A short note on \(C^{*}\)-valued contraction mappings
- Hamed H Alsulami^{1},
- Ravi P Agarwal^{2},
- Erdal Karapınar^{1, 3}Email author and
- Farshid Khojasteh^{4}
https://doi.org/10.1186/s13660-016-0992-5
© Alsulami et al. 2016
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
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 [1–5] 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.
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)
Theorem 1.2
([6])
Theorem 1.3
([6], Theorem 2.2.5)
- (1)
There exists a unique element \(b \in A^{+}\) such that \(b^{2} = a\).
- (2)
The set \(A^{+}\) is equal to \(\{a^{*} a : a \in A\}\).
- (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
- (d1)
\(\theta\leq d(x, y)\) for all \(x, y \in X \) and \(d(x, y) = \theta\iff x = y\);
- (d2)
\(d(x, y) = d(y, x)\) for all \(x, y \in X\);
- (d3)
\(d(x, y) \leq d(x, z) + d(z, y) \) for all \(x, y, z \in X\).
2 Main result
Theorem 2.1
Proof
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.
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Authors’ Affiliations
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