Fixed point results in \(C^{*}\)-algebra-valued metric spaces are direct consequences of their standard metric counterparts
- Zoran Kadelburg^{1} and
- Stojan Radenović^{2, 3}Email author
https://doi.org/10.1186/s13663-016-0544-1
© Kadelburg and Radenović 2016
Received: 18 January 2016
Accepted: 13 April 2016
Published: 23 April 2016
Abstract
Very recently, Ma et al. (Fixed Point Theory Appl. 2014:206, 2014) introduced \(C^{*}\)-algebra-valued metric spaces as a new concept. Also, Ma and Jiang (Fixed Point Theory Appl. 2015:222, 2015), generalizing this concept, introduced \(C^{*}\)-algebra-valued b-metric spaces. In both frameworks, these and other authors proved some fixed point results. We show in this paper that all these results (as well as many others) can be directly obtained as consequences of their standard metric or b-metric counterparts.
Keywords
\(C^{*}\)-algebra cone metric space over Banach algebra \(C^{*}\)-algebra-valued metric space b-metric spaceMSC
47H10 46L99 54H251 Introduction
One of the main directions in obtaining possible generalizations of fixed point results in metric spaces is introducing new types of spaces. One of such attempts was made by Ma et al. in [1], where they introduced \(C^{*}\)-algebra-valued metric spaces as a new concept and proved some related fixed point results. This line of research was continued in [2–6], where several other fixed point results were obtained in the framework of \(C^{*}\)-algebra-valued metric, as well as (more general) \(C^{*}\)-algebra-valued b-metric spaces.
However, it was observed by Alsulami et al. in [7] that these results are in fact not new. Namely, they showed that, using the Banach-Alaoglu theorem and the Gelfand representation, the basic result of [1–5] can be reduced to the existing corresponding fixed point theorems in the setting of standard metric spaces.
In this paper, we will show the same for several more results of this type, including those from [6] and those in \(C^{*}\)-algebra-valued b-metric spaces. Moreover, our method is easier since it uses just basic properties of \(C^{*}\)-algebras. Even several other known theorems can be adapted to hold true in these new kinds of spaces.
In a way, the paper can be considered as written along the lines of well-known papers [8, 9] where, in some other situations, it was also shown that a noncritical approach to ‘generalizations’ of fixed point results can lead to results which have no real meaning.
2 Preliminaries
- (i)
\((\lambda a+\mu b)^{*}=\overline{\lambda}a+\overline{\mu}b\);
- (ii)
\((ab)^{*}=b^{*}a^{*}\);
- (iii)
\(\|a^{*}a\|=\|a\|^{2}\).
In the rest of this paper, \(\mathbb{A}\) will always be a unital \(C^{*}\)-algebra with the unit i and the zero element θ. \(\mathbb{A}_{h}\) will denote the set of all self-adjoint elements a (i.e., satisfying \(a^{*}=a\)), and \(\mathbb{A}^{+}\) will be the set of positive elements of \(\mathbb{A}\), i.e., the elements \(a\in\mathbb{A}_{h}\) having the spectrum \(\sigma(a)\) contained in \([0,+\infty)\). It is easy to see that \(\mathbb{A}^{+}\) is a (closed) cone in the normed space \(\mathbb{A}\) (see, e.g., [10], Lemma 2.2.3), thus inducing a partial order ⪯ on \(\mathbb{A}_{h}\) by \(a\preceq b\) if and only if \(b-a\in \mathbb{A}^{+}\).
For further terminology and basic results in \(C^{*}\)-algebras we will refer to [10] (as we have already mentioned). In particular, we will use the following simple result.
Lemma 1
([10], Theorem 2.2.5)
- (1)
\(\mathbb{A}^{+}=\{a^{*}a : a\in\mathbb{A}\}\);
- (2)
if \(a,b\in \mathbb{A}_{h}\), \(a\preceq b\), and \(c\in\mathbb{A}\), then \(c^{*}ac\preceq c^{*}bc\);
- (3)
for all \(a,b\in\mathbb{A}_{h}\), if \(\theta\preceq a\preceq b\) then \(\|a\|\leq\|b\|\).
In the standard terminology used for cones in normed spaces (see, e.g., [11], Ch. 6), the property (2) of the previous lemma means that the cone \(\mathbb{A}^{+}\) in \(\mathbb{A}_{h}\) is normal with normal constant equal to 1 (in this case the norm is called monotone). This simple fact will be crucial for obtaining our results.
3 Main results
3.1 Fixed point results in \(C^{*}\)-algebra-valued metric spaces
In [1], Ma et al. introduced the following concept, being, in fact, a special case of previously known concepts of cone metric spaces [12] and cone metric spaces over Banach algebras [13, 14].
Definition 1
([1], Definition 2.1)
- (i)
\(d(x,y)\succeq \theta\) for all \(x,y\in X\) and \(d(x,y)=\theta\iff x=y\);
- (ii)
\(d(x,y)=d(y,x)\) for all \(x,y\in X\);
- (iii)
\(d(x,y)\preceq d(x,z)+d(z,y)\) for all \(x,y,z\in X\).
As the main result, they proved the following.
Theorem 1
([1], Theorem 2.1)
As our first contribution, we prove that Theorem 1 is not a new result.
Theorem 2
Theorem 1 is equivalent to the Banach contraction principle (BCP).
Proof
Remark 1
- (1)
the fixed point result for expansion mappings [1], Theorem 2.2;
- (2)
the Chatterjea fixed point result [1], Theorem 2.3 (with the contractive condition in the form \(d(Tx,Ty)\preceq a^{*}(d(x,Ty)+d(y,Tx))a\), \(a\in\mathbb{A}\), \(\|a\|<1/{\sqrt{2}}\));
- (3)
a fixed point result for contractions in \(C^{*}\)-algebra-valued spaces endowed with a graph [4], Theorem 2.5 (this reduces to [15], Theorem 3.1).
In fact, the same is true for several more general results, e.g., for most of the fixed point results contained in the well-known paper [16]. As an example, we prove the following.
Theorem 3
Proof
Of course, the results of Kannan, Zamfirescu, Hardy-Rogers (and many others; see [16]) follow as special cases.
Moreover, several known common fixed point results can be easily reformulated in the framework of \(C^{*}\)-algebra-valued metric spaces.
3.2 Fixed point results in \(C^{*}\)-algebra-valued b-metric spaces
In an attempt to extend further the obtained results, Ma and Jiang introduced in [2] the following concept (thus generalizing the concept of a b-metric space of Czerwik [18]).
Definition 2
([2], Definition 2.1)
- (i)
\(d(x,y)\succeq \theta\) for all \(x,y\in X\) and \(d(x,y)=\theta\) if and only if \(x=y\);
- (ii)
\(d(x,y)=d(y,x)\) for all \(x,y\in X\);
- (iii)
\(d(x,y)\preceq b[d(x,z)+d(z,y)]\) for all \(x,y,z\in X\).
In [2], as well as in [5, 6], several fixed point results were obtained in \(C^{*}\)-algebra-valued b-metric spaces. However, we will show that neither of these results is in fact new - all of them can be simply reduced to their known b-metric counterparts. As an example, we prove this for the following result from [2].
Theorem 4
([2], Theorem 2.1)
Recall the following b-metric version of BCP.
Theorem 5
([19], Theorem 2.1)
Proof
Again, it is obvious that Theorem 4 implies Theorem 5. In order to prove the opposite, it is enough to put \(D(x,y)=\|d(x,y)\|\), \(\|b\|=s\), and \(\|a\|=\lambda\in[0,1)\), whence \((X,D,s)\) becomes a complete b-metric space and the condition (3.4) reduces to the condition (3.5). This proves our claim. □
Remark 2
- (1)
the Chatterjea-type fixed point result [2], Theorem 2.2 (with the contractive condition in the form \(d(Tx,Ty)\preceq a^{*}(d(x,Ty)+d(y,Tx)) a\), with \(a\in\mathbb{A}\), \(\|a\|<1/\|b\|\sqrt{2}\));
- (2)
the Kannan-type fixed point result [2], Theorem 2.3 (with the contractive condition in the form \(d(Tx,Ty)\preceq a^{*}(d(x,Tx)+d(y,Ty)) a\), with \(a\in\mathbb{A}\), \(\|a\|<1/\sqrt{2\|b\|}\));
- (3)
the Banach-type cyclic fixed point result [6], Theorem 4.1 (with the improved condition \(\|\lambda\|<1\) instead of \(\|\lambda\|<1/\|b\|\));
- (4)
the Banach-type fixed point result for expansive mappings [6], Theorem 4.4 (the same comment);
- (5)
the Kannan-type, resp. Chatterjea-type cyclic fixed point results [6], Theorem 4.5 and Theorem 4.7 (with contractive conditions as in (2), resp. (1)).
Naturally, the same applies to several other fixed and common fixed point results in b-metric spaces.
Remark 3
We note that the conclusions of this paper do not hold in cone metric spaces over Banach algebras treated in [13, 14] and several other articles. Namely, Lemma 1(3) does not necessarily hold in arbitrary Banach algebras. Also, in the fixed point results obtained in these spaces, usually the spectral radius \(r(a)\) is used instead of the norm \(\|a\|\). Since, in general, \(r(a)<\|a\|\) (in Banach algebras which are not \(C^{*}\)-algebras), these results are more general and cannot be reduced (at least not directly) to their metric counterparts.
Declarations
Acknowledgements
The authors are highly indebted to the referees of this paper, who helped us to improve its exposition. The first author is thankful to the Ministry of Education, Science and Technological Development of Serbia, Grant No. 174002.
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
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