vert (u_{x,y}a)^{*}(u_{x,y}a) \bigr\Vert \\ =&\Vert u_{x,y}a\Vert ^{2} \\ \leq&\Vert a\Vert ^{2} \Vert u_{x,y}\Vert ^{2} \\ =&\Vert a\Vert ^{2} \bigl\Vert d(x,y)\bigr\Vert . \end{aligned}$$
Taking \(D(x,y)=\Vert d(x,y)\Vert \) and \(k=\Vert a\Vert ^{2}<1\) and applying the Banach contraction principle we deduce the desired results. □

As a result, the main result of Ma et al. [1] follows from the Banach contraction mapping principle. The other results in [1] and the fixed point theorems in [2, 3] can be derived from the existing corresponding fixed point theorems in the setting of the standard metric space in the literature.

Declarations

Acknowledgements

The basic idea appeared in the 11th International Conference on Fixed Point Theory and Applications, Istanbul, Turkey, July 20-25, 2015. So, we thank the organizers.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Nonlinear Analysis and Applied Mathematics Research Group (NAAM), King Abdulaziz University, Jeddah, Saudi Arabia
(2)
Department of Mathematics, Texas A&M University-Kingsville, Kingsville, USA
(3)
Department of Mathematics, Atilim University, Ïncek, Turkey
(4)
Young Researcher and Elite Club, Arak-Branch, Islamic Azad University, Arak, Iran

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Copyright

© Alsulami et al. 2016

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