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

Abstract

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. Let be an infinite dimensional unital Abelian complex Banach algebra satisfying the following: (i) condition (A) in Fupinwong and Dhompongsa, 2010, (ii) if is such that for each then and (iii) We prove that there exists an element in such that does not have the fixed point property. Moreover, as a consequence of the proof, we have that, for each element in with infinite spectrum and the Banach algebra generated by does not have the fixed point property.

1. 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 without a fixed point. Consequently, does not have the weak fixed point property.

In 1983, J. Elton, P. K. Lin, E. Odell, and S. Szarek [2] proved that ) has the weak fixed point property, if is a compact ordinal with

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 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 has the weak fixed point property if and only if is discrete.

Corollary 3. A von Neumann algebra has the weak fixed point property if and only if 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.

Theorem 5. Let be a metrizable compact space. Then, the following conditions are all equivalent:(1) is -almost weakly orthogonal,(2) is -weakly orthogonal,(3)

Corollary 6. Let be a compact set with 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

It can be seen that each -algebra satisfies condition (A).

In 2010, W. Fupinwong and S. Dhompongsa [5] proved that each infinite dimensional unital Abelian real Banach algebra with satisfying (i) if is such that for each then and (ii) does not have the fixed point property. Moreover, they proved the following theorem.

Theorem 7. Let be an infinite dimensional unital Abelian complex Banach algebra satisfying condition (A) and each of the following:(i)If is such that for each , then (ii) 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 , where 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 2012, W. Fupinwong [8] show that the unitality in Theorem 7 proved in [5] can be omitted.

In 2016, by using Urysohn’s lemma and Schauder-Tychonoff fixed point theorem, D. Alimohammadi [9] proved the following result.

Theorem 8. Let be a locally compact Hausdorff space. Then the following statements are equivalent:(i) is infinite set.(ii) is infinite dimensional.(iii) does not have the fixed point property.

In 2017, J. Daengsaen and W. Fupinwong [10] showed that for each infinite dimensional real Abelian Banach algebra with satisfying (i) if is such that for each then and (ii) 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 , then , and (iii) We prove that there exists an element 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 in with infinite spectrum and , the Banach algebra generated by does not have the fixed point property.

2. Preliminaries

Let be the field or Let be a Banach space over . We say that a mapping is nonexpansive if for each , where is a nonempty subset of A Banach space over 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 to be the set where is the set of all invertible elements in

The spectral radius of is defined to be

We say that a mapping is a character on an algebra over if is a nonzero homomorphism. We denote by the set of all characters on X. If is a unital Abelian Banach algebra over , it is known that is compact.

If is a complex Banach algebra, condition (A) is defined by the following.

(A) For each , there exists an element such that , for each

We denote by the unital Banach algebra of continuous functions from a topological space to where the operations are defined pointwise and the norm is the sup-norm.

The following Theorem is known as the Stone-Weierstrass approximation theorem for

Theorem 9. Let be a subalgebra of satisfying the following conditions:(i) separates the points of (ii)A annihilates no point of Then is dense in

Let be an Abelian Banach algebra over The Gelfand representation is defined by , where is defined by for each If is unital and Abelian, then , for each It is known that if is Abelian, where

The Jacobson radical of a Banach algebra over 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 is said to be semisimple if

3. 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

Lemma 10. Let be a complex unital Banach algebra satisfying Then (i)(ii) is semisimple.

Proof. Let be a complex unital Banach algebra satisfying (i) From , for each , it follows that , for each Therefore, (ii) From (i), we have and hence, for each , implies Since , for each , Therefore, is semisimple.

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.

Lemma 12. Let be an infinite dimensional complex unital Abelian Banach algebra with condition (A) and satisfy Then there exists with infinite spectrum and , for each

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 Hence for each

Lemma 13. Let be an infinite dimensional complex unital Banach algebra, and let be an element in with infinite spectrum. Then is linearly independent.

Proof. Assume that where Let with for each
From , it follows that for each Hence , for each

Lemma 14. Let be an infinite dimensional complex unital Abelian Banach algebra satisfying condition (A), and let be an element in with infinite spectrum and , for each Define Then is an infinite dimensional real unital Abelian Banach algebra with

Proof. From Lemma 13, is linearly independent in , so is infinite dimensional. To show that Let , and define by Indeed, is real-valued since , for each Obviously, is a nonzero homomorphism on So

Similarly, one can prove the following lemma.

Lemma 15. Let be an infinite dimensional complex unital Banach algebra satisfying condition (A), and let be an element in with infinite spectrum and , for each Define Then 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 be an element in with infinite spectrum and , for each Define If satisfies then is a real unital Abelian Banach algebra satisfying the following conditions:
(i) The Gelfand representation from into is a bounded isomorphism.
(ii) The inverse is also a bounded isomorphism.

Proof. (i) It follows that So is injective. We have is a subalgebra of separating the points of and annihilating no point of Moreover, is complete, so is closed. In fact, if is a Cauchy sequence in , assume to the contrary that is not Cauchy. So there exist and subsequences and of such that for each Let So , for each Since is Cauchy, Hence which is a contradiction. So we conclude that is a Cauchy sequence. Then is a convergent sequence in , say Therefore, since for each , So is complete. It follows from the Stone-Weierstrass theorem that is surjective.
(ii) is a consequence of the open mapping theorem.

Lemma 17. Let be an infinite dimensional complex unital Abelian Banach algebra satisfying condition (A), and let be an element in with infinite spectrum and , for each Define If there exists an element in with infinite spectrum and , then there exists satisfying the following conditions:
(i)
(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 with infinite spectrum and
Since is unital and Abelian, the spectrum of is Hence Let be an infinite sequence in We may assume that is strictly increasing and
Define a continuous function bySo is joining the points and , and Let It follows from Lemma 16 that Since , We have that is a strictly decreasing sequence in Moreover,

We next give the following two lemmas which are important tools for proving the main result.

Lemma 18. Let be an infinite dimensional complex unital Abelian Banach algebra satisfying condition (A) and and let be an element in with infinite spectrum and , for each Define Then there exists a sequence in such that , for each , and is a sequence of nonempty pairwise disjoint subsets of

Proof. Obviously, since From Lemma 14 and Lemma 2.10 (iii) in [5], there exists such that is infinite. We have such that is infinite. From Lemma 17, we may assume without generality that satisfies and there exists a strictly decreasing sequence of real number in , say Moreover, we may assume that
Define a continuous function bySo is joining the points and , and
Let , and define a continuous function bySo is joining the points and , and
Let Continuing in this manner, we get a sequence of points in with , for each , and is a sequence of nonempty pairwise disjoint subsets of
Moreover, , for each , since , for each

Lemma 19. Let be an infinite dimensional complex unital Abelian Banach algebra satisfying condition (A) and and let be an element in with infinite spectrum and , for each Define Assume that there exists a bounded sequence in which contains no convergent subsequences such that is finite for each Then there exists an element such that is equal to or

Proof. It follows form Lemma 14 and Lemma 16 that is an infinite dimensional real unital Abelian Banach algebra with and homeomorphic to Assume that there exists a bounded sequence in which contains no convergent subsequences and such that is finite for each From the proof of Lemma 2.10 (ii) in [5], we have where is a closed set in , is closed and open for each , and is a partition of Define by for each So is a character on There are two cases to be considered. If is in , define byIf is in , for some , we may assume without loss of generality that , and define byFor each case, the inverse image of each closed set in is closed, so Let be the Gelfand representation. Therefore, is an element in , say , such that is equal to or Moreover, since

Lemma 20. Let be an infinite dimensional complex unital Abelian Banach algebra satisfying condition (A) and and let be an element in with infinite spectrum and , for each Define and let with , and , for each Define where , and define by Assume that satisfies the following condition.
If is such that , for each , then
Then is a nonempty bounded closed convex subset of and is a nonexpansive mapping.

Proof. Obviously, is closed and convex. is nonempty since
Let Hence Therefore, is bounded.
Let , and let Define by for each So
Then From (i), we have So is nonexpansive.

4. Main Result

Theorem 21. Let be an infinite dimensional complex unital Abelian Banach algebra satisfying condition (A) and the following conditions:(i)If is such that , for each , then ,(ii) Then there exists an element in such that does not have the fixed point property.

Proof. Let be an infinite dimensional complex unital Abelian Banach algebra satisfying (i), (ii), and condition (A). It follows from Lemma 12 that there exists an element in with infinite spectrum and , for each Let From Lemma 14, is an infinite dimensional real unital Abelian Banach algebra with From Lemma 18, it follows that there is a sequence in such that for each , and , , are nonempty pairwise disjoint.
Write , define and define by From Lemma 20, is a nonempty bounded closed convex subset in and is nonexpansive for each
Assume to the contrary that has the fixed point property. For each , since is a nonempty bounded closed convex subset in , has a fixed point in , say Since is a fixed point of , Then , and thenfor each Since are pairwise disjoint, , if Thus has no convergent subsequences. Since and are homeomorphic, has no convergent subsequences. From Lemma 19, there exists an element in such that is equal to or
Write , define and define by It follows from Lemma 20 that is a nonexpansive mapping on a nonempty bounded closed convex subset in So has a fixed point in , say There are two cases to be considered.

Case 1 (). Hence ThenSo and It follows from that is a nonempty pairwise disjoint open covering of the compact set , which is a contradiction.

Case 2 ().  where
It can be seen that is a nonempty bounded closed convex subset of
Define by for each We have Define by It follows from (i) that is a nonexpansive mapping on a nonempty bounded closed convex subset of
has a fixed point since has the fixed point property. It follows that has a fixed point, say ThenSo and It follows from that is a nonempty pairwise disjoint open covering of the compact set , which is a contradiction.
So we conclude that does not have the fixed point property.

By following the proof of the above theorem, we obtain some corollaries.

Corollary 22. Let be an infinite dimensional complex unital Abelian Banach algebra satisfying condition (A) and the following conditions:
(i) If is such that , for each , then ,
(ii)
If is an element in with infinite spectrum and , then the Banach algebra generated by does not have the fixed point property.

Corollary 23. Let be a self-adjoint element in a unital Abelian -algebra , then the algebra generated by does not have the fixed point property.

Finally, we pose an interesting problem.

Problem 24. Can condition (A) be removed from Theorem 21?

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research was supported by Chiang Mai University.