Preserver Problems on Function Spaces, Operator Algebras, and Related TopicsView this Special Issue
Approximate Preservers on Banach Algebras and C*-Algebras
The aim of the present paper is to give approximate versions of Hua’s theorem and other related results for Banach algebras and C*-algebras. We also study linear maps approximately preserving the conorm between unital C*-algebras.
A well known formulation of the celebrated Hua’s theorem  asserts that every bijective additive map on a division ring such that and for every invertible element is either an automorphism or an antiautomorphism. This result was later moved to matrix algebras in  and finally extended to Banach algebras in  (see also ). In , the author called the previous relation strongly preserving invertibility.
Let and be unital Banach algebras. Recall that an additive map is a Jordan homomorphism iffor every , or equivalently, for all. Obvious examples of Jordan homomorphisms are homomorphisms and anti-homomorphisms. It is well known that every unital (i.e.,) Jordan homomorphism strongly preserves invertibility. Reciprocally, one of the results in  proves that every additive map between Banach algebras strongly preserving invertibility is a multiple of a Jordan homomorphism. In particular, if the map is unital, the map itself is a Jordan homomorphism.
There exist also other versions of Hua’s theorem involving some important kinds of generalized invertibility. Given a ringand an element , is said to be Drazin invertible if there exist and a nonnegative integer such that Such is unique whenever it exists. In this case, it is called the Drazin inverse of and it is denoted by . This notion was introduced by Drazin in  and it has proved useful in many fields of pure and applied mathematics (see for instance [6, 7]). If the previous identities are satisfied for some and , is called the group inverse of and we will denote it by . Let , , and denote the sets of all invertible, Drazin invertible, and group invertible elements in , respectively. Clearly Linear or additive maps between unital Banach algebras strongly preserving group and Drazin invertibility were introduced in [3, 4] (see also , where the author described additive maps between operator algebras of infinite-dimensional Hilbert spaces strongly preserving Drazin invertibility). If is a Jordan homomorphism between Banach algebras, it was shown in [3, Theorem 2.1] that (i) strongly preserves group invertibility, that is, for every ,(ii) strongly preserves Drazin invertibility, that is, for every .
Conversely, if is an additive map strongly preserving invertibility, group invertibility, or Drazin invertibility, and (resp., is invertible or ), then (resp., ) is a unital Jordan homomorphism and commutes with the image of , [4, Theorem 4.2]. In , the authors showed that the same holds without any hypothesis on and even whenis not necessarily unital.
Recall that an element is called regular if there exists (not necessarily unique) such that and . Notice that the first equality is a necessary and sufficient condition forto be regular, and that, and are idempotents in fulfilling and . For an element , let us consider the left and right multiplication operators and , respectively. If is regular, then so are and , and thus its ranges and are both closed.
An element has Moore-Penrose inverse when , and the associated idempotents and are self-adjoint. In the aforementioned papers by Harte and Mbekhta, it is shown that every regular element in a -algebra has a Moore-Penrose inverse, and that it is unique. For a regular element in a -algebra , will denote its Moore-Penrose inverse. The set of all Moore-Penrose invertible elements in a -algebra will be denoted by .
Let and be unital -algebras. A linear map strongly preserves Moore-Penrose invertibility if for every . Every Jordan-homomorphism strongly preserves Moore-Penrose invertibility, and the question is whether or not the converse holds. Some partial positive answers are given by Mbekhta in , and more recently by the authors of the present paper in , whenhas a rich structure of projections. The problem for general -algebras remains open. However, we can consider an alternative approach.
Recall that the class of -algebras is contained in a wider class of Banach spaces, the so called -triples, in which the concept of regularity extends the one given for -algebras. A -triple is a complex Banach spacetogether with a continuous triple product which is conjugate linear in the middle variable, and symmetric and bilinear in the outer variables satisfying that(a) whereis the operator ongiven by(b) is an hermitian operator with nonnegative spectrum;(c).
For each in a -triple , will stand for the conjugate linear operator on defined by .
An element in a -triple is called von Neumann regular if there exists (a unique) such that , , and . The elementis called the generalized inverse of and it will be denoted by . We will also note by the set of all elements in with generalized inverse. We refer to [13–17] for basics results on von Neumann regularity in -triples.
Every -algebra is a -triple via the triple product given by For a -algebra, it is well known thatand, for every regular elementin, we have.
A linear map between -triples is a triple homomorphism if for every . Every triple homomorphism between -triples strongly preserves generalized invertibility; that is, for every . In , the authors characterized the triple homomorphism between -algebras as the linear maps strongly preserving generalized invertibility. As a consequence, it is proved that a self-adjoint linear map from a unital -algebrainto a -algebra is a triple homomorphism if and only if it strongly preserves Moore-Penrose invertibility [9, Theorem 3.5].
The Hua-type descriptions belong to the framework of linear preserver problems. This has become an active research area in many topics of matrix theory, operator theory, and Banach algebras theory. Some of the most popular preserver problems are those dealing with determining the linear maps preserving properties related to invertibility. Every Jordan isomorphism between unital Banach algebras is unital and preserves invertibility in both directions [18, Proposition 1.3], or equivalently, preserves the spectrum; that is, , for every . The celebrated Kaplansky’s conjecture, , reformulated by Aupetit in , states that every unital surjective linear map between unital semisimple Banach algebras preserving invertibility in both directions is a Jordan isomorphism. Many partial positive results are known so far [18, 20–24], but the general problem is still open even in the class of -algebras. In the commutative setting, the classical Gleason-Kahane-Zelazko theorem (see ) states that a linear functional on a unital complex Banach algebra is multiplicative if and only if , for all . On the other hand, Jafarian and Sourour proved in  that every spectrum preserving surjective linear map is either an isomorphism or an anti-isomorphism, where denotes the Banach algebra of all bounded linear operators on a Banach space .
The authors of [25, 26] consider the problem of characterizing the approximately multiplicative linear functionals among all linear functionals on a commutative Banach algebra in terms of spectra. More recently, in , (see also ) Alaminos, Extremera, and Villena investigate approximate versions of Kaplansky’s problem, by providing approximate formulations of [18, 22]. They considered linear maps that approximately preserve spectrum or spectral radius on operator algebras and established the relationship between approximately preserving spectrum (resp., spectral radius) and approximately being a Jordan homomorphism (resp., weighted Jordan homomorphism).
Let and be Banach algebras and be a bounded linear map. Following [27, 29], the multiplicativity, antimultiplicativity, and Jordan-multiplicativity of can be measured by considering the following values: respectively. Obviously, is a homomorphism (anti-homomorphism, Jordan homomorphism) if and only if (resp., , ).
For a bounded linear map between -algebras, we define the triple multiplicativity and the self-adjointness of , respectively, as the following quantities:
Clearly, is a triple homomorphism if and only if , and is self-adjoint if and only if .
The aim of the present paper is to bring Hua type theorems into this framework. In order to make this possible, we have adapted some techniques from  involving ultraproducts of Banach algebras. Section 2 contains all the technical results about invertibility and coset representatives in ultraproducts of Banach algebras that we will need throughout the paper. Section 3 provides approximate versions of Hua’s theorem for invertibility and group invertibility in Banach algebras. We translate the strongly invertibility preserving condition into and the condition into for some . We prove that for every unital Banach algebras and , if in (7) or (8) then , uniformly on any set of linear maps with norms bounded above.
We show that for every unital -algebras and if in (9), then , uniformly on any set of linear maps whose norms are bounded above.
In this section, we also study linear maps that approximately preserve the conorm. Recall that the conorm of an element in a unital Banach algebra is defined as the reduced minimum modulus of the left multiplication operator by , . For a bounded linear operator on a complex Banach space , its reduced minimum modulus is given by It is well known that if and only if has closed range. In , it is shown that an element in a unital -algebra is regular if and only if . In this case,
(see [11, Theorem 2]). In , the authors characterized the linear maps between unital-algebras preserving the conorm. By [30, Theorem 3.1], if is a unital linear map such that for every , then is an isometric Jordan-homomorphism. Also, from [30, Theorem 3.2], if is a surjective linear map such that for every , then is an isometric Jordan-homomorphism multiplied by a unitary element. Hence, we replace the condition by in order to get approximate versions of [30, Theorems 3.1 and 3.2].
2. Ultraproducts of Banach Algebras: Basic Tools
Given a free ultrafilter on and a sequence of Banach spaces , the so called ultraproduct of the sequence is defined as follows: where is the Banach space of all bounded sequences , with for all , equipped with the norm and If the sequence is constant, is called the ultrapower of with respect to the ultrafilter . We will denote by the equivalence class of the sequence . The ultrapower of a Banach space is also a Banach space provided with the following norm: Of course, the ultrapower of a Banach algebra (resp., -algebra) is also a Banach algebra (resp., -algebra), with respect to the pointwise operations.
Finally, for every Banach spaces and , the canonical linear isometry given by for every and , allows us to consider as a closed subspace of . For , the canonical map gives an isometric unital homomorphism from to . The reader can see  in order to find basic results on ultraproducts.
Let be a unital Banach algebra and a free ultrafilter on . The following proposition is devoted to the description of invertible elements in through certain coset representatives. The result is probably well known, but the lack of an adequate reference moves us to include it here.
Proposition 1. Let . The following assertions are equivalent.(1)is invertible.(2)has a coset representative such that for all and is bounded.
Proof. For, just note that is an inverse for .
Reciprocally, assume that is invertible. Then, there exists such that . That is, Fix. The above identities imply that In particular, is right invertible for every and is left invertible for every . Thus, is invertible for every . Moreover, which shows that is bounded. Therefore, we can assume without loss of generality that consists of invertible elements and is bounded. (Otherwise, we choose Clearly, .)
Remark 2. It is clear that for with , we can choose a coset representative such that for all as follows: Hence, for every invertible element in , we can find a coset representative fulfilling the conditions in Proposition 1 and satisfying for all . We will name this one a normalized representative for .
3. Approximate Preservers in Banach Algebras
Let and be unital Banach algebras. Recall that, Boudi and Mbekhta proved in [4, Theorem 2.2] that an additive map strongly preserves invertibility if and only if is a unital Jordan homomorphism and commutes with the range of . Hence, for a bounded linear map between unital Banach algebras, we consider the unit-commutativity of , defined as in order to measure how close is our “approximately preserving invertibility” map to fulfilling that property.
Obviously, every bounded linear map satisfies. The next lemma shows the good behaviour of this concept with the ultraproduct of operators.
Some arguments in this section are inspired in .
Lemma 3. Let be a bounded sequence of linear maps between Banach algebras and , where is supposed to be unital. Consider . Then,
Proof. Given with , we can choose with for every . Therefore, and hence, Reciprocally, for each , there exists with such that Taking limit along we obtain
Our first main result provides an approximate version of Hua’s theorem for Banach algebras [4, Theorem 2.2] in the above mentioned.
Theorem 4. Let and be unital Banach algebras and . Then, there exists such that for every linear map with , the condition implies that
Proof. Suppose that the assertion of the theorem is false. Then, we can find and a sequence of linear maps from to such that, for every ,(i),(ii),(iii) or.
Consider that . We claim that strongly preserves invertibility. Indeed, let be an invertible element. We can suppose, without loss of generality, that . Let be its normalized representative, with , for some (see Proposition 1 and Remark 2). As we get for all . Hence, is invertible and is its inverse. This yields Thus, for every invertible element . By [4, Theorem 2.2], and , for every . We apply [27, Lemmas 3.4] and Lemma 3 to obtain, respectively, the following: Consequently, Finally, gives us the desired contradiction.
Our goal now is to achieve a group invertibility version for the previous theorem. Recall that given an additive map from a unital Banach algebra into a Banach algebra , by [9, Theorem 2.4], if strongly preserves group invertibility, then is a Jordan homomorphism and commutes with the range of . In order to take advantage of Proposition 1, our first step is to improve [9, Theorem 2.4] by showing that all the information required is located in .
Recall that the so-called Hua’s identity asserts that, if , and are invertible elements in a ring, then
Theorem 5. Let and be Banach algebras, being unital, and be an additive map such thatfor all. Then, is a Jordan homomorphism and commutes with.
Proof. A look to the arguments employed in [9, Lemma 2.1], allows us to show that preserves the cubes of the invertible elements. Indeed, given and with , as and are invertible elements, we can apply Hua’ s identity to obtain Let us assume that. Since, it follows thatis invertible in the unital Banach algebrafor, with inverse. Identity (36) applied forandgives Hence, for every such that , we get Hence, , as desired. From this last identity, reasoning as in [9, Proposition 2.3], we deduce that the following equalities hold for every : Finally, it only remains to repeat the arguments inin [9, Theorem 2.4] to conclude the proof.
Now, we can state the following result.
Theorem 6. Let and be Banach algebras where is unital and . Then, there exists such that for every linear map with , the condition implies that
Proof. First, notice that if has a coset representative where is group invertible for every and is bounded, then is group invertible and . Hence, the same arguments used in Theorem 4 produce an operatorsatisfyingfor every invertible element . Now, Theorem 5 proves thatis a Jordan homomorphism andcommutes with. Again, the final argument in Theorem 4 completes the proof.
In [32, Proposition 2.5], the authors proved, in particular, that if an additive mapbetween unital Banach algebras satisfies and is invertible, thenis a Jordan homomorphism andcommutes with. It is clear now that for a sequence of linear operatorssatisfying thatfor all , and its ultrapoductfulfillsfor every invertible. Therefore,is a Jordan homomorphism andcommutes with. This leads us to the following approximate formulation of [32, Proposition 2.5].
Theorem 7. Let and be unital Banach algebras and . Then, there existssuch that for every linear mapwith, the following condition: implies that
4. Approximate Preservers in -Algebras
The aim of this section is twofold. On the one hand, we prove that linear maps approximately preserving generalized invertibility in -algebras are close to be triple homomorphisms. On the other hand, we study linear maps approximately preserving the conorm.
4.1. Approximate Preservers of the Moore-Penrose Inverse and the Generalized Inverse
Given a-triple, the triple cube of an elementis defined as. An element satisfyingis called a tripotent. The following polarization identity allows us to write the triple product as linear combination of triple cubes: Hence, a linear map between -triples is a triple homomorphism if and only if it preserves triple cubes.
Each tripotent in gives rise to the so-called Peirce decomposition of associated to , that is, where for is theeigenspace of . The peirce spaceis a -algebra with product and involution. Moreover, the triple product induced onby this Jordan -algebra structure coincides with its original triple product.
It is proved in [16, Lemma 3.2] (compare with [13, Theorem 3.4]) that for every regular elementin a -triple, there exists a tripotentsuch thatis a self-adjoint invertible element in the -algebra. Ifis invertible, its inverse is denoted as usual by. Moreover, ifandare invertible elements in the Jordan algebrasuch thatis also invertible, then is invertible and the Hua identity holds, where(see , (11)).
Letbe a unital-algebra, and. Then, there exists a unique partial isometry, such thatis self-adjoint and invertible in the Jordan algebra, with inverseHence, for everywith, the elementis invertible in. Reciprocally, the inverses ofandinare their generalized inverses in. By the Hua identity (48), we obtain
Theorem 8. Letandbe -algebras,being unital, anda bounded linear map satisfyingfor every self-adjoint invertible element. Then, is a triple homomorphism.
Proof. Arguing as in [9, Lemma 3.1], pick a self-adjoint invertible element. We may assume that. Then, givenwith, from identities (36) and (49), we get which shows that. Once we have proved thatpreserves cubes of self-adjoint invertible elements, given a self-adjoint element, andwith, as the elementis self-adjoint and invertible, we get. Expanding this las equation we obtain for every, and. From this, we deduce thatfor every. That is,preserves triple cubes of self-adjoint elements. By [34, Theorem 20],is a triple homomorphism.
Recall that we measure how close is a linear mapbetween -algebras to being a triple homomorphism or self-adjoint by the triple multiplicativity and the self-adjointness of, respectively as follows:
Remark 9. Letandbe -algebras. It is clear that every Jordan-homomorphismis a triple homomorphism. We ask whetherand being small implybeing small.
Letbe a bounded linear map. Define for every. Then, Therefore, This implies that
As in Lemma 3, it can be shown that, for an operator, the following holds:
We will omit the proof of the next result in order to avoid repetition. The argument is analogous to the one used in the proofs of Theorems 4 and 6: assuming the contrary, we can construct a mapbetween the ultrapowers fulfillingfor every self-adjoint invertible. By Theorem 8,is a triple homomorphism.
Theorem 10. Letandbe -algebras whereis unital and. Then, there existssuch that for every linear mapwith, the condition implies that
Corollary 12. Letandbe -algebras whereis unital and. Then, there existssuch that for every linear mapwith, the conditions imply that
Proof. Let us briefly sketch the proof: assuming the contrary, there existand a sequenceof linear maps from tosuch that, for every, Then, is a self-adjoint map such that, for every invertible element. In particular,, for every invertible self-adjoint element. From Theorem 8,is a triple homomorphism (Contradiction).
4.2. Maps Approximately Preserving the Conorm
Letandbe unital-algebras. Kadison proved in  that a surjective linear mapis an isometry if and only ifis a Jordan-isomorphism multiplied by a unitary element in. In , the authors address the question of characterizing surjective linear maps preserving some spectral quantities. Given an elementof a Banach algebra, the minimum modulus and the surjectivity modulus ofare defined, respectively, by Obviously,) if and only ifis a left (resp., right) topological divisor of zero. It is well known that for any invertible element, Moreover, for every. Ifis a-algebra, then. In particular,) if and only ifis left (resp., right) invertible.
Recall also from the introduction that the conorm of an elementin a Banach algebra, is defined as For a regular elementin a-algebra,
Letandbe unital -algebras. By Theorems 3.1 and in , ifis a linear map preserving any of these spectral quantities, thenis an isometric Jordan *-homomorphism wheneveris unital, andis an isometric Jordan *-homomorphism multiplied by a unitary element, wheneveris surjective. In the next results, we show that the same holds if we just impose the preserving condition for invertible elements. Notice that we focus our attention on the conorm but identical results can be established for the minimum and surjective modulus.
Theorem 13. Letandbe unital -algebras anda unital linear map satisfying for all. Then, is a Jordan *-homomorphism.
Proof. First, let us prove thatis injective. Takesuch thatand letbe sufficiently small so thatis invertible. Then,
In particular, we get
as. Hence, by [30, Lemma 4.1], bothandare self-adjoint and, consequently,.
We claim thatis positive. Indeed, given a self-adjoint element, we know that Sinceis invertible forsmall enough, it follows that This implies thatis self-adjoint.
Moreover, givenand, there exists a neighborhoodofsuch that. If then, wheredenotes the Kato spectrum ofas follows: As for, the elementis invertible; then, we havefor every. Consequently, and. Sincefor every(see [36, Sections ]), we have just proved that for every. Beingself-adjoint, this implies thatis positive and hence,.
Arguing as in [30, Theorem 5.1], given a self-adjoint elementandsufficiently small so thatis a unitary element with spectrum strictly contained in the unit circle, since the elementis unitary. From we deduce thatas desired.
Theorem 14. Letandbe unital -algebras and a surjective linear map satisfyingfor all. Then, is a Jordan *-homomorphism multiplied by a unitary element in.
Proof. First, let us prove thatis invertible. Since,is regular. Let, andsuch that. Notice that.
Forsufficiently small such that, Hence, Reasoning in a similar way to [30, Theorem 6.2], we get and therefore for small enough . From these inequalities, we get, respectively, thatandare self-adjoint. This shows thatand thus. Consequently,, that is,is right invertible. Similarly it can be proved thatis left invertible.
Note that, as in the previous theorem,is injective. Thereforeis a unital and bijective linear map satisfying
Letbe a self-adjoint element inandsmall such thatis invertible. Takingin the previous identity, we have
It follows thatis self-adjoint and so is.
We claim thatis positive. Note that for everyand, it is clear that, with This implies, by [11, Theorem 2], the following: Hence, for every, we have So, we have shown so far the following: The first inequality can be used to show the following: in a similar way as in the previous theorem. As a consequence,is positive. In order to conclude thatis an isometric Jordan-isomorphism, it suffices to prove thatis also positive (see for instance [35, Corollary 5]). So, letbe a positive element. Asis self-adjoint,is self-adjoint. We can therefore write, where and are positive elements and. For every, we have (Recall that if, then.) The previous spectral inclusion gives
By Lemmas and in , we getand so. Consequently,is positive as desired. We conclude the proof by showing thatis unitary. Indeed, sinceis a Jordan-isomorphism and, it is clear that, for every,is invertible with inverse. Moreover,; that is, for every. Sinceis an isometry, or equivalently, for every. This yields thatis unitary.
Note that, for every invertible element, whereis a normalized representative, We are now in position to prove the main results in this section.
Theorem 15. Letandbe unital -algebras and. Then, there existssuch that for every linear mapwith, the following conditions: imply that
Proof. Note that if has a coset representativewherehas Moore-Penrose inverse for everyandis bounded, thenis Moore-Penrose invertible and.
As we did above, suppose that the assertion is false, that is, we can findand a sequenceof linear maps fromtosatisfying(i),(ii), ,(iii)or,
for every. Consider. We claim thatis unital and preserves the conorm of invertible elements. On the one hand, so. On the other hand, for an invertible elementwith norm 1, letbe its normalized representative: