3.1 Fixed point results in \(C^{*}\)algebravalued 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)
Let X be a nonempty set and let \(d:X\times X\to\mathbb{A}\) satisfy

(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 \(x,y,z\in X\).
Then d is called a \(C^{*}\)algebravalued metric on X and \((X,\mathbb{A},d)\) is called a \(C^{*}\)algebravalued metric space.
As the main result, they proved the following.
Theorem 1
([1], Theorem 2.1)
Suppose that
\((X,\mathbb{A},d)\)
is a
\(C^{*}\)algebravalued metric space and let for a mapping
\(T:X\to X\)
there exists
\(a\in\mathbb{A}\)
with
\(\a\<1\)
such that
$$ d(Tx,Ty)\preceq a^{*}d(x,y) a,\quad \textit{for all }x,y\in X. $$
(3.1)
Then
T
has a unique fixed point in X.
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
First of all, obviously, taking \(\mathbb{A}=\mathbb{R}\) (with standard operations, absolute value as the norm and involution given by \(a^{*}=a\)) in Theorem 1, the condition (3.1) reduces to
$$d(Tx,Ty)\leq a^{2}d(x,y),\quad \text{for all }x,y\in X, $$
with \(a^{2}\in[0,1)\), hence Theorem 1 reduces to BCP.
Conversely, let the conditions of Theorem 1 be satisfied. Denote
$$D(x,y)=\bigl\Vert d(x,y)\bigr\Vert \quad \text{for all }x,y\in X. $$
Then it is easy to see that \((X,D)\) is a complete (standard) metric space. In particular, the triangular inequality follows from \(\theta\preceq d(x,y)\preceq d(x,z)+d(z,y)\) and Lemma 1(3):
$$\begin{aligned} D(x,y)&=\bigl\Vert d(x,y)\bigr\Vert \leq\bigl\Vert d(x,z)+d(z,y)\bigr\Vert \\ &\leq\bigl\Vert d(x,z)\bigr\Vert +\bigl\Vert d(z,y)\bigr\Vert =D(x,z)+D(z,y). \end{aligned}$$
Moreover, \(T:X\to X\) is a (Banachtype) contraction in \((X,D)\) since (3.1) and Lemma 1(3) imply that, for all \(x,y\in X\),
$$\begin{aligned} D(Tx,Ty)&\leq\bigl\Vert a^{*}d(x,y) a\bigr\Vert \leq\bigl\Vert a^{*}\bigr\Vert \bigl\Vert d(x,y)\bigr\Vert \Vert a\Vert \\ &=\Vert a\Vert ^{2}D(x,y), \end{aligned}$$
where \(\a\^{2}\in[0,1)\). Hence, BCP implies that T has a unique fixed point. □
Remark 1
In a similar way, it is easy to show that the following results from [1, 4] can be directly reduced to their wellknown standard metric counterparts:

(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 \(C^{*}\)algebravalued 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 wellknown paper [16]. As an example, we prove the following.
Theorem 3
Let
\((X,\mathbb{A},d)\)
be a
\(C^{*}\)algebravalued metric space and
\(T:X\to X\)
be a mapping. Suppose that there exists
\(a\in\mathbb{A}\)
with
\(\a\<1\)
and that for all
\(x,y\in X\)
there exists
$$ u(x,y)\in\bigl\{ d(x,y),d(x,Tx),d(y,Ty),d(x,Ty),d(y,Tx)\bigr\} $$
(3.2)
such that
$$ d(Tx,Ty)\preceq a^{*}u(x,y) a. $$
(3.3)
Then
T
has a unique fixed point in
X.
Proof
As in the proof of Theorem 2, denote \(D(x,y)=\d(x,y)\\) for \(x,y\in X\). Then \((X,D)\) is a complete (standard) metric space. For arbitrary \(x,y\in X\), choose \(u(x,y)\) such that (3.2) and (3.3) hold. Then, by Lemma 1(3),
$$\begin{aligned} D(Tx,Ty)&=\bigl\Vert d(Tx,Ty)\bigr\Vert \leq\bigl\Vert a^{*}\bigr\Vert \cdot\bigl\Vert u(x,y)\bigr\Vert \cdot \Vert a\Vert \\ &\leq \Vert a\Vert ^{2}\max\bigl\{ \bigl\Vert d(x,y)\bigr\Vert , \bigl\Vert d(x,Tx)\bigr\Vert ,\bigl\Vert d(y,Ty)\bigr\Vert ,\bigl\Vert d(x,Ty)\bigr\Vert ,\bigl\Vert d(y,Tx)\bigr\Vert \bigr\} \\ &=\Vert a\Vert ^{2}\max\bigl\{ D(x,y),D(x,Tx),D(y,Ty),D(x,Ty),D(y,Tx) \bigr\} , \end{aligned}$$
where \(\a\^{2}\in[0,1)\). Hence, \(T:X\to X\) is a quasicontraction (in the sense of [17]) and it follows that it has a unique fixed point in X. □
Of course, the results of Kannan, Zamfirescu, HardyRogers (and many others; see [16]) follow as special cases.
Moreover, several known common fixed point results can be easily reformulated in the framework of \(C^{*}\)algebravalued metric spaces.
3.2 Fixed point results in \(C^{*}\)algebravalued bmetric 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 bmetric space of Czerwik [18]).
Definition 2
([2], Definition 2.1)
Let X be a nonempty set. A mapping \(d:X\times X\rightarrow \mathbb{A}\) is called a \(C^{*}\)algebravalued bmetric on X if there exists \(b\in \mathbb{A}\) such that \(b\succeq i\) and the following conditions are satisfied:

(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 \(x,y,z\in X\).
Then \((X,\mathbb{A},d,b)\) is called a \(C^{*}\)algebravalued bmetric space.
In [2], as well as in [5, 6], several fixed point results were obtained in \(C^{*}\)algebravalued bmetric spaces. However, we will show that neither of these results is in fact new  all of them can be simply reduced to their known bmetric counterparts. As an example, we prove this for the following result from [2].
Theorem 4
([2], Theorem 2.1)
Suppose that
\((X,\mathbb{A},d,b)\)
is a
\(C^{*}\)algebravalued
bmetric space and that for a mapping
\(T:X\rightarrow X\)
there exists
\(a\in \mathbb{A}\)
with
\(\a\<1\)
such that
$$ d(Tx,Ty) \preceq a^{*}d(x,y) a\quad \textit{for all }x,y\in X. $$
(3.4)
Then there exists a unique fixed point of
T
in X.
Recall the following bmetric version of BCP.
Theorem 5
([19], Theorem 2.1)
Let
\((X,D,s)\)
be a complete
bmetric space and let
\(T:X\to X\)
be a map such that, for some
\(\lambda\in[0,1)\)
and for all
\(x,y\in X\),
$$ D(Tx,Ty)\leq\lambda D(x,y). $$
(3.5)
Then
T
has a unique fixed point in
X.
Theorem 6
Theorem
4
is equivalent to Theorem
5.
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 bmetric space and the condition (3.4) reduces to the condition (3.5). This proves our claim. □
Remark 2
We note some other results from [2, 6] that can be reduced in the same way to wellknown results in bmetric spaces:

(1)
the Chatterjeatype 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 Kannantype 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 Banachtype cyclic fixed point result [6], Theorem 4.1 (with the improved condition \(\\lambda\<1\) instead of \(\\lambda\<1/\b\\));

(4)
the Banachtype fixed point result for expansive mappings [6], Theorem 4.4 (the same comment);

(5)
the Kannantype, resp. Chatterjeatype 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 bmetric 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.