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“To my Children Tarannom and Pouya”

Using methods from the theory of commutative graded Banach algebras, we obtain a generalization of the two-dimensional Borsuk-Ulam theorem as follows. Let

The classical Borsuk-Ulam theorem states that for every continuous function

We consider the case

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

We also give a concrete example of an

Let

Let

This can be proved by induction on order of

Note that the existence of a

Let

For a group

For a discrete group

In Proposition

Let

Let

A

Assume that

This shows that there is a

Since

Let

A part of the philosophy of noncommutative geometry is to translate the classical facts about compact topological spaces into language of (noncommutative) Banach or

For a topological space

Let

We say that two morphisms

For a unital Banach algebra

In the next theorem, which is the main result of this paper,

Without lose of generality we assume that

So

On the other hand,

Now assume that

The following corollaries are immediate consequence of the above theorem.

Let

Let

Putting

For a continuous function

Put

The following corollary is an obvious consequence of the last part of the main theorem. In this corollary and its sequel,

Let

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

For a complex Banach algebra

So considering this isomorphism, the equality of the Bass and topological stable rank for commutative

Let

In the following example, we drop simultaneously the commutativity of both grading group

Let

Now consider the 3-fold covering space which is illustrated in the Figure

Put

But could we give any such example with a finite Abelian group

Let

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.

Every linear combination of two odd words

For a contradiction assume that

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

Assume that

A family of noncommutative spheres is a family of

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

Let

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

Let

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