a note on topological divisors of zero and division algebras

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RACSAM DOI 10.1007/s13398-014-0168-4 ORIGINAL PAPER A note on topological divisors of zero and division algebras J. Carlos Marcos · Ángel Rodríguez-Palacios · M. Victoria Velasco Received: 23 September 2013 / Accepted: 11 March 2014 © Springer-Verlag Italia 2014 Abstract We prove that normed unital complex (possibly non-associative) algebras with no non-zero left topological divisor of zero are isomorphic to the field C of complex numbers. We also show the existence of a complete normed unital infinite-dimensional complex algebra with no non-zero two-sided topological divisor of zero. Keywords Non-associative Algebra · Topological divisor of zero · Division algebra Mathematics Subject Classification (2010) 17D99 · 46H70 · 46H99 1 Introduction By an algebra we mean a real or complex vector space A endowed with a bilinear mapping (a, b) ab from A × A to A, which is called the product of A. An algebra is said to be associative (respectively, commutative) if its product is associative (respectively, commuta- tive). An algebra A is said to be unital if there exists a non-zero element 1 A such that a1 = 1a = a, for every a A. Obviously, the element 1 above is unique whenever it exists, and is called the unit of A. Let A be an algebra. For a A, the left multiplication operator associated to a A is defined as the operator L a : A A given by L a (b) := ab, for every b A. Similarly, Partially supported by Junta de Andalucía grants FQM 0199 and FQM 3737, and Project MTM-2009-12067. J. Carlos Marcos I.E.S. Generalife, Calle Huerta del Rasillo 1, 18004 Granada, Spain e-mail: [email protected] A. Rodríguez-Palacios · M. V. Velasco (B ) Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain e-mail: [email protected] A. Rodríguez-Palacios e-mail: [email protected]

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Page 1: A note on topological divisors of zero and division algebras

RACSAMDOI 10.1007/s13398-014-0168-4

ORIGINAL PAPER

A note on topological divisors of zero and divisionalgebras

J. Carlos Marcos · Ángel Rodríguez-Palacios ·M. Victoria Velasco

Received: 23 September 2013 / Accepted: 11 March 2014© Springer-Verlag Italia 2014

Abstract We prove that normed unital complex (possibly non-associative) algebras with nonon-zero left topological divisor of zero are isomorphic to the field C of complex numbers. Wealso show the existence of a complete normed unital infinite-dimensional complex algebrawith no non-zero two-sided topological divisor of zero.

Keywords Non-associative Algebra · Topological divisor of zero · Division algebra

Mathematics Subject Classification (2010) 17D99 · 46H70 · 46H99

1 Introduction

By an algebra we mean a real or complex vector space A endowed with a bilinear mapping(a, b) → ab from A × A to A, which is called the product of A. An algebra is said to beassociative (respectively, commutative) if its product is associative (respectively, commuta-tive). An algebra A is said to be unital if there exists a non-zero element 1 ∈ A such thata1 = 1a = a, for every a ∈ A. Obviously, the element 1 above is unique whenever it exists,and is called the unit of A.

Let A be an algebra. For a ∈ A, the left multiplication operator associated to a ∈ A isdefined as the operator La : A → A given by La(b) := ab, for every b ∈ A. Similarly,

Partially supported by Junta de Andalucía grants FQM 0199 and FQM 3737, and Project MTM-2009-12067.

J. Carlos MarcosI.E.S. Generalife, Calle Huerta del Rasillo 1, 18004 Granada, Spaine-mail: [email protected]

A. Rodríguez-Palacios · M. V. Velasco (B)Departamento de Análisis Matemático, Facultad de Ciencias,Universidad de Granada, 18071 Granada, Spaine-mail: [email protected]

A. Rodríguez-Palaciose-mail: [email protected]

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the right multiplication operator associated to a ∈ A is the operator Ra : A → A given byRa(b) := ba, for every b ∈ A. Note that La and Ra belong to the associative algebra L(A)

of all linear operators on (the vector space of) A.

By a left-division algebra we mean a non-zero algebra such that, for every a ∈ A \ {0},the left multiplication operator La is bijective. Similarly, right-division algebras are defined.Algebras that are both left and right division algebras are called division algebras. The abovenotions are classical in the literature (see for instance [8,9,15,18]). Moreover, they coincidewhen applied to associative algebras. Indeed, we have the following.

Fact 1 ([13, p. 936]) An associative algebra A is a left-division algebra if and only if it isa right-division algebra, if and only if it is a division algebra, if and only if it is a divisionalgebra in the classical meaning that A is unital and every non-zero element in A has aninverse.

We recall that, given a unital associative algebra A, an element a ∈ A is said to be invertibleif there exists an element b ∈ A such that ab = ba = 1. If a is invertible, then the element babove is unique, is called the inverse of a, and is denoted by a−1.

A normed algebra is an algebra A endowed with a norm ‖ · ‖ such that ‖ab‖ ≤ ‖a‖‖b‖,for all a, b ∈ A. If the norm is complete, then we say that A is a complete normed algebra.Note that, up to an equivalent renorming, a normed algebra is an algebra endowed with anorm that makes continuous the product. We point out that Banach algebras in the classicalmeaning of [1,4,11,12,19] are nothing other than complete normed associative algebras.

The celebrated complex) Gelfand–Mazur theorem ([5,10]) reads as follows.

Theorem 2 (Gelfand–Mazur) Normed division associative complex algebras are isomorphicto C.

The following associativity-free Gelfand–Mazur type theorem is due to Kaplansky [9].Although originally formulated without requiring completeness, the completeness is implic-itly assumed (to be sure that the inverse of a bijective left multiplication operator is bounded).

Theorem 3 Complete normed left-division complex algebras are isomorphic to C.

§4 The question (raised for example in the introduction of [13]) whether completeness canbe removed in Theorem 3 remains an open problem, even if “left-division” is strengthenedto “division”.

Let A be a normed algebra. An element a of A is said to be a left (respectively, right)topological divisor of zero in A if there exists a sequence an of norm-one elements of Asatisfying lim aan = 0 (respectively, lim ana = 0). In this way, left (respectively, right)topological divisors of zero in A are nothing other than those elements a of A such that theoperator La (respectively Ra) is not bounded below. Elements of A which are left or right(respectively, left and right) topological divisors of zero are called one-sided topologicaldivisors of zero (respectively, two-sided topological divisors of zero) in A. In the case that Ais commutative, all notions introduced above coincide and give rise the notion of a topologicaldivisor of zero.

After a forerunner due to Šilov [16] for the complete unital case, Kaplansky [7] provedTheorem 5 immediately below. The formulation given here incorporates also a slight refine-ment pointed out in [2].

Theorem 5 (Kaplansky) Let A be a non-zero normed associative complex algebra with nonon-zero two-sided topological divisor of zero. Then A is isomorphic to C.

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A note on topological divisors of zero and division algebras

Now, the reader should keep in mind that, as a consequence of Fact 1 (respectively, of theBanach isomorphism theorem), we have the following.

Fact 6 Let A be a normed division algebra. If A is associative (respectively, complete), thenA has no non-zero one-sided topological divisor of zero.

It follows that Theorem 5 contains the Gelfand–Mazur Theorem 2. This is the reason why,sometimes, Theorem 5 is known as the (complex) Gelfand–Mazur–Kaplansky theorem.

The next example, essentially due to Urbanik and Wright [17] (see also [13]) shows thatTheorem 5 does not survive without associativity, even if the requirement of absence of non-zero two-sided topological divisors of zero is strengthened to that of absence of non-zero lefttopological divisors of zero, nor even if this last requirement is strengthened to Kaplansky’soriginal one of absence of non-zero one-sided topological divisors of zero. To this end, werecall that an absolute-valued algebra is a non-zero real or complex algebra A endowed with anorm ‖ · ‖ satisfying ‖ab‖ = ‖a‖‖b‖ for all a, b ∈ A, and note that absolute-valued algebrashave no non-zero one-sided topological divisor of zero.

Example 1 The real or complex Banach space �p (1 ≤ p < ∞) can be converted into anabsolute-valued algebra, by simply taking an injective mapping φ: N × N → N, by definingthe product of basic vectors ei , e j ∈ �p by the rule ei e j := eφ(i, j), and then by extending theproduct by bilinearity and continuity.

Now that we know that associativity cannot be removed in Theorem 5, even if the require-ment that A has no non-zero two-sided topological divisor of zero is strengthened to the onethat A has no non-zero left topological divisor of zero, our first main result (Theorem 7)asserts that, under such a strengthening of the assumption, Theorem 5 remains true in thenon-associative setting, as soon as we require in addition the existence of some elementa ∈ A such that La has dense range. As relevant consequences, C is the unique normedunital complex algebra with no non-zero left topological divisor of zero (Corollary 1), aswell as the unique normed left-division complex algebra with no non-zero left topologicaldivisor of zero (Corollary 2). We note that, via Fact 6, Corollary 2 contains Kaplansky’sTheorem 3.

Our second main result (Theorem 8) asserts that, even in the complete case, Corollary 1just reviewed does not remain true if the required absence of non-zero left topological divisorsof zero is relaxed to the one of the absence of non-zero two-sided topological divisors ofzero.

The paper concludes with a section (Sect. 3) where the main results are fully discussed,and some applications are indicated.

2 The results

Given a unital associative algebra A over K := R or C, we denote by Inv(A) the set of allinvertible elements in A, and, for a ∈ A, we define the spectrum of a relative to A, σ A(a),by the equality

σ A(a) := {λ ∈ K: a − λ1 /∈ Inv(A)}.Now let X be a vector space over K, denote by L(X) the unital associative algebra of all

linear operators on X, let IX stand for the identity mapping on X , and let T be in L(X). Thenwe have

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σL(X)(T ) = {λ ∈ K: T − λIX is not bijective}.Moreover, if X is in fact a Banach space, if L(X) stands for the complete normed unitalassociative algebra of all bounded linear operators on X , and if the operator T above isbounded, then, by the Banach isomorphism theorem, we have σ L(X)(T ) = σL(X)(T ).

Lemma 1 Let X be a complex normed space, let F, G be in L(X), and assume that G isbounded below and has dense range. Then exists λ ∈ C such that F − λG is not boundedbelow.

Proof Let X stand for the completion of X . For T ∈ L(X), let T denote the unique boundedlinear operator on X which extends T , and note that, if T is bounded below, then so is T . Bythe assumptions on G, we have that G ∈ Inv(L(X)). Therefore, by [1, Theorem 5.8] we canfind λ in the boundary of the set σ L(X)(G−1

F) relative to C. Then, by [12, pp. 278–279],G−1

F − λIX is not bounded below, and hence

F − λG = F − λG = G(G−1F − λI

X )

is not bounded below. It follows that F − λG is not bounded below.

Now, we state and prove the first main result in this paper.

Theorem 7 Let A be a non-zero normed complex algebra with no non-zero left topologicaldivisor of zero, and assume that there is a ∈ A such that La has dense range. Then A isisomorphic to C.

Proof Let b be in A. Noticing that La is bounded below and has dense range, we can applythe above lemma, to find λ ∈ C such that Lb−λa = Lb − λLa is not bounded below. SinceA has no non-zero left topological divisor of zero, we deduce that b = λa. Thus, A isone-dimensional and therefore, since A has non-zero product, it is isomorphic to C.

As straightforward consequences of Theorem 7, we derive the following two corollaries.

Corollary 1 Normed unital complex algebras with no non-zero left topological divisorof zero are isomorphic to C.

Corollary 2 Non-zero normed left-division complex algebras with no non-zero left topolog-ical divisor of zero are isomorphic to C.

Although Corollary 2 is a direct consequence of Theorem 7, we also add an autonomousproof, following the arguments of Kaplansky in [9].

Proof of Corollary 2 Suppose that A is a non-zero normed left-division complex algebrawithout non-zero left topological divisors of zero. The key idea, which must be omnipresentalong the argument, is that Lc belongs to Inv(L(A)) for every non-zero c ∈ A. Indeed, this isso because, for such a c, the operator Lc is surjective and bounded below, and hence L−1

c liesin L(A). Now, fix a non-zero element a ∈ A, let b in A, and recall that, by [1, Theorem 5.7],there exists λ ∈ C such that L−1

a Lb − λIA /∈ Inv(L(A)). Then we have

Lb−λa = Lb − λLa = La(L−1a Lb − λIA) /∈ Inv(L(A)).

This implies b − λa = 0. Thus, dim A = 1.

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Now we state and prove the second and last main result in this paper. It asserts that, evenin the complete case, Corollary 1 does not remain true if the requirement of absence of non-zero left topological divisors of zero is relaxed to the one of absence of non-zero two-sidedtopological divisors of zero. To this end, we recall the notion and standard properties of theunital hull of a normed algebra.

Let A be an algebra over K = R or C. The unital hull of A is defined as the algebra overK whose vector space is K × A and whose product is given by

(λ, x)(μ, y) := (λμ, λy + μx + xy).

We will denote by A1 the unital hull of A. A1 is always a unital algebra, whose unit is1 := (1, 0). Moreover, the mapping a → (0, a) from A to A1 identifies A with an ideal ofA1. With this identification in mind, we have

A1 = K1 ⊕ A.

Moreover, if A is (complete) normed, then A1, endowed with the norm

‖λ1 + a‖ := |λ| + ‖a‖,becomes a (complete) normed algebra over K containing isometrically A as a closed ideal.

Theorem 8 There exists an infinite-dimensional complete normed unital complex algebraA with no non-zero two-sided topological divisor of zero.

Proof Let {en : n ∈ N} be the canonical Hilbertian basis of the complex Hilbert space �2,and, according to Example 1, convert �2 into an absolute-valued algebra under the productdetermined by ei e j = e2i 3 j . Let T be the bounded linear operator on �2 determined byT (en) = 1

n en . Then T is injective and compact with ‖T ‖ = 1.Now, let B be the complete normed complex algebra consisting of the Banach space of

�2 and the product x � y := xT (y), and, for x ∈ B, let L�x denote the operator of left

multiplication by x relative to the product �. We claim that

σ L(B)(L�x ) = {0} for every x ∈ B. (1)

To prove the claim, note at first that, since L�x = Lx T and T is compact, L�

x is compact, sothat it is enough to show that L�

x has no non-zero eigen-value. For x = ∑

i λi ei ∈ B \{0}, putp(x) := min{i ∈ N: λi = 0}. Assume that the claim is not true. Then there are x = ∑

i λi ei

and y = ∑

i μi ei in B \ {0} such that L�x (y) = αy for some non-zero complex number α.

Then, since L�x (y) = ∑

i, jλi μ j

j e2i 3 j , we realize that

p(y) = p(αy) = p(L�x (y)) = 2p(x)3p(y) > p(y),

a contradiction.Now, let A stand for the normed unital hull of B. We are going to conclude the proof by

showing that A has no non-zero two-sided topological divisor of zero. Let a = λ1 + x bea two-sided topological divisor of zero in A. Since a is a left topological divisor of zero,there are sequences λn and xn in C and B, respectively, such that |λn | + ‖xn‖ = 1 for everyn ∈ N, λλn → 0, and λxn + λn x + x � xn → 0. Assume that λ = 0. Then we haveλn → 0, and, consequently, λxn + x � xn → 0 and ‖xn‖ → 1. The two last convergencesimply that −λ lies in σ L(B)(L�

x ), contradicting (1). Therefore λ = 0, and hence a = x ∈ B.Now, since x is a right topological divisor of zero in A, there are sequences μn and yn in C

and B, respectively, such that |μn | + ‖yn‖ = 1 for every n ∈ N, and μn x + yn � x → 0. By

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J. Carlos Marcos et al.

passing to subsequences if necessary, we can assume that μn converges to some μ ∈ C, and,consequently, that yn � x → −μx . Assume that x = 0. Then, since for n, m ∈ N we have

‖yn � x − ym � x‖ = ‖(yn − ym) � x‖ = ‖yn − ym‖‖T (x)‖,and T is injective, it follows that yn is a Cauchy sequence in B. Hence, by putting y :=limn→∞ yn , we have that y � x = −μx (which implies that −μ lies in σ L(B)(L�

y )) and

|μ|+‖y‖ = 1. The fact that −μ ∈ σ L(B)(L�y ) and (1) yields μ = 0, and consequently, since

‖y‖‖T (x)‖ = ‖y � x‖ = |μ|‖x‖ = 0,

also y = 0, which contradicts the equality |μ| + ‖y‖ = 1. Therefore a = x = 0, as desired.

3 Some complements

According to Fact 1, division associative algebras are unital, and left-division associativealgebras are division algebras. As the following two examples show, these facts do notremain true if associativity is removed.

Example 2 Let K stand for the real or complex field, and let A = K(x) be the algebra overK of all rational functions in the indeterminate x . We define a new product ∗ of rationalfunctions r(x), s(x) ∈ A by

r(x) ∗ s(x) := r(x−1)s(x−1).

Then (A, ∗) becomes a non-unital division algebra over K.

Example 3 Let K and A be as in Example 2. Let T : A → A be an injective but non-surjectivelinear operator (which does exist because A is infinite-dimensional). We define in A a newproduct � given by

r(x) � s(x) := T (r(x))s(x).

Then, since L�r(x) = LT (r(x)) is bijective for every r(x) ∈ A \ {0}, it follows that (A,�) is a

left-division algebra over K. But, for every s(x) ∈ A, the operator R�s(x) = Rs(x) ◦ T is not

bijective because T is not either. As a consequence, (A,�) is not a division algebra.

Example 3 has been communicated to us by A. Kaidi.In relation to Examples 2 and 3, we note that, by Theorem 3, complete normed non-unital

division complex algebras, and complete normed non-division left-division complex algebrascannot exist. Without completeness, the existence of such algebras is unknown (cf. §4).

In the case of real algebras, there are better versions of Examples 2 and 3. Indeed, consid-ering the product (z, w) → zw on the real Banach space underlying C, we obtain a completenormed non-unital division real algebra. Moreover, the existence of complete normed non-division left-division real algebras follows from the following.

Theorem 9 Absolute-valued division algebras are finite-dimensional [18], and there existinfinite-dimensional complete absolute-valued left-division real algebras ([3,13]).

By a quasi-division algebra we mean a non-zero algebra such that, for every a ∈ A \ {0},at least one of the operators La, Ra is bijective. As a matter of fact, we have the following.

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Theorem 10 ([14]) Complete normed quasi-division complex algebras have dimension ≤ 2,and moreover the dimension 2 is allowed.

Since unital algebras of dimension ≤ 2 are (associative and) commutative, and com-mutative quasi-division algebras are division algebras, the following corollary follows fromTheorems 10 and 3.

Corollary 3 Complete normed unital quasi-division complex algebras are isomorphic to C.

Now, we are going to summarize and clarify several results in this paper by discussingDiagram 11 immediately below.

§11 Let A be a non-zero complete normed algebra over K = R or C. Then, keeping inmind the Banach isomorphism theorem, the implications in the following diagram hold. Inthat diagram, the abbreviation t.d.z. stands for topological divisor of zero.

We note that conditions and implications in the first line of the above diagram are of apurely algebraic nature, and hence do not need the algebra A to be normed. On the otherhand, conditions and implications in the second line of the diagram only need the algebra Ato be normed, but not to be complete.

If the algebra A above is associative, then, by Theorem 5 and its version for real algebras(that non-zero normed associative real algebras with no non-zero two-sided divisors of zeroare isomorphic to R, C, or the algebra of Hamilton’s quaternions [7,2]), the weakest conditionin Diagram in §11 implies the strongest one in the diagram, and hence all conditions in thediagram are equivalent.

The discussion of Diagram in §11 in the general (possibly non-associative) case is morecomplicated. If K = R, then, as shown in [14], none of the implications in the diagram isreversible, the main tools being Example 1 and Theorem 9. It is also shown in [14] that,if K = C, then none of the implications in Diagram 11 is reversible, except the horizontalone on the left of the first line, which is indeed reversible. In this case, the main tools areExample 1 and Theorems 3 and 10.

We conclude the discussion of Diagram in §11 by considering the case that the completenormed algebra A is unital. In that case, by [6, Lemma 2.2], the vertical implications onthe left and on the middle are reversible. Moreover, if in addition A is complex, then, byCorollary 3, all conditions in the first line of the diagram are equivalent to the one that A isisomorphic to C. Therefore, in the complete normed unital complex case, all conditions inthe diagram are equivalent to the one that A is isomorphic to C, unless the weakest one (thatA has no non-zero two-sided topological divisor of zero), which, in view of Theorem 8, isindeed exceptional.

Acknowledgment We would like to express our gratitude to M. Cabrera and A. Kaidi for helpful commentsand suggestions concerning the topic of this paper. We also thank the referees for their suggestions to improvethe presentation of the paper.

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