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

1. Introduction

A Banach space X is said to have the fixed point property if for each nonexpansive mapping T : E [right arrow] E on a bounded closed convex subset E of X has a fixed point, to have the weak fixed point property if for each nonexpansive mapping T : E [right arrow] E on a weakly compact convex subset E of X has a fixed point.

In 1981, D. E. Alspach [1] showed that there is an isometry T : E [right arrow] E on a weakly compact convex subset E of the Lebesgue space [L.sub.1] [0, 1] without a fixed point. Consequently, [L.sub.1] [0,1] does not have the weak fixed point property.

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

In 1997, A. T. Lau, P. F. Mah, and Ali Ulger [3] proved the following theorem.

Theorem 1. Let X be a locally compact Hausdorff space. If [C.sub.0](X) has the weak fixed point property, then X is dispersed.

Moreover, by using Theorem 1, they proved the following results.

Corollary 2. Let G be a locally compact group. Then the [C.sup.*]-algebra [C.sub.0](G) has the weak fixed point property if and only if G 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 [omega]-almost weak orthogonality in the Banach lattice C(K) and proved the following results.

Theorem 4. Let X be a [omega]-almost weakly orthogonal closed subspace of C(K) where K is a metrizable compact space. Then X has the weak fixed point property.

Theorem 5. Let K be a metrizable compact space. Then, the following conditions are all equivalent:

(1) C(K) is [omega]-almost weakly orthogonal,

(2) C(K) is [omega]-weakly orthogonal,

(3) [K.sup.([omega])] = 0.

Corollary 6. Let K be a compact set with [K.sup.([omega])] = 0. Then C(K) has the weak fixed point property.

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

(A) For each x [member of] X, there exists an element y [member of] X such that [tau](y) = [bar.[tau](x)], for each [tau] [member of] [OMEGA](X).

It can be seen that each [C.sup.*]-algebra satisfies condition (A).

In 2010, W. Fupinwong and S. Dhompongsa [5] proved that each infinite dimensional unital Abelian real Banach algebra X with [OMEGA](X) [not equal to] 0 satisfying (i) if x, y [member of] X is such that [absolute value of [tau](x)] [less than or equal to] [absolute value of [tau](y)] for each [tau] [member of] [OMEGA](X) then [parallel]x[parallel] [less than or equal to] [parallel]y[parallel] and (ii) inf {r(x) : x [member of] X, [parallel]x[parallel] = 1} > 0 does not have the fixed point property. Moreover, they proved the following theorem.

Theorem 7. Let X be an infinite dimensional unital Abelian complex Banach algebra satisfying condition (A) and each of the following:

(i) If x, y [member of] X is such that [absolute value of [tau](x)] [less than or equal to] [absolute value of [tau](y)] for each [tau] [member of] [OMEGA](X), then [parallel]x[parallel] [less than or equal to] [parallel]y[parallel].

(ii) inf{r(x) : x [member of] X, [parallel]x[parallel] = 1} > 0.

Then X 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 C([OMEGA]), where [OMEGA] 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 [C.sup.*]-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 [OMEGA] be a locally compact Hausdorff space. Then the following statements are equivalent:

(i) [OMEGA] is infinite set.

(ii) [C.sub.0]([OMEGA]) is infinite dimensional.

(iii) [C.sub.0]([OMEGA]) 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 X with [OMEGA](X) [not equal to] 0 satisfying (i) if x, y [member of] X is such that [absolute value of [tau](x)] [less than or equal to] [absolute value of [tau](y)] for each [tau] [member of] [OMEGA](X) then [parallel]x[parallel] [less than or equal to] [parallel]y[parallel] and (ii) inf{r(x) : x [member of] X, [parallel]x[parallel] = 1} > 0 does not have the fixed point property.

In this paper, let X be an infinite dimensional unital Abelian complex Banach algebra satisfying (i) condition (A), (ii) if x, y [member of] X is such that [absolute value of t(x)] [less than or equal to] [absolute value of [tau](y)], for each [tau] [member of] [OMEGA](X), then [parallel]x[parallel] [less than or equal to] [parallel]y[parallel], and (iii) inf {r(x) : x [member of] X, [parallel]x[parallel] = 1} > 0. We prove that there exists an element [x.sub.0] in X such that

[mathematical expression not reproducible] (1)

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 [x.sub.0] in X with infinite spectrum and [sigma]([x.sub.0]) [subset] R, the Banach algebra [mathematical expression not reproducible] generated by [x.sub.0] does not have the fixed point property.

2. Preliminaries

Let F be the field R or C. Let X be a Banach space over F. We say that a mapping T : E [subset] X [right arrow] X is nonexpansive if

[parallel]Tx - Ty[parallel] [less than or equal to] [parallel]x - y[parallel] (2)

for each x, y [member of] E, where E is a nonempty subset of X. A Banach space X over F is said to have the fixed point property if for each nonexpansive mapping T : E [right arrow] E on a nonempty bounded closed convex subset E of X has a fixed point.

We define the spectrum of an element x of a unital Banach algebra X over F to be the set

[sigma](x) = [[sigma].sub.X](x) = {[lambda] [member of] F : [lambda]1 - x [not member of] Inv (X)}, (3)

where Inv(X) is the set of all invertible elements in X.

The spectral radius of x is defined to be

[mathematical expression not reproducible]. (4)

We say that a mapping [tau] : X [right arrow] F is a character on an algebra X over F if [tau] is a nonzero homomorphism. We denote by [OMEGA](X) the set of all characters on X. If X is a unital Abelian Banach algebra over F, it is known that [OMEGA](X) is compact.

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

(A) For each x [member of] X, there exists an element y [member of] X such that [tau](y) = [bar.[tau](x)], for each [tau] [member of] [OMEGA](X).

We denote by [C.sub.F](S) the unital Banach algebra of continuous functions from a topological space S to F 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 [C.sub.R](S).

Theorem 9. Let A be a subalgebra of [C.sub.R](S) satisfying the following conditions:

(i) A separates the points of S.

(ii) A annihilates no point of S.

Then A is dense in [C.sub.R](S).

Let X be an Abelian Banach algebra over F. The Gelfand representation [phi] : X [right arrow] [C.sub.F]([OMEGA](X)) is defined by x [right arrow] [??], where [??] is defined by

[??]([tau]) = [tau](x), (5)

for each [tau] [member of] [OMEGA](X). If X is unital and Abelian, then [sigma](x) = {[tau](x) : [tau] [member of] [OMEGA](X)}, for each x [member of] X. It is known that r(x) = [[parallel][??][parallel].sub.[infinity],X] if X is Abelian, where

[mathematical expression not reproducible]. (6)

The Jacobson radical J(X) of a Banach algebra X over F is the intersection of all regular maximal left ideals of X. It is known that if X is a unital complex Banach algebra and x [member of] J(X) then the spectral radius r(x) of x is equal to zero. A Banach algebra X over F is said to be semisimple if J(X) = {0}.

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 [mathematical expression not reproducible].

Lemma 10. Let X be a complex unital Banach algebra satisfying

inf {[[parallel][??][parallel].sub.[infinity],X]: x [member of] X, [parallel]x[parallel] = 1} > 0. (7)

Then

(i) inf{r(x) : x [member of] X, [parallel]x[parallel] = 1} > 0.

(ii) X is semisimple.

Proof. Let X be a complex unital Banach algebra satisfying

[mathematical expression not reproducible]. (8)

(i) From {[tau](x) : [tau] [member of] [OMEGA](X)} [subset] [sigma](x), for each x [member of] X, it follows that [[parallel][??][parallel].sub.[infinity],X] [less than or equal to] r(x), for each x [member of] X. Therefore,

[mathematical expression not reproducible]. (9)

(ii) From (i), we have

inf {r(x) : x [member of] X, [parallel]x[parallel] = 1} > 0, (10)

and hence, for each x [member of] X, r(x) = 0 implies x = 0. Since r(x) = 0, for each x [member of] J(X), J(X) = {0}. Therefore, X is semisimple.

The following lemma was proved in [11].

Lemma 11. Let X 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 X be an infinite dimensional complex unital Abelian Banach algebra with condition (A) and satisfy

inf {r(x) : x [member of] X, [parallel]x[parallel] = 1} > 0. (11)

Then there exists [x.sub.0] [member of] X with infinite spectrum and [tau]([x.sub.0]) [member of] R, for each [tau] [member of] [OMEGA](X).

Proof. Since X is Abelian,

[mathematical expression not reproducible]. (12)

It follows from Lemma 10 that X is semisimple. From Lemma 11, there exists an element x in X with infinite spectrum. From condition (A), there exists y [member of] X such that

[tau](y) = [bar.[tau](x)], (13)

for each t [member of] [OMEGA](X). Hence

[tau] (xy) = [tau] (x) [tau] (y) = [tau] (x) [bar.[tau](x)] [member of] R, (14)

for each [tau] [member of] [OMEGA](X).

Lemma 13. Let X be an infinite dimensional complex unital Banach algebra, and let [x.sub.0] be an element in X with infinite spectrum. Then {[x.sub.0]: n [member of] N} is linearly independent.

Proof. Assume that

[k.summation over (i=1)] [[alpha].sub.i][x.sup.i.sub.0] = 0, (15)

where [[alpha].sub.i] [member of] C. Let

{[[lambda].sub.1], [[lambda].sub.2], [[lambda].sub.3], ...} [subset or equal to] [sigma] ([x.sub.0]) (16)

with [[lambda].sub.i] [not equal to] [[lambda].sub.j] for each i [not equal to] j.

From [mathematical expression not reproducible], it follows that

[mathematical expression not reproducible], (17)

for each j [member of] N. Hence [[alpha].sub.i] = 0, for each i [member of] {1, 2, 3, ..., k}.

Lemma 14. Let X be an infinite dimensional complex unital Abelian Banach algebra satisfying condition (A), and let [x.sub.0] be an element in X with infinite spectrum and [tau]([x.sub.0]) [member of] R, for each t [member of] [OMEGA](X). Define

[mathematical expression not reproducible]. (18)

Then Z is an infinite dimensional real unital Abelian Banach algebra with [OMEGA](Z) [not equal to] 0.

Proof. From Lemma 13, {[x.sup.n.sub.0]: n [member of] N} is linearly independent in X, so Z is infinite dimensional. To show that [OMEGA](Z) = 0. Let [tau] [member of] [OMEGA](Z), and define [omega] : Z [right arrow] R by

x [right arrow] [tau](x). (19)

Indeed, [omega] is real-valued since [tau]([x.sub.0]) [member of] R, for each [tau] [member of] [OMEGA](Z). Obviously, [omega] is a nonzero homomorphism on Z. So [OMEGA](Z) [not equal to] 0.

Similarly, one can prove the following lemma.

Lemma 15. Let X be an infinite dimensional complex unital Banach algebra satisfying condition (A), and let [x.sub.0] be an element in X with infinite spectrum and [tau]([x.sub.0]) [member of] R, for each [tau] [member of] [OMEGA](X). Define

[mathematical expression not reproducible]. (2)

Then [<[x.sub.0]>.sub.R] is an infinite dimensional real nonunital Abelian Banach algebra.

Some useful properties of the real unital Abelian Banach algebra Z is shown in the following lemma.

Lemma 16. Let X be an infinite dimensional complex unital Abelian Banach algebra satisfying condition (A), and let [x.sub.0] be an element in X with infinite spectrum and [tau]([x.sub.0]) [member of] R, for each [tau] [member of] [OMEGA](X). Define

[mathematical expression not reproducible]. (21)

If X satisfies

[mathematical expression not reproducible], (22)

then Z is a real unital Abelian Banach algebra satisfying the following conditions:

(i) The Gelfand representation [phi] from Z into [C.sub.R]([OMEGA](Z)) is a bounded isomorphism.

(ii) The inverse [[phi].sup.-1] is also a bounded isomorphism.

Proof. (i) It follows that ker([phi]) = {0}. So [phi] is injective. We have [phi](Z) is a subalgebra of [C.sub.R]([OMEGA](Z)) separating the points of [OMEGA](Z) and annihilating no point of [OMEGA](Z). Moreover, [phi](Z) is complete, so [phi](Z) is closed. In fact, if {[[??].sub.n]} is a Cauchy sequence in [phi](Z), assume to the contrary that {[z.sub.n]} is not Cauchy. So there exist [[epsilon].sub.0] > 0 and subsequences {[z'.sub.n]} and {[z".sub.n]} of {[z.sub.n]} such that

[mathematical expression not reproducible], (23)

for each n [member of] N. Let [y.sub.n] = ([z'.sub.n] - [z".sub.n])/[[epsilon].sub.0]. So [parallel][y.sub.n][parallel] [greater than or equal to] 1, for each n [member of] N. Since {[mathematical expression not reproducible]} is Cauchy, [mathematical expression not reproducible]. Hence

[mathematical expression not reproducible], (24)

which is a contradiction. So we conclude that {[z.sub.n]} is a Cauchy sequence. Then {[z.sub.n]} is a convergent sequence in Z, say lim [z.sub.n] = [z.sub.0] [member of] Z. Therefore,

[mathematical expression not reproducible], (25)

since for each n [member of] N,

[mathematical expression not reproducible]. (26)

So [phi](Z) is complete. It follows from the Stone-Weierstrass theorem that [phi] is surjective.

(ii) is a consequence of the open mapping theorem.

Lemma 17. Let X be an infinite dimensional complex unital Abelian Banach algebra satisfying condition (A), and let [x.sub.0] be an element in X with infinite spectrum and [tau]([x.sub.0]) [member of] R, for each [tau] [member of] [OMEGA](X). Define

[mathematical expression not reproducible]. (27)

If there exists an element x in [<[x.sub.0]>.sub.R] with infinite spectrum [sigma](x) and [sigma](x) [subset] R, then there exists y [member of] [<[x.sub.0]>.sub.R] satisfying the following conditions:

(i) 1 [member of] [sigma](y) [subset] [0, 1].

(ii) There exists a strictly decreasing sequence in [sigma] (y).

Proof. Let X be an infinite dimensional complex unital Banach algebra. Assume that there exists an element x in [<[x.sub.0]>.sub.R] with infinite spectrum [[sigma].sub.(x)] and [[sigma].sub.(x)] [subset] R.

Since Z is unital and Abelian, the spectrum of x is [??]([OMEGA](Z)). Hence

[mathematical expression not reproducible]. (28)

Let {[a.sub.n]} be an infinite sequence in [sigma]([x.sup.2]/r([x.sup.2])). We may assume that {[a.sub.n]} is strictly increasing and [a.sub.1] > 0.

Define a continuous function g : [0, 1] [right arrow] [0, 1] by

[mathematical expression not reproducible]. (29)

So g is joining the points (0, 0) and ([a.sub.1], 1), and g(1) = 0. Let

[mathematical expression not reproducible]. (30)

It follows from Lemma 16 that z [member of] Z. Since g(0) = 0, z [member of] [<[x.sub.0]>.sub.R]. We have that {g([a.sub.n])} is a strictly decreasing sequence in [sigma](z). Moreover, 1 = g([a.sub.1]) [member of] [sigma](z) [subset] [0, 1].

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

Lemma 18. Let X be an infinite dimensional complex unital Abelian Banach algebra satisfying condition (A) and

[mathematical expression not reproducible], (31)

and let [x.sub.0] be an element in X with infinite spectrum and [tau]([x.sub.0]) [member of] R, for each [tau] [member of] [OMEGA](Z). Define

[mathematical expression not reproducible]. (32)

Then there exists a sequence {[z.sub.n]} in [<[x.sub.0]>.sub.R] such that {[tau]([z.sub.n]) : [tau] [member of] [OMEGA](Z)} [subset] [0, 1], for each n [member of] N, and [mathematical expression not reproducible] is a sequence of nonempty pairwise disjoint subsets of [OMEGA](Z).

Proof. Obviously,

[mathematical expression not reproducible], (33)

since inf{[[parallel][??][parallel].sub.[infinity],X] : x [member of] X, [parallel]x[parallel] = 1} > 0. From Lemma 14 and Lemma 2.10 (iii) in [5], there exists [mathematical expression not reproducible] such that {[tau]([z.sub.0]) : [tau] [member of] [OMEGA](Z)} is infinite. We have [mathematical expression not reproducible] such that [sigma]([z.sub.1]) is infinite. From Lemma 17, we may assume without generality that [z.sub.1] satisfies

1 [member of] [sigma] ([z.sub.1]) [subset] [0, 1] (34)

and there exists a strictly decreasing sequence of real number in [sigma]([z.sub.1]), say {[a.sub.n]}. Moreover, we may assume that [a.sub.1] < 1.

Define a continuous function [g.sub.1] : [0, 1] [right arrow] [0, 1] by

[mathematical expression not reproducible]. (35)

So [g.sub.1] is joining the points (0, 0) and ([a.sub.1], 1), and [g.sub.1] (1) [member of] ([g.sub.1]([a.sub.2]), 1).

Let [mathematical expression not reproducible], and define a continuous function [g.sub.2] : [0, 1] [right arrow] [0, 1] by

[mathematical expression not reproducible]. (36)

So [g.sub.2] is joining the points (0, 0) and ([g.sub.1]([a.sub.2]), 1), and [g.sub.2](1) [member of] ([g.sub.2]([g.sub.1]([a.sub.3])), 1).

Let [mathematical expression not reproducible]. Continuing in this manner, we get a sequence of points {[z.sub.n]} in Z with 1 [member of] {[tau]([z.sub.n]) : [tau] [member of] [OMEGA](Z)} [subset] [0, 1], for each n [member of] N, and {[mathematical expression not reproducible]} is a sequence of nonempty pairwise disjoint subsets of [OMEGA](Z).

Moreover, [z.sub.n] [member of] [<[x.sub.0]>.sub.R], for each n [member of] N, since [g.sub.n](0) = 0, for each n [member of] N.

Lemma 19. Let X be an infinite dimensional complex unital Abelian Banach algebra satisfying condition (A) and

[mathematical expression not reproducible], (37)

and let [x.sub.0] be an element in X with infinite spectrum and [tau]([x.sub.0]) [member of] R, for each [tau] [member of] [OMEGA](X). Define

[mathematical expression not reproducible]. (38)

Assume that there exists a bounded sequence {[y.sub.n]} in Z which contains no convergent subsequences such that {[tau]([y.sub.n]) : [tau] [member of] [OMEGA](Z)} is finite for each n [member of] N. Then there exists an element [z.sub.0] [member of] [<[x.sub.0]>.sub.R] such that [[tau]([z.sub.0]) : [tau] [member of] [OMEGA](Z)} is equal to {0, 1, 1/2, 2/3, 3/4, ...} or {0, 1, 1/2, 1/3, 1/4, ...}.

Proof. It follows form Lemma 14 and Lemma 16 that Z is an infinite dimensional real unital Abelian Banach algebra with [OMEGA](Z) [not equal to] 0 and homeomorphic to [C.sub.R]([OMEGA](Z)). Assume that there exists a bounded sequence {[y.sub.n]} in Z which contains no convergent subsequences and such that {[tau]([y.sub.n]) : [tau] [member of] [OMEGA](Z)} is finite for each n [member of] N. From the proof of Lemma 2.10 (ii) in [5], we have

[OMEGA] (Z) = ([[union].sub.n[member of]N][G.sub.n]) [union] F, (39)

where F is a closed set in [OMEGA](Z), [G.sub.n] is closed and open for each n [member of] N, and {F, [G.sub.1], [G.sub.2], ...} is a partition of [OMEGA](Z). Define [[tau].sub.Z] : Z [right arrow] R by

[mathematical expression not reproducible], (40)

for each [[summation].sup.k.sub.i=0] [[alpha].sub.i][x.sup.i.sub.0] [member of] Z. So [[tau].sub.Z] is a character on Z. There are two cases to be considered. If [[tau].sub.Z] is in F, define [psi] : [OMEGA](Z) [right arrow] R by

[mathematical expression not reproducible]. (41)

If [[tau].sub.Z] is in [mathematical expression not reproducible], for some [n.sub.0] [member of] N, we may assume without loss of generality that [n.sub.0] = 1, and define [psi] : [OMEGA](Z) [right arrow] R by

[mathematical expression not reproducible]. (42)

For each case, the inverse image of each closed set in [psi]([OMEGA](Z)) is closed, so [psi] [member of] [C.sub.R]([OMEGA](Z)). Let [phi] : Z [right arrow] [C.sub.R]([OMEGA](Z)) be the Gelfand representation. Therefore, [[phi].sup.-1]([psi]) is an element in Z, say [z.sub.0], such that {[tau]([z.sub.0]) : [tau] [member of] [OMEGA](Z)} is equal to {0, 1, 1/2, 2/3, 3/4, ...} or {0, 1, 1/2, 1/3, 1/4, ...}. Moreover, [z.sub.0] [member of] [<[x.sub.0]>.sub.R] since [[tau].sub.Z]([z.sub.0]) = [psi]([[tau].sub.Z]) = 0.

Lemma 20. Let X be an infinite dimensional complex unital Abelian Banach algebra satisfying condition (A) and

[mathematical expression not reproducible], (43)

and let [x.sub.0] be an element in X with infinite spectrum and [tau]([x.sub.0]) [member of] R, for each [tau] [member of] [OMEGA](X). Define

[mathematical expression not reproducible]. (44)

and let x [member of] [<[x.sub.0]>.sub.R] with [([??]).sup.-1] {1} [not equal to] 0, and 0 [less than or equal to] [tau](x) [less than or equal to] 1, for each [tau] [member of] [OMEGA](Z). Define

[mathematical expression not reproducible], (45)

where A=[([??]).sup.-1] {1}, and define T : E [right arrow] E by

Z [right arrow] xz. (46)

Assume that X satisfies the following condition.

If x, y [member of] X is such that [absolute value of [tau](x)] [less than or equal to] [absolute value of [tau](y)], for each r [member of] [OMEGA](X), then [parallel]x[parallel] [less than or equal to] [parallel]y[parallel].

Then E is a nonempty bounded closed convex subset of [<[x.sub.0]>.sub.R] and T : E [right arrow] E is a nonexpansive mapping.

Proof. Obviously, E is closed and convex. E is nonempty since x [member of] E

Let z [member of] E. Hence

[mathematical expression not reproducible]. (47)

Therefore, E is bounded.

Let [omega] [member of] [OMEGA](X), and let z, z' [member of] E. Define [tau] : Z [right arrow] R by

[mathematical expression not reproducible], (48)

for each [[summation].sup.k.sub.i=0] [[alpha].sub.i][x.sup.i.sub.0] [member of] Z. So [tau] [member of] [OMEGA](Z).

Then

[mathematical expression not reproducible]. (49)

From (i), we have

[parallel]T(z)-T(z')[parallel] [less than or equal to] [parallel]z - z'[parallel]. (50)

So T is nonexpansive.

4. Main Result

Theorem 21. Let X be an infinite dimensional complex unital Abelian Banach algebra satisfying condition (A) and the following conditions:

(i) If x, y [member of] X is such that [absolute value of [tau](x)] [less than or equal to] [absolute value of [tau](y)], for each [tau] [member of] [OMEGA](X), then [parallel]x[parallel] [less than or equal to] [parallel]y[parallel],

(ii) inf{r(x) : x [member of] X, [parallel]x[parallel] = 1} > 0.

Then there exists an element [x.sub.0] in X such that

[mathematical expression not reproducible] (51)

does not have the fixed point property.

Proof. Let X 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 [x.sub.0] in X with infinite spectrum and [tau]([x.sub.0]) [member of] R, for each [tau] [member of] [OMEGA](X). Let

[mathematical expression not reproducible]. (52)

From Lemma 14, Z is an infinite dimensional real unital Abelian Banach algebra with [OMEGA](Z) [not equal to] 0. From Lemma 18, it follows that there is a sequence {[z.sub.n]} in [<[x.sub.0]>.sub.R] such that {[tau]([z.sub.n]) : [tau] [member of] [OMEGA](Z)} [subset] [0, 1] for each n [member of] N, and [mathematical expression not reproducible], ... are nonempty pairwise disjoint.

Write [mathematical expression not reproducible], define

[mathematical expression not reproducible], (53)

and define [T.sub.n] : [E.sub.n] [right arrow] [E.sub.n] by

z [right arrow] [z.sub.n]z. (54)

From Lemma 20, [E.sub.n] is a nonempty bounded closed convex subset in [<[x.sub.0]>.sub.R] and [T.sub.n] is nonexpansive for each n [member of] N.

Assume to the contrary that [<[x.sub.0]>.sub.R] has the fixed point property. For each n [member of] N, since [E.sub.n] is a nonempty bounded closed convex subset in [<[x.sub.0]>.sub.R], [T.sub.n] has a fixed point in [E.sub.n], say [y.sub.n]. Since [y.sub.n] is a fixed point of [T.sub.n], [y.sub.n] = [z.sub.n][y.sub.n]. Then [mathematical expression not reproducible], and then

[mathematical expression not reproducible], (55)

for each n [member of] N. Since [A.sub.1], [A.sub.2], [A.sub.3], ... are pairwise disjoint, [mathematical expression not reproducible]. Thus {[??]} has no convergent subsequences. Since Z and [C.sub.R]([OMEGA](Z)) are homeomorphic, {[y.sub.n]} has no convergent subsequences. From Lemma 19, there exists an element [z.sub.0] in [<[x.sub.0]>.sub.R] such that {[tau]([z.sub.0]) : [tau] [member of] [OMEGA](Z)} is equal to {0, 1, 1/2, 2/3, 3/4, ...} or {0, 1, 1/2, 1/3, 1/4, ...}.

Write [mathematical expression not reproducible], define

[mathematical expression not reproducible], (56)

and define [T.sub.0] : [E.sub.0] [right arrow] [E.sub.0] by

z [right arrow] [z.sub.0]z. (57)

It follows from Lemma 20 that [T.sub.0] is a nonexpansive mapping on a nonempty bounded closed convex subset [E.sub.0] in [<[x.sub.0]>.sub.R]. So [T.sub.0] has a fixed point in [E.sub.0], say [y.sub.0]. There are two cases to be considered.

Case 1 ({t([z.sub.0]) : [tau] [member of] [OMEGA](Z)} = {0, 1, 1/2, 2/3, 3/4, ...}). Hence [mathematical expression not reproducible]. Then

[mathematical expression not reproducible]. (58)

So

[mathematical expression not reproducible]. (59)

and

[mathematical expression not reproducible]. (60)

It follows from

[mathematical expression not reproducible] (61)

that [mathematical expression not reproducible] is a nonempty pairwise disjoint open covering of the compact set [OMEGA] (Z)\[A.sub.0], which is a contradiction.

Case 2 ({[tau]([z.sub.0]) : [tau] [member of] [OMEGA](Z)} = |0, 1, 1/2, 1/3, 1/4, ...}).

[mathematical expression not reproducible], (62)

where [mathematical expression not reproducible].

It can be seen that E is a nonempty bounded closed convex subset of Z.

Define T : E [right arrow] E by

Z + 1 [right arrow] (-[z.sub.0] + 1)(z + 1), (63)

for each z + 1 [member of] E. We have

[mathematical expression not reproducible]. (64)

Define S : Z [right arrow] Z by

[mathematical expression not reproducible]. (65)

It follows from (i) that STS : S(E) [right arrow] S(E) is a nonexpansive mapping on a nonempty bounded closed convex subset S(E) of [<[x.sub.0]>.sub.R].

STS has a fixed point since [<[x.sub.0]>.sub.R] has the fixed point property. It follows that T has a fixed point, say [y.sub.0] + 1. Then

[mathematical expression not reproducible]. (66)

So

[mathematical expression not reproducible]. (67)

and

[mathematical expression not reproducible]. (68)

It follows from

[mathematical expression not reproducible], (69)

that [mathematical expression not reproducible] is a nonempty pairwise disjoint open covering of the compact set [OMEGA](Z) \ A, which is a contradiction.

So we conclude that [<[x.sub.0]>.sub.R] does not have the fixed point property.

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

Corollary 22. Let X be an infinite dimensional complex unital Abelian Banach algebra satisfying condition (A) and the following conditions:

(i) If x, y [member of] X is such that [absolute value of [tau](x)] [less than or equal to] [absolute value of [tau](y)], for each [tau] [member of] [OMEGA](X), then [parallel]x[parallel] [less than or equal to] [parallel]y[parallel],

(ii) inf{r(x) :x [member of] X, [parallel]x[parallel] = 1} > 0.

If [x.sub.0] is an element in X with infinite spectrum and [sigma](x) [subset] R, then the Banach algebra [mathematical expression not reproducible] generated by [x.sub.0] does not have the fixed point property.

Corollary 23. Let [x.sub.0] be a self-adjoint element in a unital Abelian [C.sup.*]-algebra X, then the algebra [mathematical expression not reproducible] generated by [x.sub.0] does not have the fixed point property.

Finally, we pose an interesting problem.

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

https://doi.org/10.1155/2018/9045790

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.

References

[1] D. E. Alspach, "A fixed point free nonexpansive map," Proceedings of the American Mathematical Society, vol. 82, no. 3, pp. 423-424, 1981.

[2] J. Elton, P.-K. Lin, E. Odell, and S. Szarek, "Remarks on the fixed point problem for nonexpansive maps," in Fixed points and nonexpansive mappings (Cincinnati, Ohio, 1982), vol. 18 of Contemp. Math., pp. 87-120, Amer. Math. Soc., Providence, RI, 1983.

[3] A. T. Lau, P F. Mah, and A. Ulger, "Fixed point property and normal structure for Banach spaces associated to locally compact groups," Proceedings of the American Mathematical Society, vol. 125, no. 7, pp. 2021-2027, 1997.

[4] T. Domnguez Benavides and M. A. Japon Pineda, "Fixed points of nonexpansive mappings in spaces of continuous functions," Proceedings of the American Mathematical Society, vol. 133, no. 10, pp. 3037-3046, 2005.

[5] W. Fupinwong and S. Dhompongsa, "The fixed point property of unital abelian Banach algebras," Fixed Point Theory and Applications, Art. ID 362829, 13 pages, 2010.

[6] D. Alimohammadi and S. Moradi, "On the fixed point property of unital uniformly closed subalgebras of C(X)" Fixed Point Theory and Applications, Art. ID 268450, 9 pages, 2010.

[7] S. Dhompongsa, W. Fupinwong, and W. Lawton, "Fixed point properties of [C.sup.*]-algebra," Journal of Mathematical Analysis and Applications, vol. 374, no. 1, pp. 22-28, 2011.

[8] W. Fupinwong, "Nonexpansive mappings on Abelian Banach algebras and their fixed points," Fixed Point Theory and Applications, 2012.

[9] D. Alimohammadi, "Nonexpansive mappings on complex [C.sup.*]algebras and thier fixed points," Int. J. Nonlinear Anal. Appl, vol. 7, pp. 21-29, 2016.

[10] J. Daengsaen and W. Fupinwong, "Fixed points of nonexpansive mappings on real Abelian Banach algebras," in Proceedings of the 22nd Annual Meeting in Mathematics (AMM), 6 pages, Thailand, 2017.

[11] I. Kaplansky, "Ring isomorphisms of Banach algebras," Canadian Journal of Mathematics. Journal Canadien de Mathematiques, vol. 6, pp. 374-381, 1954.

P. Thongin (1) and W. Fupinwong (iD) (2)

(1) PhD Degree Program in Mathematics, Faculty of Science, ChiangMai University, ChiangMai 50200, Thailand

(2) Center of Excellence in Mathematics and Applied Mathematics, Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand

Correspondence should be addressed to W. Fupinwong; g4865050@hotmail.com

Received 14 January 2018; Accepted 16 May 2018; Published 10 June 2018

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Title Annotation:Research Article
Author:Thongin, P.; Fupinwong, W.
Publication:Journal of Function Spaces
Date:Jan 1, 2018
Words:5546
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