The Fixed Point Property of a Banach Algebra Generated by an Element with Infinite Spectrum

Copyright © 2018 P.Thongin andW. Fupinwong.This is an open access article distributed under theCreativeCommonsAttribution License,whichpermits unrestricteduse, distribution, and reproduction in anymedium, provided the original work is properly cited. A Banach spaceX is said to have the fixed point property if for each nonexpansive mapping T : E → E on a bounded closed convex subset E ofX has a fixed point. LetX be an infinite dimensional unital Abelian complex Banach algebra satisfying the following: (i) condition (A) in Fupinwong and Dhompongsa, 2010, (ii) if x, y ∈ X is such that |τ(x)| ≤ |τ(y)|, for each τ ∈ Ω(X), then ‖x‖ ≤ ‖y‖, and (iii) inf{r(x) : x ∈ X, ‖x‖ = 1} > 0.We prove that there exists an element x0 inX such that ⟨x0⟩R = {∑ki=1 αixi0 : k ∈ N, αi ∈ R} does not have the fixed point property.Moreover, as a consequence of the proof, we have that, for each element x0 inXwith infinite spectrum and σ(x0) ⊂ R, the Banach algebra ⟨x0⟩ = {∑ki=1 αixi0 : k ∈ N, αi ∈ C} generated by x0 does not have the fixed point property.


Introduction
A Banach space  is said to have the fixed point property if for each nonexpansive mapping  :  →  on a bounded closed convex subset  of  has a fixed point, to have the weak fixed point property if for each nonexpansive mapping  :  →  on a weakly compact convex subset  of  has a fixed point.
In 1981, D. E. Alspach [1] showed that there is an isometry  :  →  on a weakly compact convex subset  of the Lebesgue space  1 [0, 1] without a fixed point.Consequently,  1 [0, 1] does not have the weak fixed point property.
In 1997, A. T. Lau, P. F. Mah, and Ali Ülger [3] proved the following theorem.Theorem 1.Let  be a locally compact Hausdorff space.If  0 () has the weak fixed point property, then X is dispersed.Moreover, by using Theorem 1, they proved the following results.
Corollary 2. Let  be a locally compact group.Then the  *algebra  0 () has the weak fixed point property if and only if  is discrete.

Corollary 3. A von Neumann algebra M has the weak fixed point property if and only if M is finite dimensional.
In 2005, Benavides and Pineda [4] studied the concept of -almost weak orthogonality in the Banach lattice () and proved the following results.

Theorem 4.
Let  be a -almost weakly orthogonal closed subspace of () where  is a metrizable compact space.Then  has the weak fixed point property.
If  is a complex Banach algebra, condition (A) is defined by the following.(A) For each  ∈ , there exists an element  ∈  such that () = (), for each  ∈ Ω().
It can be seen that each  * -algebra satisfies condition (A).
Then  does not have the fixed point property.
In 2010, D. Alimohammadi and S. Moradi [6] used the above result to obtain sufficient conditions to show that some unital uniformly closed subalgebras of (Ω), where Ω is a compact space, do not have the fixed point property.
In 2011, S. Dhompongsa, W. Fupinwong, and W. Lawton et al. [7] showed that a  * -algebra has the fixed point property if and only if it is finite dimensional.
In 2016, by using Urysohn's lemma and Schauder-Tychonoff fixed point theorem, D. Alimohammadi [9] proved the following result.Theorem 8. Let Ω be a locally compact Hausdorff space.Then the following statements are equivalent: (iii)  0 (Ω) does not have the fixed point property.
In this paper, let  be an infinite dimensional unital Abelian complex Banach algebra satisfying (i) condition (A), (ii) if ,  ∈  is such that |()| ≤ |()|, for each  ∈ Ω(), then ‖‖ ≤ ‖‖, and (iii) inf {() :  ∈ , ‖‖ = 1} > 0. We prove that there exists an element  0 in  such that does not have the fixed point property.Our result is a generalization of Theorem 7. And, as a consequence of the proof, we have that, for each element  0 in  with infinite spectrum and ( 0 ) ⊂ R, the Banach algebra ⟨ 0 ⟩ = {∑  =1     0 :  ∈ N,   ∈ C} generated by  0 does not have the fixed point property.

Preliminaries
Let F be the field R or C. Let  be a Banach space over F. We say that a mapping  :  ⊂  →  is nonexpansive if for each ,  ∈ , where  is a nonempty subset of .A Banach space  over F is said to have the fixed point property if for each nonexpansive mapping  :  →  on a nonempty bounded closed convex subset  of  has a fixed point.We define the spectrum of an element  of a unital Banach algebra  over F to be the set where V() is the set of all invertible elements in .
The spectral radius of  is defined to be We say that a mapping  :  → F is a character on an algebra  over F if  is a nonzero homomorphism.We denote by Ω() the set of all characters on X.If  is a unital Abelian Banach algebra over F, it is known that Ω() is compact.
If  is a complex Banach algebra, condition (A) is defined by the following.
We denote by  F () the unital Banach algebra of continuous functions from a topological space  to F where the operations are defined pointwise and the norm is the supnorm.
The following Theorem is known as the Stone-Weierstrass approximation theorem for  R ().Theorem 9. Let  be a subalgebra of  R () satisfying the following conditions: (i)  separates the points of .
(ii) A annihilates no point of .
Then  is dense in  R ().
Let  be an Abelian Banach algebra over F. The Gelfand representation  :  →  F (Ω()) is defined by   → x, where x is defined by for each  ∈ Ω().If  is unital and Abelian, then () = {() :  ∈ Ω()}, for each  ∈ .It is known that The Jacobson radical () of a Banach algebra  over F is the intersection of all regular maximal left ideals of .It is known that if  is a unital complex Banach algebra and  ∈ () then the spectral radius () of  is equal to zero.A Banach algebra  over F is said to be semisimple if () = {0}.

Lemmas
First of all, we study the relationship between the sup-norm and the spectral radius, and we prove some properties of the spectral radius on a complex unital Banach algebra satisfying inf{‖ x‖ ∞, :  ∈ , ‖‖ = 1} > 0.
The following lemma was proved in [11].
Lemma 11.Let  be an infinite dimensional semisimple complex Banach algebra.Then there exists an element with an infinite spectrum.
As a consequence of condition (A), Lemma 10 and Lemma 11, we obtain the following results, Lemma 12 and Lemma 13, immediately.
Proof.Since  is Abelian, It follows from Lemma 10 that  is semisimple.From Lemma 11, there exists an element  in  with infinite spectrum.From condition (A), there exists  ∈  such that for each  ∈ Ω().Hence for each  ∈ Ω().
Lemma 13.Let  be an infinite dimensional complex unital Banach algebra, and let  0 be an element in  with infinite spectrum.Then {  0 :  ∈ N} is linearly independent.
Proof.Assume that where with   ̸ =   for each  ̸ = .
Lemma 14.Let  be an infinite dimensional complex unital Abelian Banach algebra satisfying condition (A), and let  0 be an element in  with infinite spectrum and ( 0 ) ∈ R, for each  ∈ Ω().Define Then  is an infinite dimensional real unital Abelian Banach algebra with Ω() ̸ = 0.
Similarly, one can prove the following lemma.
Lemma 15.Let  be an infinite dimensional complex unital Banach algebra satisfying condition (A), and let  0 be an element in  with infinite spectrum and ( 0 ) ∈ R, for each  ∈ Ω().Define Then ⟨ 0 ⟩ R is an infinite dimensional real nonunital Abelian Banach algebra.
Some useful properties of the real unital Abelian Banach algebra  is shown in the following lemma.
Lemma 16.Let  be an infinite dimensional complex unital Abelian Banach algebra satisfying condition (A), and let  0 be an element in  with infinite spectrum and ( 0 ) ∈ R, for each  ∈ Ω().Define then  is a real unital Abelian Banach algebra satisfying the following conditions: (i) The Gelfand representation  from  into  R (Ω()) is a bounded isomorphism.
(ii) The inverse  −1 is also a bounded isomorphism.
Proof (ii) There exists a strictly decreasing sequence in ().
Proof.Let  be an infinite dimensional complex unital Banach algebra.Assume that there exists an element  in ⟨ 0 ⟩ R with infinite spectrum () and () ⊂ R.
We next give the following two lemmas which are important tools for proving the main result.