Approximate Preservers on Banach Algebras and C*-Algebras

Abstract

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.

1. Preliminaries

A well known formulation of the celebrated Hua’s theorem [1] asserts that every bijective additive map 𝑇∶𝐾→𝐾 on a division ring 𝐾 such that 𝑇(1)=1 and 𝑇(𝑥⁻¹)=𝑇(𝑥)⁻¹ for every invertible element 𝑥 is either an automorphism or an antiautomorphism. This result was later moved to matrix algebras in [2] and finally extended to Banach algebras in [3] (see also [4]). In [3], the author called the previous relation strongly preserving invertibility.

Let 𝐴 and 𝐵 be unital Banach algebras. Recall that an additive map 𝑇∶𝐴→𝐵 is a Jordan homomorphism if𝑇(𝑎𝑏+𝑏𝑎)=𝑇(𝑎)𝑇(𝑏)+𝑇(𝑏)𝑇(𝑎)for every 𝑎,𝑏∈𝐴, or equivalently, 𝑇(𝑎²)=𝑇(𝑎)² for all𝑎∈𝐴. Obvious examples of Jordan homomorphisms are homomorphisms and anti-homomorphisms. It is well known that every unital (i.e.,𝑇(1)=1) Jordan homomorphism strongly preserves invertibility. Reciprocally, one of the results in [4] 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 ring𝐴and an element 𝑎∈𝐴, 𝑎 is said to be Drazin invertible if there exist 𝑏∈𝐴 and a nonnegative integer 𝑘 such that 𝑏𝑎𝑏=𝑏,𝑎^𝑘𝑏𝑎=𝑎^𝑘,𝑎𝑏=𝑏𝑎. (1) Such 𝑏 is unique whenever it exists. In this case, it is called the Drazin inverse of 𝑎 and it is denoted by 𝑏=𝑎^𝐷. This notion was introduced by Drazin in [5] 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 𝑏∈𝐴 and 𝑘=1, 𝑏 is called the group inverse of 𝑎 and we will denote it by 𝑎^𝐺. Let 𝐴⁻¹, 𝐴^𝐷, and 𝐴^𝐺 denote the sets of all invertible, Drazin invertible, and group invertible elements in 𝐴, respectively. Clearly 𝐴⁻¹⊂𝐴^𝐺⊂𝐴^𝐷. (2) Linear or additive maps between unital Banach algebras strongly preserving group and Drazin invertibility were introduced in [3, 4] (see also [8], where the author described additive maps between operator algebras of infinite-dimensional Hilbert spaces strongly preserving Drazin invertibility). If 𝑇∶𝐴→𝐵 is a Jordan homomorphism between Banach algebras, it was shown in [3, Theorem 2.1] that (i)𝑇 strongly preserves group invertibility, that is, 𝑇(𝑎^𝐺)=𝑇(𝑎)^𝐺 for every 𝑎∈𝐴^𝐺,(ii)𝑇 strongly preserves Drazin invertibility, that is, 𝑇(𝑎^𝐷)=𝑇(𝑎)^𝐷 for every 𝑎∈𝐴^𝐷.

Conversely, if 𝑇∶𝐴→𝐵 is an additive map strongly preserving invertibility, group invertibility, or Drazin invertibility, and 𝑇(1)=1 (resp., 𝑇(1) is invertible or 1∈𝑇(𝐴)), then 𝑇 (resp., 𝑇(1)𝑇) is a unital Jordan homomorphism and 𝑇(1) commutes with the image of 𝑇, [4, Theorem 4.2]. In [9], the authors showed that the same holds without any hypothesis on 𝑇(1) and even when𝐵is not necessarily unital.

Recall that an element 𝑎∈𝐴 is called regular if there exists 𝑏∈𝐴 (not necessarily unique) such that 𝑎=𝑎𝑏𝑎 and 𝑏=𝑏𝑎𝑏. Notice that the first equality 𝑎=𝑎𝑏𝑎 is a necessary and sufficient condition for𝑎to be regular, and that, 𝑝=𝑎𝑏 and 𝑞=𝑏𝑎 are idempotents in 𝐴 fulfilling 𝑎𝐴=𝑝𝐴 and 𝐴𝑎=𝐴𝑞. For an element 𝑎∈𝐴, let us consider the left and right multiplication operators 𝐿_𝑎∶𝑥↦𝑎𝑥 and 𝑅_𝑎∶𝑥↦𝑥𝑎, respectively. If 𝑎 is regular, then so are 𝐿_𝑎 and 𝑅_𝑎, and thus its ranges 𝑎𝐴=𝐿_𝑎(𝐴) and 𝐴𝑎=𝑅_𝑎(𝐴) are both closed.

Regular elements in a unital 𝐶^∗-algebra 𝐴 were studied by Harte and Mbekhta in [10, 11].

An element 𝑎∈𝐴 has Moore-Penrose inverse𝑏 when 𝑎=𝑎𝑏𝑎, 𝑏=𝑏𝑎𝑏and the associated idempotents 𝑎𝑏 and 𝑏𝑎 are self-adjoint. In the aforementioned papers by Harte and Mbekhta, it is shown that every regular element in a 𝐶^∗-algebra has a Moore-Penrose inverse, and that it is unique. For a regular element 𝑎 in a 𝐶^∗-algebra 𝐴, 𝑎^† will denote its Moore-Penrose inverse. The set of all Moore-Penrose invertible elements in a 𝐶^∗-algebra 𝐴 will be denoted by 𝐴^†.

Let 𝐴 and 𝐵 be unital 𝐶^∗-algebras. A linear map 𝑇∶𝐴→𝐵strongly preserves Moore-Penrose invertibility if 𝑇(𝑎^†)=𝑇(𝑎)^† for every 𝑎∈𝐴^†. 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 [3], and more recently by the authors of the present paper in [12], when𝐴has a rich structure of projections. The problem for general 𝐶^∗-algebras remains open. However, we can consider an alternative approach.

Recall that the class of 𝐶^∗-algebras is contained in a wider class of Banach spaces, the so called 𝐽𝐵^∗-triples, in which the concept of regularity extends the one given for 𝐶^∗-algebras. A 𝐽𝐵^∗-triple is a complex Banach space𝐸together with a continuous triple product{⋅,⋅,⋅}∶𝐸×𝐸×𝐸→𝐸, which is conjugate linear in the middle variable, and symmetric and bilinear in the outer variables satisfying that(a)𝐿(𝑎,𝑏)𝐿(𝑥,𝑦)=𝐿(𝑥,𝑦)𝐿(𝑎,𝑏)+𝐿(𝐿(𝑎,𝑏)𝑥,𝑦)−𝐿(𝑥,𝐿(𝑏,𝑎)𝑦), where𝐿(𝑎,𝑏)is the operator on𝐸given by𝐿(𝑎,𝑏)𝑥={𝑎,𝑏,𝑥};(b)𝐿(𝑎,𝑎) is an hermitian operator with nonnegative spectrum;(c)‖𝐿(𝑎,𝑎)‖=‖𝑎‖².

For each 𝑥 in a 𝐽𝐵^∗-triple 𝐸, 𝑄(𝑥) will stand for the conjugate linear operator on 𝐸 defined by 𝑄(𝑥)(𝑦)={𝑥,𝑦,𝑥}.

An element 𝑎 in a 𝐽𝐵^∗-triple 𝐸 is called von Neumann regular if there exists (a unique) 𝑏∈𝐸 such that 𝑄(𝑎)(𝑏)=𝑎, 𝑄(𝑏)(𝑎)=𝑏, and 𝑄(𝑎)𝑄(𝑏)=𝑄(𝑏)𝑄(𝑎). The element𝑏is called the generalized inverse of 𝑎 and it will be denoted by 𝑎^∧. We will also note by 𝐸^∧ the set of all elements in 𝐸 with generalized inverse. We refer to [13–17] for basics results on von Neumann regularity in 𝐽𝐵^∗-triples.

Every 𝐶^∗-algebra is a 𝐽𝐵^∗-triple via the triple product given by =1𝑥,𝑦,𝑧2𝑥𝑦^∗𝑧+𝑧𝑦^∗𝑥. (3) For a 𝐶^∗-algebra𝐴, it is well known that𝐴^†=𝐴^∧and, for every regular element𝑎in𝐴, we have𝑎^∧=(𝑎^†)^∗.

A linear map 𝑇∶𝐸→𝐹 between 𝐽𝐵^∗-triples is a triple homomorphism if 𝑇𝑥,𝑦,𝑧=𝑇(𝑥),𝑇𝑦,𝑇(𝑧), (4) for every 𝑥,𝑦,𝑧∈𝐸. Every triple homomorphism 𝑇∶𝐸→𝐹 between 𝐽𝐵^∗-triples strongly preserves generalized invertibility; that is, 𝑇(𝑥^∧)=𝑇(𝑥)^∧ for every 𝑥∈𝐸^∧. In [9], the authors characterized the triple homomorphism between 𝐶^∗-algebras as the linear maps strongly preserving generalized invertibility. As a consequence, it is proved that a self-adjoint linear map from a unital 𝐶^∗-algebra𝐴into a 𝐶^∗-algebra 𝐵 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 𝑇∶𝐴→𝐵 between unital Banach algebras is unital and preserves invertibility in both directions [18, Proposition 1.3], or equivalently, preserves the spectrum; that is, 𝜎(𝑇(𝑎))=𝜎(𝑎), for every 𝑎∈𝐴. The celebrated Kaplansky’s conjecture, [19], reformulated by Aupetit in [20], states that every unital surjective linear map 𝑇 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 𝐶^∗-algebras. In the commutative setting, the classical Gleason-Kahane-Zelazko theorem (see [23]) states that a linear functional 𝜑 on a unital complex Banach algebra 𝐴 is multiplicative if and only if 𝜑(𝑎)∈𝜎(𝑎), for all 𝑎∈𝐴. On the other hand, Jafarian and Sourour proved in [22] that every spectrum preserving surjective linear map 𝑇∶ℬ(𝑋)→ℬ(𝑌) is either an isomorphism or an anti-isomorphism, where ℬ(𝑋) denotes the Banach algebra of all bounded linear operators on a Banach space 𝑋.

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 [27], (see also [28]) 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 𝐴 and 𝐵 be Banach algebras and 𝑇∶𝐴→𝐵 be a bounded linear map. Following [27, 29], the multiplicativity, antimultiplicativity, and Jordan-multiplicativity of 𝑇 can be measured by considering the following values: (𝑇)𝑇()(𝑎)𝑇(𝑏‖𝑎‖=‖𝑏‖},(𝑇)𝑇()(𝑏)𝑇(𝑎‖𝑎‖=‖𝑏‖},(𝑇)𝑇𝑎mult=sup{‖𝑎𝑏−𝑇)‖∶𝑎,𝑏∈𝐴,=1amult=sup{‖𝑎𝑏−𝑇)‖∶𝑎,𝑏∈𝐴,=1jmult=sup²−𝑇(𝑎)²∶𝑎∈𝐴,‖𝑎‖=1, (5) respectively. Obviously, 𝑇 is a homomorphism (anti-homomorphism, Jordan homomorphism) if and only if mult(𝑇)=0 (resp., amult(𝑇)=0, jmult(𝑇)=0).

For a bounded linear map 𝑇∶𝐴→𝐵 between 𝐶^∗-algebras, we define the triple multiplicativity and the self-adjointness of 𝑇, respectively, as the following quantities: (𝑇)𝑇−{𝑇(𝑎)(𝑏)(𝑐∶‖𝑎‖=‖𝑏‖=‖𝑐‖},(𝑇)𝑇𝑎tmult=sup{‖({𝑎,𝑏,𝑐}),𝑇,𝑇)}‖=1sa=sup^∗∗−𝑇(𝑎)∶‖𝑎‖=1. (6)

Clearly, 𝑇∶𝐴→𝐵 is a triple homomorphism if and only if tmult(𝑇)=0, and 𝑇 is self-adjoint if and only if sa(𝑇)=0.

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 [27] 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 𝑇(𝑎⁻¹)=𝑇(𝑎)⁻¹ into sup_{‖𝑎‖=1,𝑎∈𝐴−1}𝑇𝑎⁻¹−𝑇(𝑎)⁻¹<𝜀, (7) and the condition 𝑇(𝑎^𝐺)=𝑇(𝑎)^𝐺 into sup_{‖𝑎‖=1,𝑎∈𝐴𝐺}𝑇𝑎^𝐺−𝑇(𝑎)^𝐺<𝜀, (8) for some 𝜀>0. We prove that for every unital Banach algebras 𝐴 and 𝐵, if 𝜀→0 in (7) or (8) then jmult(𝑇(1)𝑇)→0, uniformly on any set of linear maps 𝑇∶𝐴→𝐵 with norms bounded above.

Section 4 includes an approximate formulation of [9, Theorem 3.5]. The condition 𝑇(𝑎^†)=𝑇(𝑎)^† for every 𝑎∈𝐴^† is replaced by sup_{‖𝑎‖=1,𝑎∈𝐴†}𝑇𝑎^†−𝑇(𝑎)^†<𝜀. (9)

We show that for every unital 𝐶^∗-algebras 𝐴 and 𝐵 if 𝜀→0 in (9), then tmult(𝑇)→0, 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 𝑐(𝑎)of an element 𝑎 in a unital Banach algebra is defined as the reduced minimum modulus of the left multiplication operator by 𝑎, 𝑐(𝑎)=𝛾(𝐿_𝑎)[11]. For a bounded linear operator 𝑇 on a complex Banach space 𝑋, its reduced minimum modulus is given by 𝛾(𝑇)=inf{‖𝑇𝑥‖∶dist(𝑥,ker(𝑇))≥1}. (10) It is well known that 𝛾(𝑇)>0 if and only if 𝑇 has closed range. In [10], it is shown that an element 𝑎 in a unital 𝐶^∗-algebra 𝐴 is regular if and only if 𝑐(𝑎)>0. In this case, 𝑐(𝑎)=𝑎^†−1 (11)

(see [11, Theorem 2]). In [30], the authors characterized the linear maps between unital𝐶^∗-algebras preserving the conorm. By [30, Theorem 3.1], if 𝑇∶𝐴→𝐵 is a unital linear map such that 𝑐(𝑇(𝑎))=𝑐(𝑎) for every 𝑎∈𝐴, then 𝑇 is an isometric Jordan∗-homomorphism. Also, from [30, Theorem 3.2], if 𝑇∶𝐴→𝐵 is a surjective linear map such that 𝑐(𝑇(𝑎))=𝑐(𝑎) for every 𝑎∈𝐴, then 𝑇 is an isometric Jordan∗-homomorphism multiplied by a unitary element. Hence, we replace the condition 𝑐(𝑇(𝑎))=𝑐(𝑎) by sup_‖𝑎‖=1|𝑐(𝑇(𝑎))−𝑐(𝑎)|<𝜀, (12) 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 𝒰 on ℕ and a sequence of Banach spaces {𝑋_𝑛}_𝑛∈ℕ, the so called ultraproduct of the sequence is defined as follows: 𝑋_𝑛^𝒰ℓ∶=^∞ℕ,𝑋_𝑛𝒩_𝒰, (13) where ℓ^∞(ℕ,𝑋_𝑛) is the Banach space of all bounded sequences {𝑥_𝑛}_𝑛∈ℕ, with 𝑥_𝑛∈𝑋_𝑛 for all 𝑛∈ℕ, equipped with the ℓ_∞ norm and 𝒩_𝒰∶=𝑥_{𝑛𝑛∈ℕ}∈ℓ^∞ℕ,𝑋_𝑛∶lim_𝒰𝑥_𝑛=0. (14) If the sequence {𝑋_𝑛}_𝑛∈ℕ={𝑋} is constant, 𝑋^𝒰∶=ℓ^∞(ℕ,𝑋)/𝒩_𝒰, is called the ultrapower of 𝑋 with respect to the ultrafilter 𝒰. We will denote by 𝐱=[𝑥_𝑛] the equivalence class of the sequence {𝑥_𝑛}_𝑛∈ℕ. The ultrapower of a Banach space is also a Banach space provided with the following norm: ‖𝐱‖∶=lim_𝒰𝑥_𝑛. (15) Of course, the ultrapower 𝐴^𝒰 of a Banach algebra (resp., 𝐶^∗-algebra) is also a Banach algebra (resp., 𝐶^∗-algebra), with respect to the pointwise operations.

Finally, for every Banach spaces 𝑋 and 𝑌, the canonical linear isometry ℬ(𝑋,𝑌)^𝒰→ℬ(𝑋^𝒰,𝑌^𝒰) given by 𝐓(𝐱)=𝑇_𝑛𝑥_𝑛, (16) for every 𝐓=[𝑇_𝑛]∈ℬ(𝑋,𝑌)^𝒰 and 𝐱=[𝑥_𝑛]∈𝑋^𝒰, allows us to consider ℬ(𝑋,𝑌)^𝒰 as a closed subspace of ℬ(𝑋^𝒰,𝑌^𝒰). For 𝑋=𝑌, the canonical map gives an isometric unital homomorphism from ℬ(𝑋)^𝒰 to ℬ(𝑋^𝒰). The reader can see [31] in order to find basic results on ultraproducts.

Let 𝐴 be a unital Banach algebra and 𝒰 a free ultrafilter on ℕ. The following proposition is devoted to the description of invertible elements in 𝐴^𝒰 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 𝐚∈𝐴^𝒰. The following assertions are equivalent.(1)𝐚is invertible.(2)𝐚has a coset representative [𝑢_𝑛] such that 𝑢_𝑛∈𝐴⁻¹ for all 𝑛∈ℕ and {𝑢_𝑛⁻¹}_𝑛∈ℕ is bounded.

Proof. For(2)⇒(1), just note that [𝑢_𝑛⁻¹]∈𝐴^𝒰 is an inverse for [𝑢_𝑛].
Reciprocally, assume that 𝐚=[𝑎_𝑛] is invertible. Then, there exists 𝐛=[𝑏_𝑛]∈𝐴^𝒰 such that 𝐚𝐛=𝐛𝐚=𝟏. That is, lim_𝒰𝑎_𝑛𝑏_𝑛−𝟏=0,lim_𝒰𝑏_𝑛𝑎_𝑛−𝟏=0. (17) Fix0<𝛿<1. The above identities imply that 𝑅∶=𝑛∈ℕ∶𝑎_𝑛𝑏_𝑛𝑏−𝟏<𝛿∈𝒰,𝐿∶=𝑛∈ℕ∶_𝑛𝑎_𝑛−𝟏<𝛿∈𝒰. (18) In particular, 𝑎_𝑛 is right invertible for every 𝑛∈𝑅 and 𝑎_𝑛 is left invertible for every 𝑛∈𝐿. Thus, 𝑎_𝑛 is invertible for every 𝑛∈𝐼∶=𝑅∩𝐿∈𝒰. Moreover, 𝑎_𝑛⁻¹=𝑏_𝑛𝑎_𝑛𝑏_𝑛⁻¹≤𝑏_𝑛𝑎_𝑛𝑏_𝑛⁻¹≤𝑏_𝑛1−1−𝑎_𝑛𝑏_𝑛≤𝑏_𝑛,1−𝛿 (19) which shows that {𝑎_𝑛⁻¹∶𝑛∈𝐼} is bounded. Therefore, we can assume without loss of generality that {𝑎_𝑛}_𝑛∈ℕ consists of invertible elements and {𝑎_𝑛⁻¹}_𝑛∈ℕ is bounded. (Otherwise, we choose 𝑎^′_𝑛=𝑎_𝑛,if𝑛∈𝐼,1,if𝑛∉𝐼. (20) Clearly, [𝑎_𝑛]=[𝑎^′_𝑛].)

Remark 2. It is clear that for 𝐚=[𝑎_𝑛]∈𝐴^𝒰 with ‖𝐚‖=1, we can choose a coset representative 𝐚=[𝑏_𝑛] such that ‖𝑏_𝑛‖=1 for all 𝑛∈ℕ as follows: 𝑏_𝑛=𝑎_𝑛𝑎_𝑛,if𝑎_𝑛≠0,1,if𝑎_𝑛=0. (21) Hence, for every invertible element 𝐚 in 𝐴^𝒰, we can find a coset representative 𝐚=[𝑎_𝑛] fulfilling the conditions in Proposition 1 and satisfying ‖𝑎_𝑛‖=‖𝐚‖ for all 𝑛∈ℕ. We will name this one a normalized representative for 𝐚.

3. Approximate Preservers in Banach Algebras

Let 𝐴 and 𝐵 be unital Banach algebras. Recall that, Boudi and Mbekhta proved in [4, Theorem 2.2] that an additive map 𝑇∶𝐴→𝐵 strongly preserves invertibility if and only if 𝑇(1)𝑇 is a unital Jordan homomorphism and 𝑇(1) commutes with the range of 𝑇. Hence, for a bounded linear map 𝑇∶𝐴→𝐵 between unital Banach algebras, we consider the unit-commutativity of 𝑇, defined as ucomm(𝑇)=sup_‖𝑎‖=1‖𝑇(𝑎)𝑇(1)−𝑇(1)𝑇(𝑎)‖, (22) in order to measure how close is our “approximately preserving invertibility” map to fulfilling that property.

Obviously, every bounded linear map satisfiesucomm(𝑇)≤2‖𝑇‖². The next lemma shows the good behaviour of this concept with the ultraproduct of operators.

Some arguments in this section are inspired in [27].

Lemma 3. Let{𝑇_𝑛}_𝑛∈ℕ be a bounded sequence of linear maps between Banach algebras 𝐴 and 𝐵, where 𝐴 is supposed to be unital. Consider 𝐓=[𝑇_𝑛]∶𝐴^𝒰→𝐵^𝒰. Then, lim_𝒰𝑢𝑐𝑜𝑚𝑚𝑇_𝑛=𝑢𝑐𝑜𝑚𝑚(𝐓). (23)

Proof. Given 𝐚∈𝐴^𝒰 with ‖𝐚‖=1, we can choose 𝐚=[𝑎_𝑛] with ‖𝑎_𝑛‖=1 for every 𝑛∈ℕ. Therefore, ‖𝐓(𝐚)𝐓(𝟏)−𝐓(𝟏)𝐓(𝐚)‖=lim_𝒰𝑇_𝑛𝑎_𝑛𝑇_𝑛(1)−𝑇_𝑛(1)𝑇_𝑛𝑎_𝑛≤lim_𝒰ucomm𝑇_𝑛, (24) and hence, ucomm(𝐓)≤lim_𝒰ucomm𝑇_𝑛. (25) Reciprocally, for each 𝑛∈ℕ, there exists 𝑎_𝑛∈𝐴 with ‖𝑎_𝑛‖=1 such that ucomm𝑇_𝑛−1𝑛<𝑇_𝑛𝑎_𝑛𝑇_𝑛(1)−𝑇_𝑛(1)𝑇_𝑛𝑎_𝑛. (26) Taking limit along 𝒰 we obtain lim_𝒰ucomm𝑇_𝑛≤lim_𝒰𝑇_𝑛𝑎_𝑛𝑇_𝑛(1)−𝑇_𝑛(1)𝑇_𝑛𝑎_𝑛=‖𝐓(𝐚)𝐓(𝟏)−𝐓(𝟏)𝐓(𝐚)‖≤ucomm(𝐓). (27)

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 𝐴 and 𝐵 be unital Banach algebras and 𝐾,𝜀>0. Then, there exists 𝛿>0 such that for every linear map 𝑇∶𝐴→𝐵 with ‖𝑇‖<𝐾, the condition sup_{‖𝑎‖=1,𝑎∈𝐴−1}𝑇𝑎⁻¹−𝑇(𝑎)⁻¹<𝛿 (28) implies that 𝑗𝑚𝑢𝑙𝑡(𝑇(𝟏)𝑇)<𝜀,𝑢𝑐𝑜𝑚𝑚(𝑇)<𝜀. (29)

Proof. Suppose that the assertion of the theorem is false. Then, we can find 𝐾₀,𝜀₀>0 and a sequence {𝑇_𝑛}_𝑛∈ℕ of linear maps from 𝐴 to 𝐵 such that, for every 𝑛∈ℕ,(i)‖𝑇_𝑛‖<𝐾₀,(ii)sup_‖𝑎‖=1‖𝑇_𝑛(𝑎⁻¹)−𝑇_𝑛(𝑎)⁻¹‖<1/𝑛,(iii)jmult(𝑇_𝑛(1)𝑇_𝑛)≥𝜀₀ orucomm(𝑇_𝑛)≥𝜀₀.
Consider that 𝐓=[𝑇_𝑛]∶𝐴^𝒰→𝐵^𝒰. We claim that 𝐓 strongly preserves invertibility. Indeed, let 𝐚∈𝐴^𝒰 be an invertible element. We can suppose, without loss of generality, that ‖𝐚‖=1. Let [𝑎_𝑛] be its normalized representative, with ‖𝑎_𝑛⁻¹‖<𝛼, for some 𝛼>0 (see Proposition 1 and Remark 2). As 𝑇_𝑛𝑎_𝑛⁻¹≤𝑇_𝑛𝑎_𝑛⁻¹<𝐾₀𝑇𝛼,_𝑛𝑎_𝑛⁻¹−𝑇_𝑛𝑎_𝑛⁻¹<1, (30) we get 𝑇_𝑛𝑎_𝑛⁻¹<𝐾₀𝛼+1, (31) for all 𝑛∈ℕ. Hence, 𝐓(𝐚) is invertible and [𝑇_𝑛(𝑎_𝑛)⁻¹] is its inverse. This yields 𝐓𝐚^−𝟏−𝐓(𝐚)⁻¹=lim_𝒰𝑇_𝑛𝑎_𝑛⁻¹−𝑇_𝑛𝑎_𝑛⁻¹≤lim_𝒰1𝑛=0. (32) Thus, 𝐓(𝐚^−𝟏)=𝐓(𝐚)⁻¹ for every invertible element 𝐚∈𝐴^𝒰. By [4, Theorem 2.2], 𝐓(𝟏)𝐓(𝐚²)=(𝐓(𝟏)𝐓(𝐚))² and 𝐓(𝟏)𝐓(𝐚)=𝐓(𝐚)𝐓(𝟏), for every 𝐚∈𝐴^𝒰. We apply [27, Lemmas 3.4] and Lemma 3 to obtain, respectively, the following: 0=jmult(𝐓(𝟏)𝐓)=lim_𝒰jmult𝑇_𝑛(1)𝑇_𝑛,0=ucomm(𝐓)=lim_𝒰ucomm𝑇_𝑛. (33) Consequently, 𝐼=𝑛∈ℕ∶jmult𝑇_𝑛(1)𝑇_𝑛<𝜀₀𝑇∈𝒰,𝐽=𝑛∈ℕ∶ucomm_𝑛<𝜀₀∈𝒰. (34) Finally, 𝐼∩𝐽∈𝒰gives us the desired contradiction.

Our goal now is to achieve a group invertibility version for the previous theorem. Recall that given an additive map 𝑇∶𝐴→𝐵 from a unital Banach algebra 𝐴 into a Banach algebra 𝐵, by [9, Theorem 2.4], if 𝑇 strongly preserves group invertibility, then 𝑇(1)𝑇 is a Jordan homomorphism and 𝑇(1) commutes with the range of 𝑇. 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 𝐴⁻¹.

Recall that the so-called Hua’s identity asserts that, if 𝑎,𝑏, and 𝑎−𝑏⁻¹ are invertible elements in a ring, then 𝑎⁻¹−𝑎−𝑏⁻¹⁻¹⁻¹=𝑎−𝑎𝑏𝑎. (35)

Theorem 5. Let 𝐴 and 𝐵 be Banach algebras, 𝐴 being unital, and 𝑇∶𝐴→𝐵 be an additive map such that𝑇(𝑎⁻¹)=𝑇(𝑎)^𝐺for all𝑎∈𝐴⁻¹. Then, 𝑇(1)𝑇is a Jordan homomorphism and 𝑇(1) commutes with𝑇(𝐴).

Proof. A look to the arguments employed in [9, Lemma 2.1], allows us to show that 𝑇 preserves the cubes of the invertible elements. Indeed, given 𝑢∈𝐴⁻¹ and 𝜆∈ℚ with 0<|𝜆|<‖𝑢⁻¹‖⁻², as 𝜆⁻¹𝑢 and 𝑢−𝜆𝑢⁻¹ are invertible elements, we can apply Hua’ s identity to obtain 𝑢⁻¹−𝑢−𝜆𝑢⁻¹⁻¹⁻¹=𝑢−𝑢𝜆⁻¹𝑢𝑢=𝑢−𝜆⁻¹𝑢³. (36) Let us assume that𝑇(𝑢)≠0. Since𝑇(𝑢)∈𝐵^𝐺, it follows that𝑇(𝑢)is invertible in the unital Banach algebra𝑝𝐵𝑝for𝑝=𝑇(𝑢)𝑇(𝑢)^𝐺, with inverse𝑇(𝑢)^𝐺. Identity (36) applied for𝑇(𝑢)and0<|𝜆|<‖𝑇(𝑢)^𝐺‖⁻²gives 𝑇(𝑢)−𝜆⁻¹𝑇(𝑢)³=𝑇(𝑢)^𝐺−𝑇(𝑢)−𝜆𝑇(𝑢)^𝐺𝐺𝐺. (37) Hence, for every 𝜆∈ℚ such that 0<|𝜆|<min{‖𝑢⁻¹‖⁻²,‖𝑇(𝑢)^𝐺‖⁻²}, we get 𝑇(𝑢)−𝜆⁻¹𝑇(𝑢)³=𝑇𝑢⁻¹−𝑇𝑢−𝜆𝑢^−1𝐺𝐺=𝑇𝑢⁻¹−𝑢−𝜆𝑢^−1−1𝐺=𝑇𝑢⁻¹−𝑢−𝜆𝑢⁻¹⁻¹⁻¹=𝑇𝑢−𝜆⁻¹𝑢³=𝑇(𝑢)−𝜆⁻¹𝑇𝑢³. (38) Hence, 𝑇(𝑢³)=𝑇(𝑢)³, as desired. From this last identity, reasoning as in [9, Proposition 2.3], we deduce that the following equalities hold for every 𝑥∈𝐴: 3𝑇(𝑥)=𝑇(𝟏)²𝑇(𝑥)+𝑇(𝑥)𝑇(𝟏)²(𝟏)𝑇(𝑥)𝑇(𝟏),𝑥+𝑇3𝑇²=𝑇(𝑥)²𝑇(𝟏)+𝑇(𝟏)𝑇(𝑥)²+𝑇(𝑥)𝑇(𝟏)𝑇(𝑥). (39) Finally, it only remains to repeat the arguments in(2)⇒(3)in [9, Theorem 2.4] to conclude the proof.

Now, we can state the following result.

Theorem 6. Let 𝐴 and 𝐵 be Banach algebras where 𝐴 is unital and 𝐾,𝜀>0. Then, there exists 𝛿>0 such that for every linear map 𝑇∶𝐴→𝐵 with ‖𝑇‖<𝐾, the condition sup_{‖𝑎‖=1,𝑎∈𝐴𝐺}𝑇𝑎^𝐺−𝑇(𝑎)^𝐺<𝛿 (40) implies that 𝑗𝑚𝑢𝑙𝑡(𝑇(𝟏)𝑇)<𝜀,𝑢𝑐𝑜𝑚𝑚(𝑇)<𝜀. (41)

Proof. First, notice that if 𝐛∈𝐴^𝒰 has a coset representative 𝐛=[𝑏_𝑛] where 𝑏_𝑛 is group invertible for every 𝑛∈ℕ and {𝑏^𝐺_𝑛}_𝑛∈ℕ is bounded, then 𝐛is group invertible and 𝐛^𝐺=[𝑏^𝐺_𝑛]. Hence, the same arguments used in Theorem 4 produce an operator𝐓=[𝑇_𝑛]∶𝐴^𝒰→𝐵^𝒰satisfying𝐓(𝐚⁻¹)=𝐓(𝐚)^𝐺for every invertible element 𝐚∈𝐴^𝒰. Now, Theorem 5 proves that𝐓(𝟏)𝐓is a Jordan homomorphism and𝐓(𝟏)commutes with𝐓(𝐴^𝒰). Again, the final argument in Theorem 4 completes the proof.

In [32, Proposition 2.5], the authors proved, in particular, that if an additive map𝑇∶𝐴→𝐵between unital Banach algebras satisfies 𝑇(𝑎)𝑇𝑎⁻¹=𝑇(1)²,forevery𝑎∈𝐴⁻¹, (42) and 𝑇(1)is invertible, then𝑇(1)⁻¹𝑇is a Jordan homomorphism and𝑇(1)commutes with𝑇(𝐴). It is clear now that for a sequence of linear operators𝑇_𝑛∶𝐴→𝐵satisfying that‖𝑇_𝑛‖,‖𝑇_𝑛(1)⁻¹‖<𝐾for all 𝑛∈ℕ, and 𝑇_𝑛(𝑎)𝑇_𝑛𝑎⁻¹−𝑇_𝑛(1)²<1𝑛∀𝑛∈ℕ, (43) its ultrapoduct𝐓∶𝐴^𝒰→𝐵^𝒰fulfills𝐓(𝐚)𝐓(𝐚⁻¹)=𝐓(𝟏)²for every invertible𝐚∈𝐴^𝒰. Therefore,𝐓(𝟏)⁻¹𝐓is a Jordan homomorphism and𝐓(𝟏)commutes with𝐓(𝐴^𝒰). This leads us to the following approximate formulation of [32, Proposition 2.5].

Theorem 7. Let 𝐴 and 𝐵be unital Banach algebras and 𝐾,𝜀>0. Then, there exists𝛿>0such that for every linear map𝑇∶𝐴→𝐵with‖𝑇‖,‖𝑇(1)⁻¹‖<𝐾, the following condition: sup_{𝑎∈𝐴−1}𝑇(𝑎)𝑇𝑎⁻¹−𝑇(1)²<𝛿 (44) implies that 𝑗𝑚𝑢𝑙𝑡𝑇(1)⁻¹𝑇<𝜀,𝑢𝑐𝑜𝑚𝑚(𝑇)<𝜀. (45)

4. Approximate Preservers in 𝐶^∗-Algebras

The aim of this section is twofold. On the one hand, we prove that linear maps approximately preserving generalized invertibility in 𝐶^∗-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 a𝐽𝐵^∗-triple𝐸, the triple cube of an element𝑥∈𝐸is defined as𝑥^[3]∶={𝑥,𝑥,𝑥}. An element satisfying𝑒^[3]=𝑒is called a tripotent. The following polarization identity allows us to write the triple product as linear combination of triple cubes: 𝑥,𝑦,𝑧=_{𝛼2=1𝛽4=1}𝛼𝛽𝑥+𝛼𝑦+𝛽𝑧^[3],for𝑥,𝑦,𝑧∈𝐸. (46) Hence, a linear map between 𝐽𝐵^∗-triples is a triple homomorphism if and only if it preserves triple cubes.

Each tripotent 𝑒 in 𝐸 gives rise to the so-called Peirce decomposition of 𝐸 associated to 𝑒, that is, 𝐸=𝐸₂(𝑒)⊕𝐸₁(𝑒)⊕𝐸₀(𝑒), (47) where for 𝑖=0,1,2,𝐸_𝑖(𝑒)is the𝑖/2eigenspace of 𝐿(𝑒,𝑒). The peirce space𝐸₂(𝑒)is a 𝐽𝐵^∗-algebra with product 𝑥⋅𝑦∶={𝑥,𝑒,𝑦} and involution𝑥^♯∶={𝑒,𝑥,𝑒}. Moreover, the triple product induced on𝐸₂(𝑒)by 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 element𝑎in a 𝐽𝐵^∗-triple𝐸, there exists a tripotent𝑒∈𝐸such that𝑎is a self-adjoint invertible element in the 𝐽𝐵^∗-algebra𝐸₂(𝑒). If𝑎is invertible, its inverse is denoted as usual by𝑎⁻¹. Moreover, if𝑎and𝑏are invertible elements in the Jordan algebra𝐽=(𝐽,∘)such that𝑎−𝑏⁻¹is also invertible, then𝑎⁻¹+(𝑏⁻¹−𝑎)⁻¹ is invertible and the Hua identity 𝑎⁻¹+𝑏⁻¹−𝑎⁻¹⁻¹=𝑎−𝑈_𝑎(𝑏) (48) holds, where𝑈_𝑎(𝑥)=2𝑎∘(𝑎∘𝑥)−𝑎²∘𝑥(see [33], (11)).

Let𝐴be a unital𝐶^∗-algebra, and𝑢∈𝐴^∧⧵{0}. Then, there exists a unique partial isometry𝑒, such that𝑢is self-adjoint and invertible in the Jordan algebra(𝐸_𝐴)₂(𝑒)=𝑒𝑒^∗𝐴𝑒^∗𝑒, with inverse𝑢^∧.Hence, for every𝜆∈ℂwith0<|𝜆|<‖𝑢^∧‖⁻², the element𝑢−𝜆𝑢^∧is invertible in𝑒𝑒^∗𝐴𝑒^∗𝑒. Reciprocally, the inverses of𝑢−𝜆𝑢^∧and𝑢^∧−(𝑢−𝜆𝑢^∧)^∧in𝑒𝑒^∗𝐴𝑒^∗𝑒are their generalized inverses in𝐴. By the Hua identity (48), we obtain 𝑢−𝜆⁻¹𝑢^[3]=𝑢^∧−𝑢−𝜆𝑢^∧∧∧. (49)

Theorem 8. Let𝐴and𝐵be 𝐶^∗-algebras,𝐴being unital, and𝑇∶𝐴→𝐵a bounded linear map satisfying𝑇(𝑥⁻¹)=𝑇(𝑥)^∧for every self-adjoint invertible element𝑥∈𝐴. Then, 𝑇is a triple homomorphism.

Proof. Arguing as in [9, Lemma 3.1], pick a self-adjoint invertible element𝑢∈𝐴. We may assume that𝑇(𝑢)≠0. Then, given𝜆∈ℂwith0<|𝜆|<min{‖𝑢⁻¹‖⁻²,‖𝑇(𝑢)^∧‖⁻²}, from identities (36) and (49), we get 𝑇(𝑢)−𝜆⁻¹𝑇(𝑢)^[3]=𝑇(𝑢)^∧−𝑇(𝑢)−𝜆𝑇(𝑢)^∧∧∧=𝑇(𝑢)−𝜆⁻¹𝑇𝑢³, (50) which shows that𝑇(𝑢³)=𝑇(𝑢)^[3]. Once we have proved that𝑇preserves cubes of self-adjoint invertible elements, given a self-adjoint element𝑎∈𝐴, and𝛼∈ℝwith|𝛼|>‖𝑎‖, as the element𝑎+𝛼is self-adjoint and invertible, we get𝑇((𝑎+𝛼)³)=𝑇(𝑎+𝛼)^[3]. Expanding this las equation we obtain 𝑇𝑎³+3𝛼𝑇𝑎²+3𝛼²𝑇(𝑎)+𝛼³𝑇(1)=𝑇(𝑎)^[3]+𝛼³𝑇(1){𝑇(𝑎)(𝑎)(1{𝑇(𝑎)(1)(𝑎+2𝛼,𝑇,𝑇)}+𝛼,𝑇,𝑇)}+2𝛼²{𝑇(1),𝑇(1),𝑇(𝑎)}+𝛼²{𝑇(1),𝑇(𝑎),𝑇(1)}, (51) for every𝑎∈𝐴, and|𝛼|>‖𝑎‖. From this, we deduce that𝑇(𝑎³)=𝑇(𝑎)^[3]for every𝑎∈𝐴. That is,𝑇preserves triple cubes of self-adjoint elements. By [34, Theorem 20],𝑇is a triple homomorphism.

Recall that we measure how close is a linear map𝑇∶𝐴→𝐵between 𝐶^∗-algebras to being a triple homomorphism or self-adjoint by the triple multiplicativity and the self-adjointness of𝑇, respectively as follows: (𝑇)𝑇−{𝑇(𝑎)(𝑏)(𝑐∶‖𝑎‖=‖𝑏‖=‖𝑐‖},(𝑇)tmult∶=sup{‖({𝑎,𝑏,𝑐}),𝑇,𝑇)}‖=1sa∶=sup_‖𝑎‖=1𝑇𝑎^∗∗−𝑇(𝑎). (52)

Remark 9. Let𝐴and𝐵be 𝐶^∗-algebras. It is clear that every Jordan∗-homomorphism𝑇∶𝐴→𝐵is a triple homomorphism. We ask whetherjmult(𝑇)and sa(𝑇)being small implytmult(𝑇)being small.
Let𝑇∶𝐴→𝐵be a bounded linear map. Define 𝑇_𝐽()()(𝑎)(𝑏),𝑇𝑎,𝑏∶=𝑇𝑎∘𝑏−𝑇∘𝑇_𝑇()−{𝑇(𝑎)(𝑏)(𝑐,𝑇𝑎,𝑏,𝑐∶=𝑇({𝑎,𝑏,𝑐}),𝑇,𝑇)}_∗(𝑎)∶=𝑇𝑎^∗∗−𝑇(𝑎), (53) for every𝑎,𝑏,𝑐∈𝐴. Then, 𝑇({𝑎,𝑏,𝑐})=𝑇𝑎∘𝑏^∗∘𝑐+𝑇𝑎∘𝑏^∗∘𝑐−𝑇(𝑎∘𝑐)∘𝑏^∗=𝑇𝑎∘𝑏^∗∘𝑇(𝑐)+𝑇_𝐽𝑎∘𝑏^∗(𝑎)𝑏,𝑐+𝑇∘𝑇^∗∘𝑐+𝑇_𝐽𝑎,𝑏^∗()𝑏∘𝑐−𝑇𝑎∘𝑐∘𝑇^∗−𝑇_𝐽𝑎∘𝑐,𝑏^∗=𝑇(𝑎)∘𝑇𝑏^∗∘𝑇(𝑐)+𝑇_𝐽𝑎,𝑏^∗∘𝑇(𝑐)+𝑇_𝐽𝑎∘𝑏^∗,𝑐+𝑇(𝑎)∘𝑇𝑏^∗(𝑐)(𝑎)∘𝑇+𝑇∘𝑇_𝐽𝑏^∗,𝑐+𝑇_𝐽𝑎,𝑏^∗−(𝑇(𝑎)(𝑐𝑏∘𝑐∘𝑇))∘𝑇^∗−𝑇_𝐽(𝑎,𝑐)∘𝑇𝑏^∗−𝑇_𝐽𝑎∘𝑐,𝑏^∗=𝑇(𝑎),𝑇𝑏^∗∗,𝑇(𝑐)+𝑇_𝐽𝑎,𝑏^∗∘𝑇(𝑐)+𝑇_𝐽𝑎∘𝑏^∗,𝑐+𝑇(𝑎)∘𝑇_𝐽𝑏^∗,𝑐+𝑇_𝐽𝑎,𝑏^∗∘𝑐−𝑇_𝐽(𝑎,𝑐)∘𝑇𝑏^∗−𝑇_𝐽𝑎∘𝑐,𝑏^∗. (54) Therefore, 𝑇_𝑇()=𝑇(𝑎)𝑎,𝑏,𝑐,𝑇_∗(𝑏),𝑇(𝑐)+𝑇_𝐽𝑎,𝑏^∗∘𝑇(𝑐)+𝑇_𝐽𝑎∘𝑏^∗(𝑎),𝑐+𝑇∘𝑇_𝐽𝑏^∗,𝑐+𝑇_𝐽𝑎,𝑏^∗∘𝑐−𝑇_𝐽(𝑎,𝑐)∘𝑇𝑏^∗−𝑇_𝐽𝑎∘𝑐,𝑏^∗. (55) This implies that tmult(𝑇)≤‖𝑇‖²sa(𝑇)+3(‖𝑇‖+1)jmult(𝑇). (56)

As in Lemma 3, it can be shown that, for an operator𝐓=[𝑇_𝑛], the following holds: lim_𝒰tmult𝑇_𝑛=tmult(𝐓),lim_𝒰sa𝑇_𝑛=sa(𝐓). (57)

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 map𝐓=[𝑇_𝑛]∶𝐴^𝒰→𝐵^𝒰between the ultrapowers fulfilling𝐓(𝐱⁻¹)=𝐓(𝐱)^∧for every self-adjoint invertible𝐱∈𝐴^𝒰. By Theorem 8,𝐓is a triple homomorphism.

Theorem 10. Let𝐴and𝐵be 𝐶^∗-algebras where𝐴is unital and𝐾,𝜀>0. Then, there exists𝛿>0such that for every linear map𝑇∶𝐴→𝐵with‖𝑇‖<𝐾, the condition sup_{‖𝑎‖=1,𝑎=𝑎∗}𝑇𝑎^∧−𝑇(𝑎)^∧<𝛿 (58) implies that 𝑡𝑚𝑢𝑙𝑡(𝑇)<𝜀. (59)

Remark 11. Notice that in Theorem 8 it can be also obtained that𝑇(1)^∗𝑇is a Jordan∗-homomorphism. Hence, the hypothesis of Theorem 10 also yields tojmult(𝑇(1)^∗𝑇)<𝜀andsa(𝑇(1)^∗𝑇)<𝜀.

Corollary 12. Let𝐴and𝐵be 𝐶^∗-algebras where𝐴is unital and𝐾,𝜀>0. Then, there exists𝛿>0such that for every linear map𝑇∶𝐴→𝐵with‖𝑇‖<𝐾, the conditions sup_{‖𝑎‖=1,𝑎∈𝐴†}𝑇𝑎^†−𝑇(𝑎)^†<𝛿,sa(𝑇)<𝛿 (60) imply that tmult(𝑇)<𝜀. (61)

Proof. Let us briefly sketch the proof: assuming the contrary, there exist𝐾₀,𝜀₀>0and a sequence{𝑇_𝑛}_𝑛∈ℕof linear maps from 𝐴 to𝐵such that, for every𝑛∈ℕ, 𝑇_𝑛<𝐾₀,sup_‖𝑎‖=1𝑇_𝑛𝑎^†−𝑇_𝑛(𝑎)^†<1𝑛,sa𝑇_𝑛<1𝑛,tmult𝑇_𝑛≥𝜀₀. (62) Then, 𝐓=[𝑇_𝑛]∶𝐴^𝒰→𝐵^𝒰is a self-adjoint map such that𝐓(𝐚⁻¹)=𝐓(𝐚)^†, for every invertible element𝐚∈𝐴^𝒰. In particular,𝐓(𝐚⁻¹)=𝐓(𝐚)^∧, for every invertible self-adjoint element𝐚∈𝐴^𝒰. From Theorem 8,𝐓is a triple homomorphism (Contradiction).

4.2. Maps Approximately Preserving the Conorm

Let𝐴and𝐵be unital𝐶^∗-algebras. Kadison proved in [35] that a surjective linear map𝑇∶𝐴→𝐵is an isometry if and only if𝑇is a Jordan∗-isomorphism multiplied by a unitary element in𝐵. In [30], the authors address the question of characterizing surjective linear maps preserving some spectral quantities. Given an element𝑎of a Banach algebra𝐴, the minimum modulus and the surjectivity modulus of𝑎are defined, respectively, by 𝑚(𝑎)‖‖𝑥‖},𝑞(𝑎)‖‖𝑥‖}.∶=inf{‖𝑎𝑥∶𝑥∈𝐴,=1∶=inf{‖𝑥𝑎∶𝑥∈𝐴,=1 (63) Obviously,𝑚(𝑎)=0(resp.,𝑞(𝑎)=0) if and only if𝑎is a left (resp., right) topological divisor of zero. It is well known that for any invertible element𝑎∈𝐴, 𝑚(𝑎)=𝑞(𝑎)=𝑎⁻¹⁻¹. (64) Moreover, 𝑚(𝑎)𝑚(𝑏)()≤‖𝑎‖𝑚(𝑏),|𝑚(𝑎)(𝑏≤‖‖,≤𝑚𝑎𝑏−𝑚)|𝑎−𝑏 (65) for every𝑎,𝑏∈𝐴. If𝐴is a𝐶^∗-algebra, then𝑚(𝑎)=𝑞(𝑎^∗). In particular,𝑚(𝑎)>0(resp.,𝑞(𝑎)>0) if and only if𝑎is left (resp., right) invertible.

Recall also from the introduction that the conorm of an element𝑎in a Banach algebra𝐴, is defined as 𝑐(𝑎)‖𝐿∶=inf‖𝑎𝑥∶dist𝑥,ker_𝑎≥1,if𝑎≠0,∞,if𝑎=0. (66) For a regular element𝑎in a𝐶^∗-algebra𝐴, 𝑐(𝑎)=𝑎^†−1. (67)

Let𝐴and𝐵be unital 𝐶^∗-algebras. By Theorems 3.1 and  3.2 in [30], if𝑇∶𝐴→𝐵is a linear map preserving any of these spectral quantities, then𝑇is an isometric Jordan *-homomorphism whenever𝑇is unital, and𝑇is an isometric Jordan *-homomorphism multiplied by a unitary element, whenever𝑇is 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. Let𝐴and𝐵be unital 𝐶^∗-algebras and𝑇∶𝐴→𝐵a unital linear map satisfying 𝑐(𝑇(𝑥))=𝑐(𝑥)for all𝑥∈𝐴⁻¹. Then, 𝑇is a Jordan *-homomorphism.

Proof. First, let us prove that𝑇is injective. Take𝑎₀∈𝐴such that𝑇(𝑎₀)=0and let𝛼∈ℂbe sufficiently small so that1+𝛼𝑎₀is invertible. Then, 1=𝑐(𝑇(1))=𝑐𝑇1+𝛼𝑎₀=𝑐1+𝛼𝑎₀. (68) In particular, we get 1≤𝑐1+𝑖𝑡𝑎₀,1≤𝑐1−𝑡𝑎₀, (69) as𝑡→0. Hence, by [30, Lemma 4.1], both𝑖𝑎₀and𝑎₀are self-adjoint and, consequently,𝑎₀=0.
We claim that𝑇is positive. Indeed, given a self-adjoint element𝑎∈𝐴, we know that 1+𝑜(𝑡)≤𝑐(1+𝑖𝑡𝑎),(as𝑡→0). (70) Since1+𝑖𝑡𝑎is invertible for𝑡∈ℝsmall enough, it follows that 1+𝑜(𝑡)≤𝑐(𝟏+𝑖𝑡𝑎)=𝑐(1+𝑖𝑡𝑇(𝑎)),(as𝑡→0). (71) This implies that𝑇(𝑎)is self-adjoint.
Moreover, given𝑥∈𝐴and𝜆∉𝜎(𝑥), there exists a neighborhood𝑈_𝜆of𝜆such that𝑈_𝜆∩𝜎(𝑥)=∅. If𝜆∉𝜎(𝑥), then𝜆∉𝜎_𝐾(𝑥), where𝜎_𝐾(𝑥)denotes the Kato spectrum of𝑥as follows: 𝜎_𝐾(𝑎)∶=𝜆∈ℂ∶lim_𝜇→𝜆𝑐𝑎−𝜇=0. (72) As for𝜇∈𝑈_𝜆, the element𝑥−𝜇is invertible; then, we have𝑐(𝑇(𝑥)−𝜇)=𝑐(𝑥−𝜇)for every𝜇∈𝑈_𝜆. Consequently, lim_𝜇→𝜆𝑐𝑇(𝑥)−𝜇>0, (73) and𝜆∉𝜎_𝐾(𝑇(𝑥)). Since𝜕𝜎(𝑎)⊆𝜎_𝐾(𝑎)⊆𝜎(𝑎)for every𝑎∈𝐴(see [36, Sections 12,13]), we have just proved that 𝜕𝜎(𝑇(𝑥))⊆𝜎_𝐾(𝑇(𝑥))⊆𝜎(𝑥), (74) for every𝑥∈𝐴. Being𝑇self-adjoint, this implies that𝑇is positive and hence,‖𝑇‖=1.
Arguing as in [30, Theorem 5.1], given a self-adjoint element𝑎∈𝐴and𝑡sufficiently small so that𝑢=𝑒^𝑖𝑡𝑎is a unitary element with spectrum strictly contained in the unit circle𝕋, since (𝑇(𝑢‖𝑇(𝑢𝑇(𝑢)𝜕𝜎))⫋𝕋,)‖≤1,₋₁=𝑐(𝑇(𝑢))⁻¹=𝑐(𝑢)⁻¹=1, (75) the element𝑇(𝑢)is unitary. From 1=𝑇(𝑢)𝑇(𝑢)^∗=𝑇(𝑢)𝑇𝑢^∗=(𝑎)−1𝟏+𝑖𝑡𝑇2𝑡²𝑇𝑎²×(𝑎)−1+⋯𝟏−𝑖𝑡𝑇2𝑡²𝑇𝑎²+⋯, (76) we deduce that𝑇(𝑎²)=𝑇(𝑎)²as desired.

Theorem 14. Let𝐴and𝐵be unital 𝐶^∗-algebras and 𝑇∶𝐴→𝐵 a surjective linear map satisfying𝑐(𝑇(𝑥))=𝑐(𝑥)for all𝑥∈𝐴⁻¹. Then, 𝑇is a Jordan *-homomorphism multiplied by a unitary element in𝐵.

Proof. First, let us prove that𝑏=𝑇(1)is invertible. Since𝑐(𝑇(1))=𝑐(1)=1,𝑏is regular. Let𝑦=1−𝑏𝑏^†, and𝑥∈𝐴such that𝑦=𝑇(𝑥). Notice that𝑏^∗𝑦=𝑦^∗𝑏=0.
For𝛼∈ℂsufficiently small such that1+𝛼𝑥∈𝐴⁻¹, 𝑐(1+𝛼𝑥)²=𝑐𝑏+𝛼𝑦²=𝑐𝑏𝑏^∗+|𝛼|²𝑦𝑦^∗. (77) Hence, lim_|𝛼|→0𝑐𝑏𝑏^∗+|𝛼|²𝑦𝑦^∗=lim_|𝛼|→0𝑐(1+𝛼𝑥)²=1=𝑐𝑏𝑏^∗. (78) Reasoning in a similar way to [30, Theorem 6.2], we get 𝑐(1+𝛼𝑥)²=𝑐𝑏𝑏^∗+|𝛼|𝑦𝑦^∗≥1−|𝛼|²𝑦², (79) and therefore 𝑐(1+𝑖𝑡𝑥)≥1−𝑡²𝑦^21/2,𝑐(1−𝑡𝑥)≥1−𝑡²𝑦^21/2, (80) for small enough 𝑡∈ℝ. From these inequalities, we get, respectively, that𝑥and𝑖𝑥are self-adjoint. This shows that𝑥=0and thus𝑦=0. Consequently,1=𝑏𝑏^†, that is,𝑏is right invertible. Similarly it can be proved that𝑏is left invertible.
Note that, as in the previous theorem,𝑇is injective. Therefore𝑆∶=𝑏⁻¹𝑇is a unital and bijective linear map satisfying 𝑚(𝑆(𝑥))=𝑚𝑏⁻¹𝑇(𝑥)≤𝑏⁻¹𝑚(𝑇(𝑥(𝑇(𝑥(𝑇(𝑥(𝑥)))=𝑚))≤𝑐))=𝑐,∀𝑥∈𝐴⁻¹. (81)
Let𝑦be a self-adjoint element in𝐵and𝑡∈ℝsmall such that1+𝑖𝑡𝑆⁻¹(𝑦)is invertible. Taking𝑥=1+𝑖𝑡𝑆⁻¹(𝑦)in the previous identity, we have 𝑚1+𝑖𝑡𝑦≤𝑐1+𝑖𝑡𝑆⁻¹𝑦. (82)
It follows that𝑆⁻¹is self-adjoint and so is𝑆.
We claim that𝑆is positive. Note that for every𝑥∈𝐴^†and𝑢∈𝐴⁻¹, it is clear that𝑢𝑥∈𝐴^†, with (𝑢𝑥)𝑥^†𝑢⁻¹()𝑥𝑢𝑥=𝑢𝑥,_†𝑢⁻¹(𝑢𝑥)𝑥^†𝑢⁻¹=𝑥^†𝑢⁻¹. (83) This implies, by [11, Theorem 2], the following: 1𝑥^†𝑢⁻¹()≤𝑥≤𝑐𝑢𝑥_†𝑢⁻¹𝑢𝑥𝑢𝑥𝑥^†𝑢⁻¹𝑥^†𝑢⁻¹≤‖𝑢‖𝑢⁻¹𝑥^†𝑢⁻¹. (84) Hence, for every𝑥∈𝐴⁻¹, we have 𝑐(𝑆(𝑥))=𝑐𝑏⁻¹𝑇(𝑥≥1)𝑇(𝑥)^†𝑏≥1‖𝑏‖𝑇(𝑥)^†=‖𝑏‖⁻¹𝑐(𝑇(𝑥,𝑐(𝑆(𝑥𝑏))))=𝑐⁻¹𝑇(𝑥)≤𝑏⁻¹‖𝑇(1)‖𝑇(𝑥)^†𝑏≤‖𝑏‖∣𝑏⁻¹²𝑐(𝑇(𝑥)). (85) So, we have shown so far the following: ‖𝑏‖⁻¹𝑐(𝑥)≤𝑐(𝑆(𝑥))≤‖𝑏‖𝑏⁻¹²𝑐(𝑥),∀𝑥∈𝐴⁻¹. (86) The first inequality can be used to show the following: 𝜕𝜎(𝑆(𝑥))⊂𝜎_𝐾(𝑆(𝑥))⊆𝜎(𝑥), (87) in a similar way as in the previous theorem. As a consequence,𝑆is positive. In order to conclude that𝑆is an isometric Jordan∗-isomorphism, it suffices to prove that𝑆⁻¹is also positive (see for instance [35, Corollary 5]). So, letℎ=𝑆(𝑎)be a positive element. As𝑆⁻¹is self-adjoint,𝑎is self-adjoint. We can therefore write𝑎=𝑥−𝑦, where𝑥 and 𝑦are positive elements and𝑥𝑦=𝑦𝑥=0. For every𝜇∈ℂ, we have 𝜕𝜎𝑆(𝑥)+𝜇𝑆𝑦⊂𝜎_𝐾𝑆(𝑥)+𝜇𝑆𝑦⊂𝜎𝑥+𝜇𝑦⊂ℝ∪𝜇ℝ. (88) (Recall that if𝑤𝑧=𝑧𝑤=0, then𝜎(𝑤+𝑧)⧵{0}=(𝜎(𝑤)⧵{0})∪(𝜎(𝑧)⧵{0}).) The previous spectral inclusion gives 𝜎𝑆(𝑥)+𝜇𝑆𝑦⊂ℝ∪𝜇ℝ. (89)
By Lemmas B and C in [37], we get𝑆(𝑦)=0and so𝑦=0. Consequently,𝑎is positive as desired. We conclude the proof by showing that𝑏is unitary. Indeed, since𝑆is a Jordan∗-isomorphism and𝑇(𝑥)=𝑏𝑆(𝑥), it is clear that, for every𝑥∈𝐴⁻¹,𝑇(𝑥)is invertible with inverse𝑇(𝑥)⁻¹=𝑆(𝑥⁻¹)𝑏⁻¹. Moreover,𝑐(𝑇(𝑥))=𝑐(𝑥); that is, 𝑇(𝑥)⁻¹=𝑥⁻¹, (90) for every𝑥∈𝐴⁻¹. Since𝑆is an isometry, 𝑆𝑥⁻¹=𝑥⁻¹=𝑆𝑥⁻¹𝑏⁻¹, (91) or equivalently, 𝑦=𝑦𝑏⁻¹, (92) for every𝑦∈𝐵⁻¹. This yields that𝑏is unitary.

Note that, for every invertible element𝐚∈𝐴^𝒰, where𝐚=[𝑎_𝑛]is a normalized representative, 𝑐(𝐚)=𝐚⁻¹⁻¹=lim_𝒰𝑎_𝑛⁻¹⁻¹=lim_𝒰𝑐𝑎_𝑛. (93) We are now in position to prove the main results in this section.

Theorem 15. Let𝐴and𝐵be unital 𝐶^∗-algebras and𝐾,𝜀>0. Then, there exists𝛿>0such that for every linear map𝑇∶𝐴→𝐵with‖𝑇‖<𝐾, the following conditions: sup_{‖𝑎‖=1,𝑎∈𝐴−1}|𝑐(𝑇(𝑎))−𝑐(𝑎)|<𝛿,‖𝑇(1)−1‖<𝛿 (94) imply that 𝑗𝑚𝑢𝑙𝑡(𝑇)<𝜀,𝑠𝑎(𝑇)<𝜀. (95)

Proof. Note that if 𝐛∈𝐵^𝒰has a coset representative𝐛=[𝑏_𝑛]where𝑏_𝑛has Moore-Penrose inverse for every𝑛∈ℕand{𝑏^†_𝑛}_𝑛∈ℕis bounded, then𝐛is Moore-Penrose invertible and𝐛^†=[𝑏^†_𝑛].
As we did above, suppose that the assertion is false, that is, we can find𝐾₀,𝜀₀>0and a sequence{𝑇_𝑛}_𝑛∈ℕof linear maps from𝐴to𝐵satisfying(i)‖𝑇_𝑛‖≤𝐾₀,(ii)sup_{‖𝑎‖=1,𝑎∈𝐴}⁻¹|𝑐(𝑇_𝑛(𝑎))−𝑐(𝑎)|<1/𝑛, ‖𝑇_𝑛(1)−1‖<1/𝑛,(iii)jmult(𝑇_𝑛)≥𝜀₀orsa(𝑇_𝑛)≥𝜀₀,
for every𝑛∈ℕ. Consider𝐓=[𝑇_𝑛]∶𝐴^𝒰→𝐵^𝒰. We claim that𝐓is unital and preserves the conorm of invertible elements. On the one hand, ‖𝐓(𝟏)−𝟏‖=lim_𝒰𝑇_𝑛(1)−1≤lim_𝒰1𝑛=0, (96) so𝐓(𝟏)=𝟏. On the other hand, for an invertible element𝐚∈𝐴^𝒰with norm 1, let[𝑎_𝑛]be its normalized representative:‖𝑎_𝑛‖=1and‖𝑎_𝑛⁻¹‖<𝛼for some positive𝛼. We know that 𝑐𝑎_𝑛−1𝑛<𝑐𝑇_𝑛𝑎_𝑛,forevery𝑛∈ℕ. (97) Hence𝑇_𝑛(𝑎_𝑛)∈𝐴^†, with‖𝑇_𝑛(𝑎_𝑛)^†‖<2𝛼, for𝑛>2𝛼. That is,{𝑇_𝑛(𝑎_𝑛)}is Moore-Penrose invertible almost everywhere, with their respective Moore-Penrose inverses uniformly bounded in norm. This ensures that𝐓(𝐚)is Moore-Penrose invertible and𝑐(𝐓(𝐚))=lim_𝒰𝑐(𝑇_𝑛(𝑎_𝑛)). Finally, Consider the following: |𝑐(𝐓(𝐚))−𝑐(𝐚)|=lim_𝒰𝑐𝑇_𝑛𝑎_𝑛−𝑐𝑎_𝑛≤lim_𝒰1𝑛=0. (98) By Theorem 13,𝐓is a Jordan *-homomorphism, which gives the contradiction.

Let 𝑋 and 𝑌 be complex Banach spaces, and letℬ(𝑋,𝑌)be the algebra of all bounded linear operators from𝑋to𝑌. Recall that the surjectivity modulus of 𝑇 is given by𝑞(𝑇)∶=sup{𝜀≥0∶𝜀𝐵_𝑌⊆𝑇(𝐵_𝑋)}, whereas usual𝐵_𝑋denotes the closed unit ball of𝑋. Note that𝑞(𝑇)>0 if and only if 𝑇 is surjective, and𝑞(𝑇)=inf{‖𝑆𝑇‖∶𝑆∈ℬ(𝑋),‖𝑆‖=1} (see [36, Theorem II.9.11]).

Theorem 16. Let𝐴and𝐵be unital 𝐶^∗-algebras and𝐾,𝜀>0. Then, there exists𝛿>0such that for every linear map𝑇∶𝐴→𝐵with‖𝑇‖<𝐾and 𝑞(𝑇)>𝐾⁻¹, the following condition: sup_{‖𝑎‖=1,𝑎∈𝐴−1}|𝑐(𝑇(𝑎))−𝑐(𝑎)|<𝛿 (99) implies that 𝑗𝑚𝑢𝑙𝑡𝑇(𝟏)^∗𝑇<𝜀,𝑠𝑎𝑇(𝟏)^∗𝑇<𝜀. (100)

Proof. If we assume the contrary, hence, as above we find𝐾₀,𝜀₀>0and a sequence{𝑇_𝑛}_𝑛∈ℕof linear maps from𝐴to𝐵satisfying(i)‖𝑇_𝑛‖≤𝐾₀,(ii)sup_{‖𝑎‖=1,𝑎∈𝐴}⁻¹|𝑐(𝑇_𝑛(𝑎))−𝑐(𝑎)|<1/𝑛, 𝑞(𝑇_𝑛)≥𝐾₀⁻¹,(iii)jmult(𝑇_𝑛(1)^∗𝑇_𝑛)≥𝜀₀orsa(𝑇_𝑛(1)^∗𝑇_𝑛)≥𝜀₀,
for every𝑛∈ℕ.
As in the previous theorem, the map𝐓=[𝑇_𝑛]preserves the conorm of all invertible elements. Moreover, 𝑞(𝐓)=lim_𝒰𝑞𝑇_𝑛≥𝐾₀⁻¹>0, (101) and thus𝐓is surjective. By Theorem 14,𝐓(𝟏)is unitary and𝐓(𝟏)^∗𝐓is a unital Jordan *-isomorphism.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors were partially supported by I+D MEC projects no. MTM 2011-23843, MTM 2010-17687, and Junta de Andalucía Grants FQM 375 and FQM 3737. The second author is also supported by a Plan Propio de Investigación grant from University of Almería. The authors would like to thank the referees for their comments and suggestions.