C∗-algebra-valued metric spaces and related fixed point theorems

R E S E A R C H

Open Access

C

∗

-algebra-valued metric spaces and related

fixed point theorems

Zhenhua Ma

_{, Lining Jiang}

_{and Hongkai Sun}

*_{Correspondence:}

jianglining@bit.edu.cn

1_{School of Mathematics and}

Statistics, Beijing Institute of Technology, Beijing, 100081, China Full list of author information is available at the end of the article

Abstract

Based on the concept and properties ofC∗-algebras, the paper introduces a concept ofC∗-algebra-valued metric spaces and gives some fixed point theorems for

self-maps 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

Keywords: C∗-algebra;C∗-algebra-valued metric; contractive mapping; expansive mapping; fixed point theorem

1 Introduction

We begin with the concept ofC∗-algebras.

Suppose thatAis a unital algebra with the unitI. An involution onAis a conjugate-linear mapa→a∗onAsuch thata∗∗=aand (ab)∗=b∗a∗for alla,b∈A. The pair (A,∗) is called a∗-algebra. A Banach∗-algebra is a∗-algebraAtogether with a complete sub-multiplicative norm such thata∗=a(∀a∈A). AC∗-algebra is a Banach∗-algebra such thata∗a=a[, ].

Notice that the seeming mild requirement on aC∗-algebra above is in fact very strong. Moreover, the existence of the involutionC∗-algebra theory can be thought of as infinite-dimensional real analysis. Clearly that under the norm topology, L(H), the set of all bounded linear operators on a Hilbert spaceH, is aC∗-algebra.

As we have known, the Banach contraction principle is a very useful, simple and clas-sical 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 gener-alized 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 [] started the investigations concerning a fixed point theory in ordered metric spaces. Later, many authors followed this research by introduc-ing and investigatintroduc-ing the different types of contractive mappintroduc-ings. For example in [] Ca-balleroet al.considered contractive like mapping in ordered metric spaces and applied their results in ordinary differential equations. In , Huang and Zhang [] 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

ric spaces [–]. In [], the authors studied the operator-valued 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 operator-valued metric spaces, and give some related fixed point theorems for self-maps with contractive or expansive conditions on such spaces.

The paper is organized as follows: Based on the concept and properties ofC∗-algebras, we first introduce a concept ofC∗-algebra-valued metric spaces. Moreover, some fixed point theorems for mappings satisfying the contractive or expansive conditions on

C∗-algebra-valued 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,Awill denote an unitalC∗-algebra with a unitI. SetAh={x∈ A:x=x∗}. We call an elementx∈Aa positive element, denote it byxθ, ifx∈Ah andσ(x)⊂R+= [,∞), whereσ(x) is the spectrum ofx. Using positive elements, one

can define a partial orderingonAh as follows:xyif and only ify–xθ, whereθ means the zero element inA. From now on, byA+we denote the set{x∈A:xθ}and

|x|= (x∗x)_.

Remark . WhenAis a unitalC∗-algebra, then for anyx∈A+we havexI⇔ x ≤

[, ].

With the help of the positive element inA, one can give the definition of aC∗ -algebra-valued metric space.

Definition . LetXbe a nonempty set. Suppose the mappingd:X×X→Asatisfies:

() θd(x,y)for allx,y∈Xandd(x,y) =θ⇔x=y; () d(x,y) =d(y,x)for allx,y∈X;

() d(x,y)d(x,z) +d(z,y)for allx,y,z∈X.

Then d is called aC∗-algebra-valued metric onX and (X,A,d) is called aC∗ -algebra-valued metric space.

It is obvious that C∗-algebra-valued metric spaces generalize the concept of metric spaces, replacing the set of real numbers byA+.

Definition . Let (X,A,d) be aC∗-algebra-valued metric space. Suppose that{xn} ⊂X andx∈X. If for anyε>  there isNsuch that for alln>N,d(xn,x) ≤ε, then{xn}is said to be convergent with respect toAand{xn}converges toxandxis the limit of{xn}. We denote it bylimn→∞xn=x.

If for anyε>  there isNsuch that for alln,m>N,d(xn,xm) ≤ε, then{xn}is called a Cauchy sequence with respect toA.

It is obvious that ifXis a Banach space, then (X,A,d) is a completeC∗-algebra-valued metric space if we set

d(x,y) =x–yI.

The following are nontrivial examples of completeC∗-algebra-valued metric space.

Example . LetX=L∞(E) andH=L(E), where E is a Lebesgue measurable set. By

L(H) we denote the set of bounded linear operators on Hilbert spaceH. ClearlyL(H) is a

C∗-algebra with the usual operator norm. Defined:X×X→L(H) by

d(f,g) =π|f–g| (∀f,g∈X),

whereπh:H→His the multiplication operator defined by πh(ϕ) =h·ϕ,

forϕ∈H. Thendis aC∗-algebra-valued metric and (X,L(H),d) is a completeC∗ -algebra-valued metric space.

Indeed, it suffices to verity the completeness. Let{fn}∞n=inXbe a Cauchy sequence with

respect toL(H). Then for a givenε> , there is a natural numberN(ε) such that for all

n,m≥N(ε),

d(fn,fm)=π|fn–fm|=fn–fm∞≤ε,

then{fn}∞n= is a Cauchy sequence in the spaceX. Thus, there is a functionf ∈Xand a

natural numberN(ε) such thatfn–f∞≤εifn≥N.

It follows that

d(fn,f)=π|fn–f|=fn–f∞≤ε.

Therefore, the sequence{fn}∞n=converges to the functionf inXwith respect toL(H), that

is, (X,L(H),d) is complete with respect toL(H).

Example . LetX=RandA=M(R). Define

d(x,y) =diag|x–y|,α|x–y|,

wherex,y∈Randα≥ is a constant. It is easy to verifydis aC∗-algebra-valued metric and (X,M(R),d) is a completeC∗-algebra-valued metric space by the completeness ofR.

Now we give the definition of aC∗-algebra-valued contractive mapping onX.

Definition . Suppose that (X,A,d) is aC∗-algebra-valued metric space. We call a map-pingT:X→Xis aC∗-algebra-valued contractive mapping onX, if there exists anA∈A

withA<  such that

Theorem . If(X,A,d)is a complete C∗-algebra-valued metric space and T is a contrac-tive mapping,there exists a unique fixed point in X.

Proof It is clear that ifA=θ,T maps theXinto a single point. Thus without loss of gen-erality, one can suppose thatA=θ.

Choosex∈Xand setxn+=Txn=Tn+x,n= , , . . . . For convenience, byBwe denote

the elementd(x,x) inA.

Notice that in aC∗-algebra, ifa,b∈A+andab, then for anyx∈Abothx∗axandx∗bx

are positive elements andx∗axx∗bx[]. Thus

d(xn+,xn) =d(Txn,Txn–)A∗d(xn,xn–)A

A∗d(xn–,xn–)A

· · ·

A∗nd(x,x)An

=A∗nBAn.

So forn+  >m,

d(xn+,xm)d(xn+,xn) +d(xn,xn–) +· · ·+d(xm+,xm)

A∗nBAn+· · ·+A∗mBAm

= n

k=m

A∗kBAk

= n

k=m

A∗kB_B_Ak

= n

k=m

B_Ak∗_B_Ak

= n

k=m

B_Ak

n

k=m

B_Ak I

n

k=m

B_Ak_I

B

n

k=m

AkI

B A m

 –AI→θ (m→ ∞).

Since

θ d(Tx,x)d(Tx,Txn) +d(Txn,x)

A∗d(x,xn)A+d(xn+,x)→θ (n→ ∞),

hence,Tx=x,i.e.,xis a fixed point ofT.

Now suppose thaty(=x) is another fixed point ofT, since θd(x,y) =d(Tx,Ty)A∗d(x,y)A,

we have

≤d(x,y)=d(Tx,Ty)

≤A∗d(x,y)A

≤A∗d(x,y)A

=Ad(x,y) < d(x,y).

It is impossible. Sod(x,y) =θ andx=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 . LetX be a nonempty set. We call a mappingT is aC∗-algebra-valued expansion mapping onX, ifT:X→Xsatisfies:

() T(X) =X;

() d(Tx,Ty)A∗d(x,y)A,∀x,y∈X,

whereA∈Ais an invertible element andA–_{< .}

Theorem . Let(X,A,d)be a complete C∗-algebra-valued metric space.Then for the expansion mapping T,there exists a unique fixed point in X.

Proof Firstly,Tis injective. Indeed, for anyx,y∈Xwithx=y, ifTx=Ty, we have θ=d(Tx,Ty)A∗d(x,y)A.

SinceA∗d(x,y)A∈A+,A∗d(x,y)A=θ. AlsoAis invertible,d(x,y) =θ, which is impossible.

ThusTis injective.

Next, we will showThas a unique fixed point inX. In fact, sinceTis invertible and for anyx,y∈X,

d(Tx,Ty)A∗d(x,y)A.

In the above formula, substitutex,ywithT–_x,T–_{y, respectively, and we get}

This means

A∗–d(x,y)A–dT–x,T–y,

and thus

A–∗d(x,y)A–dT–x,T–y.

Using Theorem ., there exists a uniquexsuch thatT–_{x=x, which means there has a}

unique fixed pointx∈Xsuch thatTx=x. Before introducing another fixed point theorem, we give a lemma first. Such a result can be found in [, ].

Lemma . Suppose thatAis a unital C∗-algebra with a unit I.

() Ifa∈A+witha<_,thenI–ais invertible anda(I–a)–< ; () suppose thata,b∈Awitha,bθandab=ba,thenabθ;

() byAwe denote the set{a∈A:ab=ba,∀b∈A}.Leta∈A,ifb,c∈Awithbcθ andI–a∈A₊is a invertible operator,then

(I–a)–b(I–a)–c.

Notice that in aC∗-algebra, ifθa,b, one cannot conclude thatθab. Indeed, consider theC∗-algebraM(C) and seta=

_{ }  

,b=_{– } –, thenab=_{– }– . Clearlya,b∈M(C)+,

whileabis not.

Theorem . Let(X,A,d)be a complete C∗-valued metric space.Suppose the mapping T:X→X satisfies for all x,y∈X

d(Tx,Ty)Ad(Tx,y) +d(Ty,x),

where A∈A₊andA< 

.Then there exists a unique fixed point in X.

Proof Without loss of generality, one can suppose that A =θ. Notice that A∈ A₊,

A(d(Tx,y) +d(Ty,x)) is also a positive element.

Choosex∈X, setxn+=Txn=Tn+x,n= , , . . . , byBwe denote the elementd(x,x)

inA. Then

d(xn+,xn) =d(Txn,Txn–)

Ad(Txn,xn–) +d(Txn–,xn)

=Ad(Txn,Txn–) +d(Txn–,Txn–)

Ad(Txn,Txn–) +d(Txn–,Txn–)

=Ad(Txn,Txn–) +Ad(Txn–,Txn–)

Thus,

(I–A)d(xn+,xn)Ad(xn,xn–).

SinceA∈A₊withA< 

, one have (I–A) –_∈A

+and furthermoreA(I–A)–∈A+

withA(I–A)–<  by Lemma .. Therefore,

d(xn+,xn)A(I–A)–d(xn,xn–) =td(xn,xn–),

wheret=A(I–A)–_.

Forn+  >m,

d(xn+,xm)d(xn+,xn) +d(xn,xn–) +· · ·+d(xm+,xm)

tn+tn–+· · ·+tmd(x,x)

= n

k=m

tk_tk_B_B

= n

k=m

B_tk_tk_B

= n

k=m

tk_B∗_tk_B

= n

k=m

tk_B

n

k=m

tk_B I

n

k=m

B_tk_I

B

n

k=m

tkI

B t

m

 –tI→θ (m→ ∞).

This implies that {xn}is a Cauchy sequence with respect toA. By the completeness of (X,A,d), there existsx∈Xsuch thatlim_n→∞xn=x,i.e.limn→∞Txn–=x. Since

d(Tx,x)d(Tx,Txn) +d(Txn,x)

Ad(Tx,xn) +d(Txn,x)

+d(xn+,x)

Ad(Tx,x) +d(x,xn) +d(xn+,x)

+d(xn+,x).

This is equivalent to

(I–A)d(Tx,x)Ad(x,xn) +d(xn+,x)

Then

d(Tx,x) ≤ A(I–A)–d(x,xn)+d(xn+,x)

+(I–A)–d(xn+,x)

→ (n→ ∞).

This implies thatTx=x i.e.,xis a fixed point ofT. Now ify(=x) is another fixed point ofT, then

θ d(x,y) =d(Tx,Ty)

Ad(Tx,y) +d(Ty,x) =Ad(x,y) +d(y,x)

i.e.,

d(x,y)A(I–A)–_d(x,y).

SinceA(I–A)–_{< ,}

≤d(x,y)=d(Tx,Ty)

≤A(I–A)–d(x,y)

≤A(I–A)–d(x,y) < d(x,y).

This means that

d(x,y) =θ ⇔ x=y.

Therefore the fixed point is unique and the proof is complete.

3 Applications

As applications of contractive mapping theorem on completeC∗-algebra-valued metric spaces, existence and uniqueness results for a type of integral equation and operator equa-tion are given.

Example . Consider the integral equation

x(t) =

E

Kt,s,x(s)ds+g(t), t∈E,

whereEis a Lebesgue measurable set. Suppose that

() there exists a continuous functionϕ:E×E→Randk∈(, )such that K(t,s,u) –K(t,s,v)≤kϕ(t,s)(u–v),

fort,s∈Eandu,v∈R; () sup_t∈EE|ϕ(t,s)|ds≤.

Then the integral equation has a unique solutionx∗inL∞(E).

Proof LetX=L∞(E) andH=L_{(E). Setdas Example ., thendis aC}∗_{-algebra-valued} metric and (X,L(H),d) is a completeC∗-algebra-valued metric space with respect toL(H).

LetT:L∞(E)→L∞(E) be

Tx(t) =

E

Kt,s,x(s)ds+g(t), t∈E.

SetA=kI, thenA∈L(H)+andA=k< . For anyh∈H, d(Tx,Ty)= sup

h=

(π|Tx–Ty|h,h)

= sup h=

E

E

Kt,s,x(s)–Kt,s,y(s)ds

h(t)h(t) dt

≤ sup h=

E E

Kt,s,x(s)–Kt,s,y(s)dsh(t)dt

≤ sup h=

E E

kϕ(t,s)x(s) –y(s)dsh(t)dt

≤ksup h=

E E

ϕ(t,s)dsh(t)dt· x–y∞

≤ksup t∈E

E

ϕ(t,s)ds· sup h=

E

h(t)dt· x–y∞

≤kx–y∞ =Ad(x,y).

SinceA< , the integral equation has a unique solutionx∗inL∞(E).

Example . Suppose thatHis a Hilbert space,L(H) is the set of linear bounded operators onH. LetA,A, . . . ,An∈L(H), which satisfy

∞

n=An<  andX∈L(H),Q∈L(H)+.

Then the operator equation

X– ∞

n=

A∗_nXAn=Q

has a unique solution inL(H).

Choose a positive operatorT∈L(H). ForX,Y∈L(H), set

d(X,Y) =X–YT.

It is easy to verify thatd(X,Y) is aC∗-algebra-valued metric and (L(H),d) is complete sinceL(H) is a Banach space.

Consider the mapF:L(H)→L(H) defined by

F(X) = ∞

n=

A∗_nXAn+Q.

Then

dF(X),F(Y)=F(X) –F(Y)T

=

∞

n=

A∗_n(X–Y)An

T

∞

n=

AnX–YT

=αd(X,Y)

=α_I∗_d(X,Y)α_I.

Using Theorem ., there exists a unique fixed point X in L(H). Furthermore, since

∞

n=A∗nXAn+Qis a positive operator, the solution is a Hermitian operator.

As a special case of Example ., one can consider the following matrix equation, which can also be found in []:

X–A∗_XA–· · ·–A∗mXAm=Q,

whereQis a positive definite matrix andA, . . . ,Amare arbitraryn×nmatrices. Using Example ., there exists a unique Hermitian matrix solution.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

Author details

1_{School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, 100081, China.}2_{Department of}

Mathematics and Physics, Hebei Institute of Architecture and Civil Engineering, Zhangjiakou, 075024, China.

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 (YJXTD-2014-A08).

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