Abstract and Applied Analysis
Volume 2012, Article ID 729745, 11 pages
http://dx.doi.org/10.1155/2012/729745
Research Article

A Banach Algebraic Approach to the Borsuk-Ulam Theorem

Faculty of Mathematics and Computer Science, Damghan University, Damghan, Iran

Received 1 October 2011; Accepted 12 December 2011

Academic Editor: David Perez-Garcia

Copyright © 2012 Ali Taghavi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Using methods from the theory of commutative graded Banach algebras, we obtain a generalization of the two-dimensional Borsuk-Ulam theorem as follows. Let be a homeomorphism of order , and let be an th root of the unity, then, for every complex valued continuous function on , the function must vanish at some point of . We also discuss some noncommutative versions of the Borsuk-Ulam theorem.

“To my Children Tarannom and Pouya”

1. Introduction

The classical Borsuk-Ulam theorem states that for every continuous function , there always exist a point such that . If we define , we obtain an equivalent statement as follows. For every odd continuous function , there exists a point such that .

We consider the case and identify with the complex numbers . Let be the Banach algebra of all continuous complex-valued functions on with the -graded structure: where is the space of all even functions and is the space of all odd functions, and the decomposition is the standard decomposition of functions to even and odd functions. Then the two-dimensional Borsuk-Ulam theorem says that a homogenous element of with nonzero degree, namely, an odd function, is not invertible.

In this paper we are mainly interested in invertible elements of a graded unital Banach algebra which are homogenous of nontrivial degree. Some natural questions about such elements are as follows. If any such element is invertible, can it be connected to the identity in the space of invertible elements? What can be said about the relative position of their spectrum with respect to the origin?

As we will see in the main theorem of this paper, for a commutative Banach algebra without nontrivial idempotent, which is graded by a finite Abelian group, a nontrivial homogenous element cannot be connected to the identity. On the other hand, using an standard lifting lemma in the theory of covering spaces, we conclude that an invertible element of has a logarithm, then it lies in the same connected component of the identity. This shows that an odd element of cannot be invertible, so it would give us a Banach algebraic proof of the Borsuk-Ulam theorem, in dimension two. The purpose of this paper is to translate the classical Borsuk-Ulam theorem into the language of noncommutative geometry.

We also give a concrete example of an -graded structure for , the reduced algebra of the free group on two generators, such that a nontrivial homogenous element lies in the same component of the identity. This would show that the commutativity of grading group and graded algebra are necessary conditions in our main theorem. Finally we give a question as a weak version of the Kaplansky-Kadison conjecture. This question naturally arises from our main result; see Question 5 at the end of the paper.

2. Preliminaries

Let be a unital complex Banach algebra, and let be a finite group with neutral element . A G-graded structure for is a decomposition where each is a Banach subspace of and . An element is called a homogenous element; it is called nontrivial homogenous if where . When and , an element of is called an odd element. A morphism is called a graded morphism provided that for all . Let be a -graded Banach algebra, and let be a normal subgroup of . Then there is an induced -graded structure for with The following proposition is used in the proof of our main theorem.

Proposition 2.1. Let be Abelian, and let be a nontrivial homogenous element of a -graded Banach algebra, then for some positive integer , there is a -graded structure for such that is a nontrivial homogenous element of as a -graded algebra.

Proof. This can be proved by induction on order of as follows. The first step of the induction is obvious since the only group of order 2 is . Let be a finite Abelian group and where . If is not a cyclic group, then there is a subgroup which does not contain , so is a nontrivial homogenous element of the induced -graded structure for . Now the order of is strictly less than the order of . So an induction argument on order of completes the proof.

Note that the existence of a -graded structure for a Banach algebra is equivalent to existence of a bounded multiplicative operator with . For any such operator, we choose a root of unity and observe that the decomposition is a -graded structure. Conversely for the grading , the following multiplicative operator satisfies

Example 2.2. Let be a compact topological space and be an orderhomeomorphism of , that is . Define with , where is the space of all continuous functions on . Then is a bounded multiplicative operator on Banach algebra with , so we would havea-graded structure for . As a particular example, the rotation of circle by is an order homeomorphism of circle. Then we naturally obtain a -graded structure for .

For a group , a -partition for is a partition of into disjoint subsets , such that . This is equivalent to say that is a normal subgroup of whose quotient is isomorphic to . This is an obvious consequence of the normal property for subgroups; see . So every group which has a normal subgroup such that is isomorphic to posses a -graded structure. For example, for every group , the group has as a normal subgroup, and is isomorphic to . In this case the partition for is .

For a discrete group , we denote by the group algebra of with complex coefficients, namely, the space of all linear combinations where ’s are complex numbers. let be the Hilbert space of all such that . acts on with . This action defines a unitary representation of on . We extend this action by linearity to . So each element of can be considered as an element of , the space of all bounded operators on , and every element of can be considered as a unitary operator on . The reduced algebra of , , is the closure of , with respect to operator norm defined on .

In Proposition 2.1, we shall prove that a -partition for a Group gives a -graded structure for . For the proof of the Proposition 2.1, we need to the following technical lemma.

Lemma 2.3. Let be a Banach space with a dense linear subspace F. Assume is a direct sum of its linear subspaces ’s. Suppose that the original norm of is equivalent to the direct limit norm . Then .

Proof of Lemma. Let where is in , the closure of . Then there are sequences of elements of which converges to . So . Since the original norm of is equivalent to the direct sum norm, we conclude that each sequence converges to zero. So for all . From continuity of norm, we have the equivalency of norm and direct sum norm on the space . So this direct sum is a topological direct sum. let be given, there is a sequence , , which converges to . This shows that each sequence is a cauchy sequence which converges to an element . So . This completes the proof of lemma.

Proposition 2.4. A -partition structure for a group gives a -graded structure for .

Proof. Assume that is a -partition for , then and where For and , we have belongs to . Assume and . Then Let and be given. We apply the definition of norm operator on operator , then we conclude that there is a with such that
This shows that there is a such that
Since , so , for . We conclude that the operator norm on is equivalent to direct sum norm . Now if we put , , and apply the above lemma to these spaces, the proof of the proposition would be completed.

Remark 2.5. Let and be two groups where is finite. Similar to above we can define an -partition for . It is a decomposition such that .This is equivalent to say that is a normal subgroup of whose quotient is isomorphic to . In the same manner as above we can prove that an -partition for a group , gives an -graded structure for . Moreover it is clear from the definition that an element which is not in , can be considered as a nontrivial homogenous element of -graded algebra .

A part of the philosophy of noncommutative geometry is to translate the classical facts about compact topological spaces into language of (noncommutative) Banach or algebras; see  or . According to the Gelfand-Naimark theorem, there is a natural contravariant functor from the category of compact Hausdorff topological space to the category of unital complex algebras. This functor assigns to the commutative algebra of all complex-valued continuous functions on . It also assigns to a continuous function , the algebra morphism by . Conversely every unital commutative algebra is isomorphic to for some compact topological space , and every morphism from to is equal to for some continuous function . In particular constant maps from to correspond to morphism from to with one-dimensional range. Most of the statements about topological spaces have an algebraic analogy in the world of algebras. For example it can be easily shown that a topological space is connected if and only if the algebra has no nontrivial idempotent, where a nontrivial idempotent in an algebra is an element such that and . So nonexistence of nontrivial idempotent for a (noncommutative) Banach algebra is interpreted as “noncommutative connectedness.”

For a topological space , considering the above functor, it is natural to identify , with , where is the unit interval. So in order to obtain a homotopy theory, for a Banach algebra , we define the Banach algebra as follows.

Definition 2.6. Let be a Banach algebra, we denote by , the Banach algebra of all continuous with the standard operations and norm. We define ; with , for .

We say that two morphisms are homotopic if there is a morphism such that , where . A morphism is called null homotopic if it is homotopic to a morphism with one-dimensional range. Obviously this is a natural Banach algebraic analogy of classical null homotopicity.

For a unital Banach algebra with unit element 1, the element is simply shown by where is an scalar. For an element , we denote for all such that is not invertible. The spectral radius of , which is denoted by , is defined as the infimum of all such that the disc of radius around the origin contains .

3. Main Result

In the next theorem, which is the main result of this paper, is an arbitrary positive integer, is a finite Abelian group, and is a unital complex Banach algebra.

Main Theorem 1. Let be G-graded Banach algebra with no nontrivial idempotents. Let be a nontrivial homogenous element. Then 0 belongs to the convex hull of the spectrum .
Further, if is commutative and a is invertible, then and 1 do not lie in the same connected component of the space of invertible elements .

Proof. Without lose of generality we assume that . So we have a multiplicative operator with . Let be a nontrivial homogenous element so where is a root of the unity. Assume for contrary that the convex hull of does not contain 0. Then can be included in a disc with center , for some , which is a subset of a branch of logarithm. Using holomorphic functional calculus as described in [4, Chapter 10], we can find a convergent series where where ’s are complex numbers.
So . Thus and are power series in . In particular and , for all and . So we have . Then is a subset of the set of all th roots of unity. On the other hand, since has no nontrivial idempotent, the spectrum of each element must be connected. So we can assume that ; otherwise we multiply with an appropriate root of unity. Put . Then is a quasinilpotent element of ; that is, its spectral radius . We have where is a quasinilpotent too because and note that for commuting elements and ; see [4, page 302]. Moreover, we have where is a quasinilpotent.
On the other hand, , then for some quasinilpotent . Thus we have . We obtain from (3.1) and (3.2) and the latest equation for some quasinilpotent . Then is a single point different from zero, since , so for some quasinilpotent . Then is a quasinilpotent element that can be expanded as a power series in . Note that we emphasize on this power series expansion only for commutativity purpose. We have , so by induction we obtain Since is multiplicative and has a power series expansion in , we conclude that ’s are commuting quasinilpotent elements so their sum is quasinilpotent too; see [4, page 302]. This implies that is quasinilpotent which is a contradiction. This completes the first part of the theorem.
Now assume that is commutative and is a nontrivial homogenous element such that is in the same connected component as the identity. Then there exists an element with . Obviously the same argument as above, but without needing to expansion of b as a power series in a, leads to a contradiction. So the proof is complete.
The following corollaries are immediate consequence of the above theorem.

Corollary 3.1. Let be a compact locally path connected and simply connected space, and let be a homeomorphism of order . Assume that is an th root of the unity. Then, for every continuous function , there is a point such that .

Proof. Let be the commutative Banach algebra of continuous functions with the usual structures. Since is connected, has no nontrivial idempotent. Define the continuous automorphism by . satisfies , so we have a -graded structure for in the form . Put , then , so is a nontrivial homogenous element of . If for all , then is an invertible element of which is not in the same connected component as the identity, by the above theorem. On the other hand, consider the covering space . Since is simply connected and locally path connected, there is a lifting of ; that is, with , using the standard lifting lemma in the theory of covering space; see [5, proposition 1.33]. So is a logarithmic element and must be in the same connected component as the identity. This contradicts to the fact that cannot be connected to the identity. This completes the proof.

Putting , and , we obtain the classical two-dimensional Borsuk Ulam theorem as follows.

Corollary 3.2 (Borsuk-Ulam Theorem). For a continuous function , there must exist a point with .

Example 3.3. Put , and let be a continuous function. Then there is a point such that . To prove this, consider the fourth-order homeomorphism of with . Now apply Corollary 3.1 with .
The following corollary is an obvious consequence of the last part of the main theorem. In this corollary and its sequel, is the connected component of the identity.

Corollary 3.4. Let be an idempotentless commutative Banach algebra which is graded by a finite Abelian group such that a nontrivial homogenous element is invertible. Then is an infinite group.

4. Further Questions and Remarks

In this section we present some questions which naturally arise from the main theorem and the corollaries of the previous sections.

First we discuss about a pure algebraic analogy of Corollary 3.4. For this purpose, we need some elements of stable rank theory and -theory for both Banach algebras and complex algebras. We say that a commutative Banach algebra has topological stable rank one if is dense in ; see . Let be a commutative complex algebra. We say that a pair is invertible if there is a pair such that . We say has Bass stable rank one if, for every invertible pair , there is an element such that is invertible; see . It is mentioned in  that, for a commutative algebra, the Bass stable rank coincide with the topological stable rank.

For a complex Banach algebra , let be the space of all by invertible matrices with entries in . There is a natural topology on this space, and there is a natural embedding of into which sends a matrices to diag(B,1), so we have an inductive system with ’s. Put . Then has a natural inductive limit topology and algebraic structure. Define where is the connected component of the identity. This Abelian group is the standard functor defined on the category of the Banach algebras. For a complex algebra , there is a pure algebraic functor defined as the quotient of by its commutator. On the other hand, it is well known that for an stable rank one Banach algebra , is naturally isomorphic to ; see .

So considering this isomorphism, the equality of the Bass and topological stable rank for commutative algebras, and the above corollary, it is natural to ask the next pure algebraic question.

Question 1. Let be an idempotentless involutive and commutative complex algebra with Bass stable rank one which is graded by a finite Abelian group. Assume that a nontrivial homogenous element of is invertible. Does this imply that is an infinite group?
In the following example, we drop simultaneously the commutativity of both grading group and the graded idempotent less algebra in the main theorem. We observe that in this case the theorem is no longer valid.

Example 4.1. Let be the free group on two generators and . We shall see that there is a normal subgroup of which does not contain and its quotient is isomorphic to , the permutation group on three elements. Assuming the existence of such subgroup , we obtain an -graded structure for for which , as an element of , is a nontrivial homogenous element. On the other hand, it is well known that this algebra has no nontrivial idempotent and , is in the same connected component as the identity; see . To prove the existence of such subgroup , we first note that , the free group on four generators, can be considered as an index three subgroup of which does not contain . This can be proved using certain covering space of figure-8 space as illustrated in [5, page 58]. Let be a covering space with . Then the induced map is an injective map. is isomorphic to whose index, as a subgroup of , is equal to the cardinal of a fibre of the covering space. Moreover, a loop lies in the range of if and only if the unique lifting of with starting point is a loop with base point , see [5, proposition 1.31].

Now consider the 3-fold covering space which is illustrated in the Figure 1. The fundamental group of the total space is , and the fundamental group of the base space is . The loop is not in the range of projecting map of the covering because its lifting with base point ends to a different point .

Put and . Then has a subgroup of index 3 which does not contain a commutator element . We obtain a morphism with the standard action of on the set of left cosets of . Then is a normal subgroup of which is contained in and does not contain . is isomorph to a subgroup of ; on the other hand, is not Abelian since does not contain a commutator element . This shows that is a normal subgroup of in which quotient is isomorphic to and does not contain the commutator element . So the above construction gives an example of a noncommutative Banach algebra , without nontrivial idempotent, which is graded by a non-Abelian finite group , such that a nontrivial homogenous element lies in the same connected component as the identity.

But could we give any such example with a finite Abelian group ? In other words, can we drop the hypothesis of commutativity of Banach algebra from the second part of the main theorem. This is a motivation for the next question which can be considered as a noncommutative analogy of the two-dimensional Borsuk-Ulam theorem.

Question 2. Let be a Banach algebra without nontrivial idempotent which is equipped with a -graded structure, where is a finite Abelian group. Can one prove that the connected component of the identity has null intersection with nontrivial homogenous elements? As a particular case, put , with the -graded structure corresponding to the -partition of to the union of odd and even words. Can a linear combination of odd words be connected to the identity? Is the decomposition of this algebra to even and odd words, the only -graded structure for , up to graded isomorphism?

There is an affirmative answer to a particular case of the second part of the above question. We thank professor Valette for his affirmative answer in this case. This affirmative answer is mentioned in the following proposition.

Proposition 4.2. Every linear combination of two odd words and in cannot be connected to the identity.

Proof. For a contradiction assume that can be connected to the identity in the space of invertible elements, where and are two complex numbers. With a small perturbation, we can assume that . (Note that with this perturbation the connected component does not change.) Now put , so lies in the same connected component as the identity with . Then the curve is a curve which lies in the space of invertible elements of . Because is a unitary element, its spectrum does not contain an element where . So as an odd element lies in the same connected component as the identity. But the only words which can be connected to the identity are members of the commutator subgroup of . This contradiction shows that a linear combination of two odd words can not be connected to the identity.

What is a Banach algebraic formulation of the higher dimensional Borsuk-Ulam theorem? In order to obtain a noncommutative version of this theorem, we restate the classical case as follows.

Let be odd continuous real-valued functions on , then the function is not an invertible element of or equivalently is not in the same connected component as the identity (since every invertible element can be connected to the identity). In fact ’s are self-adjoint elements of which are odd elements of the standard -graded structure of . Now a relevant noncommutative version of this statement can be presented as the next question. So it seems natural to ask that for what type of noncommutative spheres the answer to the next question is affirmative?

Question 3. Assume that is a -graded noncommutative -sphere and are self-adjoint elements of which are odd elements of this graded algebra. Is it true to say that is either non-invertible or is not in the same connected component as the identity?

Remark 4.3. A family of noncommutative spheres is a family of algebras with some relations as a natural generalization of the algebra of continuous functions on -sphere. For more information on noncommutative spheres, see  or .

Another candidate for the noncommutative analogy of the Borsuk-Ulam theorem can be presented as follows.

Consider the equivalent statement of the Borsuk-Ulam theorem which says that an odd continuous map is not null homotopic; namely, it is not homotopic to a constant map. We try to translate this statement into the language of Banach or algebras. The antipodal map defines an order two automorphism ; with which naturally gives a -graded structure for . Similarly an odd map defines a morphism which satisfies . This means that is a graded morphism. So we ask, for what type of noncommutative spheres (spaces), the answer to the next question is affirmative?

Question 4. Let be a noncommutative sphere with a nontrivial -graded structure, and let be a graded morphism. Is it true to say that is not a null homotopic morphism?

Final Remark on the Main Theorem
We explain that the main theorem gives us a weaker version of the Kaplansky-Kadison conjecture. This conjecture says that, for a torsion free group , has no nontrivial idempotent; see . Now as a consequence of our theorem, we present the following question as a weaker version of the Kaplansky-Kadison conjecture.

Question 5 (A Weak Version of the Kaplansky-Kadison Conjecture). Let be a torsion-free group and equipped with a -graded structure. Is it possible that the convex hull of the spectrum of a nontrivial homogenous element does not contain the origin?

Acknowledgment

The author would like to thank the referee for very valuable suggestions.

References

1. I. N. Herstein, Topics in Algebra, Wiley, 1975.
2. A. Connes, “Noncommutative differential geometry,” Institut des Hautes Études Scientifiques, no. 62, pp. 257–360, 1985.
3. J. M. Gracia-Bondía, J. C. Várilly, and H. Figueroa, Elements of Non-Commutative Geometry, Birkhäuser, Boston, Mass, USA, 2001.
4. W. Rudin, Functional Analysis, McGraw-Hill, New York, NY, USA, 2nd edition, 1991.
5. A. Hatcher, Algebraic Topology, Cambridge University Press, 2002.
6. M. A. Rieffel, “Dimension and stable rank in the K-theory of ${C}^{\ast }$-algebras,” Proceedings of the London Mathematical Society, vol. 46, no. 2, pp. 301–333, 1983.
7. H. Bass, “K-theory and stable algebra,” Institut des Hautes Études Scientifiques, no. 22, pp. 5–60, 1964.
8. K. Dykema, U. Haagerup, and M. Rørdam, “The stable rank of some free product ${C}^{\ast }$-algebras,” Duke Mathematical Journal, vol. 90, no. 1, pp. 95–121, 1997.
9. O. Bratteli, G. A. Elliott, D. E. Evans, and A. Kishimoto, “Noncommutative spheres,” International Journal of Mathematics, vol. 2, no. 2, pp. 139–166, 1991.
10. T. Natsume, “Some non-commutative spheres,” in Quantum Field Theory and Noncommutative Geometry, vol. 662 of Lecture Notes in Physics, pp. 57–66, Springer, Berlin, Germany, 2005.
11. A. Valette, Introduction to the Baum-Connes Conjecture, Birkhäuser, Basel, Switzerland, 2002.