() CUBO A Mathematical Journal Vol.17, No¯ 01, (29–40). March 2015 Maps preserving Fredholm or semi-Fredholm elements relative to some ideal Mohadeseh Rostamani Department of Pure Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran. mohadeseh.rostamani@gmail.com Shirin Hejazian Department of Pure Mathematics, Ferdowsi University of Mashhad, Tusi Mathematical Research Group (TMRG), Mashhad, Iran. hejazian@um.ac.ir ABSTRACT We consider the Calkin algebra CR(A) and the Fredholm theory in a Banach algebra A, relative to some fixed ideal F of A. Our aim is to study linear maps between unital Banach algebras A and B which are surjective up to the inessential elements relative to F, and preserve Fredholm or semi-Fredholm elements in both directions or equivalently different relatively essential spectral sets such as essential spectrum, left or right essential spectrum, the boundary of essential spectrum or the full essential spectrum. We characterize such mappings when one of CR(A) or CR(B) is commutative and also investigate similar problems when A is assumed to be a unital C∗-algebra of real rank zero and B is an arbitrary Banach algebra. RESUMEN Consideramos el álgebra de Calkin CR(A) y la teoŕıa de Fredholm en un álgebra de Ba- nach A relativa a algún ideal fijo F de A. Nuestra meta es estudiar aplicaciones lineales entre álgebras de Banach unitales A y B las cuales son sobrejectivas salvo los elementos no esenciales relativos a F y preservan los elementos de Fredholm o semi-Fredholm en ambas direcciones o equivalentemente conjuntos espectrales esenciales relativos difer- entes tales como el espectro esencial izquierdo o derecho, la frontera del espectro esencial o el espectro esencial completo. Caracterizamos dichas aplicaciones cuando uno de los CR(A) o CR(B) es conmutativo e investigamos problemas similares cuando A se asume que es una C∗-álgebra unital de rango real cero y B es una álgebra de Banach cualquiera. Keywords and Phrases: Linear preservers, Fredholm element, semi-Fredholm element, inessen- tial ideal, relative Calkin algebra. 2010 AMS Mathematics Subject Classification: 47B49, 47A10. 30 Mohadeseh Rostamani & Shirin Hejazian CUBO 17, 1 (2015) 1 Introduction Let A and B be algebras, and let P be a property. We say that a linear map θ : A → B (i) preserves property P, if for each a ∈ A, θ(a) has property P whenever a has property P; (ii) preserves property P in both directions, if for each a ∈ A, θ(a) has property P if and only if a has this property. In operator theory, one of the most active research areas is linear preservers. The main problems in this subject concern with characterizing those linear maps that leave certain properties invariant. In the context of operator algebras, some of the most well known preserving problems deal with characterizing linear maps on the algebra of all bounded linear operators acting on a Banach space which preserve Fredholm, semi-Fredholm or generalized invertible operators. New contributions to the study of linear preserver problems in L(H), the algebra of all bounded linear operators on an infinite dimensional complex Hilbert space H, have been recently made by several authors, see [3, 4, 5, 12, 13, 14]. In [11], Kim and Park investigated linear maps φ on a unital C∗-algebra A of real rank zero that are surjective up to some fixed closed ideal I and π(a) is invertible in A/I if and only if π(φ(A)) is invertible in A/I, where π : A → A/I is the canonical quotient map. In this paper we consider Fredholm theory in a Banach algebra A relative to some fixed ideal F, c.f. [16], and investigate linear preservers between Banach algebras. In Section 2 we give some preliminaries concerning Fredholm theory in Banach algebras and the ideal of inessential elements relative to a fixed ideal. We will also consider the relative Calkin algebra of a Banach algebra with respect to some fixed ideal. Section 3 is devoted to investigating linear maps between two unital Banach algebras A and B which are surjective up to inessential elements and preserve any of the sets of Fredholm, left semi-Fredholm, right semi-Fredholm or semi-Fredholm elements in both directions (equivalently, preserves different spectral sets). We characterize such mappings whenever one of the relative Calkin algebras in A or B is assumed to be commutative. In Section 4 we consider similar linear preservers between unital C*-algebras or from a unital C*-algebra of real rank zero to an arbitrary unital Banach algebra. In each case we show that a linear map which is surjective up to inessential ideal and preserves a certain essential spectral set, induces a continuous Jordan isomorphism between the corresponding relative Calkin algebras. 2 Preliminaries Throughout this section A is a unital Banach algebra with unit element 1 and F is a fixed two sided ideal of A. The elements of F are called finite elements of A. Set I(A) = k(h(F)) := ⋂ {P ∈ ΠA; F ⊆ P}, where ΠA denotes the set of all primitive ideals of A. By [15, Theorem 4.3], I(A) = {a ∈ A; a + F ∈ rad(A/F)}. CUBO 17, 1 (2015) Maps preserving Fredholm or semi-Fredholm elements . . . 31 Clearly I(A) is a closed two sided ideal of A containing F. We call I(A) the inessential ideal of A (relative to F), and the elements of I(A) are called the inessential elements relative to F. Definition 2.1. An element a ∈ A is said to be left (resp. right) semi-Fredholm (relative to F), if it is left (resp. right) invertible modulo F, and a is called Fredholm if it is invertible modulo F. An element a ∈ A is a semi-Fredholm or an Atkinson element if it is either left or right semi-Fredholm. We denote by Φl(A) (resp. Φr(A)) the set of all left (resp. right) semi-Fredholm elements of A. Also we denote by Φ(A) := Φl(A) ∩ Φr(A) and Ψ(A) := Φl(A) ∪ Φr(A) the sets of Fredholm and semi-Fredholm elements of A, respectively. By [2, BA.2.4] an element a ∈ A is left (resp. right) invertible modulo F if and only if it is left (resp. right) invertible modulo I(A). Since I(A) is a closed ideal it is easy to see that Φ(A),Φl(A) and Φr(A) are open multiplicative semigroups of A and adding I(A) to each of these classes leaves it stable. For more about Fredholm theory in Banach algebras see [2, 15, 16]. Definition 2.2. The quotient algebra A/I(A), denoted by CR(A), is called the relative Calkin algebra of A with respect to F. The relative Calkin algebra of A is semisimple; see [16, Theorem 3.2 (vi)]. We recall that if A has minimal left ideals the smallest left ideal containing all of them is called the left socle of A. The right socle is similarly defined in terms of right ideals. If A has both minimal left and minimal right ideals, and if the left socle coincides with the right socle, it is called the socle of A denoted by soc(A). In this case we say for brevity that soc(A) exists. If A has no minimal left or minimal right ideals we put soc(A) = {0}. Clearly the socle, if it exists, is a non-zero ideal of A. If A is a semiprime algebra which has minimal left ideals, then soc(A) exists (see [2, BA.3.3]). When A is a semisimple Banach algebra, as a special case in Definition 2.2, we may assume F to be the socle of A. In this case the relative Calkin algebra coincides with the generalized Calkin algebra as in [4]. For x,y ∈ A let x ◦ y = x + y − xy denote the quasi product of x and y. Set Q(A) := {x ∈ A; there exist u,v ∈ A such that u ◦ x,x ◦ v ∈ F}. By [16, Theorem 3.2 (viii)], I(A) is the largest ideal of A contained in Q(A). Lemma 2.3. Let A be a unital Banach algebra. Then I(A) = {x ∈ A; u + x ∈ Φ(A) for all u ∈ Φ(A)} = {x ∈ A; u + x ∈ Φl(A) for all u ∈ Φl(A)} = {x ∈ A; u + x ∈ Φr(A) for all u ∈ Φr(A)} = {x ∈ A; u + x ∈ Ψ(A) for all u ∈ Ψ(A)}. 32 Mohadeseh Rostamani & Shirin Hejazian CUBO 17, 1 (2015) Proof. The first equality is [16, Theorem 3.5]. We prove the second equality; the others are proved similarly. Let J = {x ∈ A;u + x ∈ Φl(A) for all u ∈ Φl(A)}. Clearly I(A) ⊆ J. A modification of the proof of [16, Theorem 3.5] shows that J is an ideal of A. We show that J ⊆ Q(A). Let x ∈ J, then 1 − x ∈ Φl(A). Thus there exists y ∈ A such that (1 − y)(1 − x) − 1 ∈ F. Since F ⊆ J we have y ∈ J. Therefore there exists z ∈ A such that (1−z)(1−y) −1 ∈ F. This shows that 1−y+F is both left and right invertible in A/F with right inverse 1−x+F. It follows that 1−x+F is invertible in A/F and so x ∈ Q(A), therefore J ⊆ Q(A). Now [16, Theorem 3.2 (viii)] implies that J = I(A). In [4, Lemma 3.2] a different method has been applied to prove the same equalities as in Lemma 2.3 when A is assumed to be semisimple and F is assumed to be the socle of A. However, it should be emphasized that semisimplicity has been used just to guarantee the existence of the socle. For any a ∈ A σ(a) := {λ ∈ C : a − λ1 is not invertible}, σl(a) := {λ ∈ C : a − λ1 is not left invertible}, σr(a) := {λ ∈ C : a − λ1 is not right invertible}, denote the spectrum, the left spectrum and the right spectrum of a, respectively. Also ∂σ(a) and r(a) are the boundary of the spectrum and the spectral radius of a, respectively. We recall that for every compact set K ⊆ C, the polynomial convex hull ηK, is defined by ηK = {z ∈ C : |p(z)| ≤ sup ξ∈K |p(ξ)|, for each polynomial p}. For an element a in a Banach algebra, the polynomial convex hull ησ(x) of σ(x) is called the full spectrum of a. Let πA : A → CR(A) denote the canonical quotient map. For a ∈ A, σe(a) := σ(πA(a)), σle(a) := σl(πA(a)), σre(a) := σr(πA(a)), ησe(a) := ησ(πA(a)) and re(a) := r(πA(a)) are called, respectively, the essential spectrum, the left essential spectrum, the right essential spectrum, the full essential spectrum and the essential spectral radius of a relative to F. Let Λ(.) denote any of the spectral sets σe(.),σle(.), σre(.),σle(.) ∩ σre(.),∂σe(.) and ησe(.). A linear map θ : A → B is said to be Λ(.)-preserving if Λ(θ(x)) = Λ(x) for all x ∈ A, and θ is called essentially spectrally bounded if there exists a positive constant M such that re(θ(x)) ≤ Mre(x) for all x ∈ A. If re(θ(x)) = re(x) for all x ∈ A, θ is called an essential spectral isometry. Throughout this paper we will always suppose the following. Assumption A. We assume that A and B are unital Banach algebras and denote by 1 the unit element both in A and B. We consider arbitrary fixed ideals F and F′ in A and B, respectively. The inessential ideals in A and B relative to F and F′ are denoted by I(A) and I(B), respectively. CR(.) CUBO 17, 1 (2015) Maps preserving Fredholm or semi-Fredholm elements . . . 33 denotes the relative Calkin algebra in the corresponding Banach algebra. Note that when we use the terms Fredholm, left semi-Fredholm, right semi-Fredholm or semi-Fredholm, we always mean relative to the specific fixed ideal considered in the corresponding algebra. For a linear map θ : A → B, N(θ) will denote the null space of θ. If θ(1) = 1, θ is called unital. We say that θ is surjective up to inessential ideal if B = θ(A) + I(B). If θ(a2) = θ(a)2 for all a ∈ A (equivalently, θ(ab + ba) = θ(a)θ(b) + θ(b)θ(a) for all a,b ∈ A), then θ is said to be a Jordan homomorphism. A Jordan isomorphism is a bijective Jordan homomorphism. 3 Fredholm and semi-Fredholm preservers We consider Assumption A and begin with the following lemma. Note that there is no assumption of semisimplicity neither for A nor for B. The proof of the following lemma is a modification of a part of the proof of [3, Theorem 1.1]. We give the proof for the sake of convenience. Lemma 3.1. If the linear map θ : A → B is surjective up to the inessential ideal and essentially spectrally bounded then θ(I(A)) ⊆ I(B). Proof. Assume that there is a positive constant M such that re(θ(a)) ≤ Mre(a) for all a ∈ A. Take u ∈ I(A) and let b ∈ B be an arbitrary element. Since θ is surjective up to the inessential ideal, there exist x ∈ A and y ∈ I(B) such that b = θ(x) + y. For every λ ∈ C, we have r(λπB(θ(u)) + πB(b)) = r(πB(λθ(u) + b)) = re(λθ(u) + b) = re(θ(λu + x) + y) = re(θ(λu + x)) ≤ Mre(λu + x) = Mre(x). The function λ 7→ r(λπB(θ(u)) + πB(b)) is subharmonic on C, and so by Liouville’s Theorem r(πB(θ(u)) + πB(b)) = r(πB(b)). Zemanek’s characterization of the radical (see [1, Theorem 5.3.1]), implies that πB(θ(u)) ∈ rad(CR(B)) = {0} and thus θ(u) ∈ I(B). Lemma 3.2. Let the linear map θ : A → B be surjective up to the inessential ideal. If θ preserves any of the sets of Fredholm, left semi-Fredholm, right semi-Fredholm or semi-Fredholm elements in both directions, then (i) θ(I(A)) ⊆ I(B); (ii) N(θ) ⊆ I(A). Proof. (i) If θ(1) = 1 (mod I(B)), then re(θ(a)) = re(a) for all a ∈ A and the result follows by Lemma 3.1. Otherwise, let S(.) denote any of the sets of Fredholm, left semi-Fredholm, right semi- Fredholm or semi-Fredholm elements in the corresponding Banach algebra. Let x ∈ I(A). Then we show that θ(x) ∈ I(B). Suppose u ∈ S(B). Since θ is surjective up to the inessential ideal I(B), 34 Mohadeseh Rostamani & Shirin Hejazian CUBO 17, 1 (2015) there exist u′ ∈ A and x′ ∈ I(B) such that u = θ(u′)+x′. It follows that θ(u′) = u−x′ ∈ S(B) and so u′ ∈ S(A). Now u+θ(x) = θ(u′)+x′ +θ(x) = θ(u′ +x)+x′ ∈ S(B). Thus, for every u ∈ S(B), u + θ(x) ∈ S(B). Lemma 2.3 implies that θ(x) ∈ I(B), which establishes that θ(I(A)) ⊆ I(B). (ii) Let S(.) be as above. If x ∈ N(θ) and u ∈ S(A), then θ(x + u) = θ(u) ∈ S(B). Thus, for all u ∈ S(A), x + u ∈ S(A) and by Lemma 2.3, x ∈ I(A). Remark 3.3. By [1, Corollary 5.2.3] a semisimple Banach algebra A is commutative if and only if the spectral radius is subadditive on A, that is, there exists M > 0 such that r(x + y) ≤ M(r(x) + r(y)) (x,y ∈ A). Now, let A and B be semisimple Banach algebras and let θ : A → B be a surjective linear map which preserves spectral radius. It is easy to see that commutativity of either of A or B, implies commutativity of the other one. Theorem 3.4. Let at least one of CR(A) or CR(B) be commutative. Suppose that θ : A → B is a linear map which is surjective up to the inessential ideal. Then the following assertions are equivalent. (i) θ preserves the set of Fredholm elements in both directions; (ii) θ(I(A)) ⊆ I(B) and the induced map θ̂ : CR(A) → CR(B), θ̂ ◦ πA = πB ◦ θ, is a continuous isomorphism multiplied by an invertible element in CR(B). Proof. (i) ⇒ (ii). By Lemma 3.2, θ(I(A)) ⊆ I(B)); thus θ induces a linear map θ̂ : CR(A) → CR(B) such that θ̂ ◦ πA = πB ◦ θ. Clearly θ̂ is surjective. Since θ(1) is a Fredholm element, πB(θ(1)) is invertible in CR(B). Take s ∈ B such that πB(s) = πB(θ(1)) −1 in CR(B). Consider ψ := LπB(s) ◦ θ̂ : CR(A) → CR(B), where the linear mapping LπB(s) is the left multiplication by πB(s) in CR(B). Then ψ(πA(x)) = πB(sθ(x)) for all x ∈ A. Now it is easy to see that ψ is surjective, ψ(πA(1)) = πB(1) and πA(x) ∈ Inv(CR(A)) ⇐⇒ x ∈ Φ(A) ⇐⇒ θ(x) ∈ Φ(B) ⇐⇒ πB(θ(x)) ∈ Inv(CR(B)) ⇐⇒ πB(s)πB(θ(x)) ∈ Inv(CR(B)) ⇐⇒ ψ(πA(x)) ∈ Inv(CR(B)). Thus ψ is unital and preserves the set of invertible elements in both directions and so it preserves the spectrum. The Banach algebras CR(A) and CR(B) are semisimple, so by Remark 3.3 both CR(A) and CR(B) are commutative. It follows from [1, Theorem 4.1.17] that ψ is an isomorphism which is continuous by semisimplicity of CR(B). The assertion (ii) ⇒ (i) is obvious. We recall that a C∗-algebra A is said to be of real rank zero, if the set of all invertible selfadjoint elements of A is dense in the set of selfadjoint elements. Suppose that A is a C*-algebra of real rank zero and B is a semisimple Banach algebra. Let I(A) and I(B) denote the inessential ideals CUBO 17, 1 (2015) Maps preserving Fredholm or semi-Fredholm elements . . . 35 relative to soc(A) and soc(B), respectively. It is proved in [4, Corollary 3.5] that if θ : A → B is a linear map which is surjective up to I(B) and if θ preserves any of the sets Φl(A), Φr(A) or Ψ(A) in both directions, then θ(I(A)) ⊆ I(B) and the induced map θ̂ : C(A) → C(B) is a continuous Jordan isomorphism multiplied by an invertible element in C(B). Here we prove a similar result in a different setting. Proposition 3.5. Let A, B satisfy Assumption A. Suppose that θ : A → B is a linear map which is surjective up to the inessential ideal. If θ preserves any of the sets of left semi-Fredholm, right semi- Fredholm or semi-Fredholm elements in both directions, then θ(I(A)) ⊆ I(B) and θ preserves the set of Fredholm elements in both directions. Moreover, if one of CR(A) or CR(B) is commutative, then the induced map θ̂ : CR(A) → CR(B) is a continuous isomorphism multiplied by an invertible element in CR(B). Proof. By assumption and Lemma 3.2, θ̂ : CR(A) → CR(B) is well defined. Since CR(A),CR(B) are semisimple, [4, Theorem 2.2] implies that θ̂ preserves invertibility in both directions. Therefore θ preserves the set of Fredholm elements in both directions. The last assertion follows from Theorem 3.4. We will need the following theorem in the sequel. We recall that a linear map θ : A → B preserves idempotents if for each idempotent e in A, θ(e) is an idempotent in B. Theorem 3.6. [10, Corollary 2.3] Let A and B be two semisimple Banach algebras and let Λ(.) denote any of the spectral sets σ(.),σl(.),σr(.),σl(.)∩σr(.),∂σ(.) and ησ(.). Suppose that θ : A → B is a surjective linear map. If Λ(θ(x)) = Λ(x) for every x ∈ A, then θ is a bijective linear map preserving idempotents. In the above theorem, since Λ(θ(x)) = Λ(x) for one of the spectral sets, we have θ(1) = 1. Remark 3.7. Let K ⊆ C be compact. It is well known that ∂η(K) is the boundary of the unbounded component of C \ K which is called the outer boundary of K. Thus max{|z| : z ∈ K} = max{|z| : z ∈ ∂η(K)}. Theorem 3.8. Let CR(A) or CR(B) be commutative. Suppose that θ : A → B is a linear map which is surjective up to the inessential ideal. The following assertions are equivalent. (i) θ preserves the set of Fredholm elements in both directions and θ(1) = 1(mod I(B)). (ii) θ(I(A)) ⊆ I(B) and the induced map θ̂ : CR(A) → CR(B), θ̂ ◦ πA = πB ◦ θ, is a continuous unital isomorphism. (iii) θ is σe-preserving; (iv) θ is σle-preserving; (v) θ is σre-preserving; 36 Mohadeseh Rostamani & Shirin Hejazian CUBO 17, 1 (2015) (vi) θ is (σle ∩ σre)-preserving; (vii) θ preserves the set of left semi-Fredholm elements in both directions and θ(1) = 1(mod I(B)). (viii) θ preserves the set of right semi-Fredholm elements in both directions and θ(1) = 1(mod I(B)). (ix) θ preserves the set of semi-Fredholm elements in both directions and θ(1) = 1(mod I(B)). (x) θ is ∂σe-preserving; (xi) θ is ησe-preserving. (xii) θ is an essential spectral isometry and θ(1) = 1(mod I(B)). Proof. (i) ⇒ (ii). Since θ(1) = 1 (mod I(B)), we have the result from Theorem 3.4. (ii) ⇒ (iii). For every x ∈ A, we have σe(θ(x)) = σ(πB(θ(x))) = σ(θ̂(πA(x))) = σ(πA(x)) = σe(x), and the result holds. (iii) ⇒ (i). Since θ is σe-preserving, it preserves Fredholm elements in both directions. Lemma 3.2 implies that θ(I(A)) ⊆ I(B) and by the hypothesis the induced map θ̂ : CR(A) → CR(B) is spectrum preserving. It follows from Theorem 3.6 that θ̂ is unital, that is θ(1) − 1 ∈ I(B). It is easy to see that (ii) implies all other assertions. (iv) ⇒ (vii). If θ is σle-preserving then θ preserves the set of left semi-Fredholm elements in both directions. By Lemma 3.2, θ̂ is well defined and by the hypothesis it preserves the left spectrum. So by Theorem 3.6, θ̂ is unital. The inverse implication is obvious. The assertions (v) ⇐⇒ (viii), (vi) ⇐⇒ (ix) are proved similarly. (vii) ⇒ (i). If θ preserves the set of left semi-Fredholm elements in both directions then by Proposition 3.5, θ preserves the set of Fredholm elements in both directions. The same reasoning gives (viii) ⇒ (i) and (ix) ⇒ (i). If each of assertions (x),(xi) and (xii) holds then by Lemma 3.1, θ̂ is well defined and from Theorem 3.6, θ̂ is bijective, unital and also a spectral isometry. Thus both CR(A) and CR(B) are commutative. Now from [1, Theorem 4.1.17], θ̂ is an isomorphism which is continuous by semisimplicity of C(B) and (ii) follows. 4 Essential spectral preservers on C∗-algebras Let A be a Banach algebra. We recall that an element a ∈ A is called regular if there exists b ∈ A such that a = aba and b = bab; b is said to be a generalized inverse for a. Note that if a and b in A satisfy a = aba, then a is regular and b′ = bab is a generalized inverse for a. This shows, in particular, that the generalized inverse of a regular element is not unique. For an element a in a Banach algebra A, the regular set of a denoted by reg(a) is the set of all λ ∈ C such that there CUBO 17, 1 (2015) Maps preserving Fredholm or semi-Fredholm elements . . . 37 exists a neighborhood Uλ of λ, and an analytic function f : Uλ → A, such that f(µ) is a generalized inverse of a − µ1 for all µ ∈ Uλ. The generalized spectrum (also called Saphar spectrum) of a is given by σg(a) := C \ reg(a). The conorm or the reduced minimum modulus of a ∈ A is given by γ(a) := { inf{‖ax‖ : dist(x,ker(La)) ≥ 1}, if a 6= 0 ∞, if a = 0, (4.1) where La denotes the operator x 7→ ax on A. If F is a fixed ideal in A then the essential conorm and the generalized essential spectrum of a relative to F is given by γe(a) := γ(πA(a)) and σge(a) := σg(πA(a)), respectively. Here πA denotes the canonical quotient map from A onto the relative Calkin algebra CR(A). It is easy to see that if A is a C ∗-algebra then σge(a) = {λ ∈ C : limµ→λ γe(a − µ) = 0}, and that ∂σe(a) ⊆ σge(a) ⊆ σe(a). It is clear from the definition of γ(.) that, if θ preserves essential conorm that is, γe(θ(a)) = γe(a) for all a ∈ A, and θ(1) = 1 (mod I(B)), then σge(θ(a)) = σge(a) for all a ∈ A. We recall that a unital C∗-algebra A is said to be finite if a∗a = 1 implies that aa∗ = 1. It is easy to see that in a finite C∗-algebra every one-sided invertible element is invertible. C∗-algebras with finite traces and those with dense subset of invertible elements are examples of finite C∗- algebras. The quotient of a finite C∗ algebra A need not be finite. A unital C∗ algebra A is said to be residually finite if every quotient of A is finite; [6, V.2.1.3]. Note that, in this section A and B as Banach algebras, satisfy Assumption A. Theorem 4.1. Let A and B be unital C∗-algebras and let θ : A → B be a ∗-preserving linear map which is surjective up to the inessential ideal. Then in the following assertions, (i) − (iv) are equivalent and (v) − (vii) imply (ii). Moreover, if A and B are residually finite C∗-algebras, then all of the following conditions are equivalent. (i) θ is σe-preserving; (ii) θ(I(A)) ⊆ I(B) and the induced map θ̂ : CR(A) → CR(B), is an isometric Jordan isomor- phism; (iii) θ preserves Fredholm elements in both directions and θ(1) = 1(mod I(B)); (iv) γe(θ(a)) = γe(a) for all a ∈ A and θ(1) = 1(mod I(B)). (v) θ is σle-preserving; (vi) θ is σre-preserving; (vii) θ is (σle ∩ σre)-preserving. Proof. (i) ⇒ (iii). Since θ is σe-preserving, it preserves Fredholm elements in both directions and it follows from Lemma 3.2 that θ̂ : CR(A) → CR(B) is well defined and is unital by Theorem 3.6. (iii) ⇒ (ii). It follows from Lemma 3.2 that θ̂ is well defined and by the assumption it is unital 38 Mohadeseh Rostamani & Shirin Hejazian CUBO 17, 1 (2015) and preserves invertibility in both directions. It is easy to see that θ̂ is ∗-preserving. Therefore, θ̂ is a Jordan isomorphism by [9] and is an isometry by [8, Lemma 4.3]. (ii) ⇒ (i) is obvious. (iv) ⇒ (ii). If (iv) holds then σge(θ(a)) = σge(a) for all a ∈ A and thus θ is an essential spectral isometry and so θ̂ is well defined by Lemma 3.1. Moreover, θ̂ is injective and preserves the conorm. Thus by [8, Theorem 3.1], θ̂ is an isometric Jordan isomorphism. (ii) ⇒ (iv) follows easily. Now suppose that (v) holds. Then θ preserves the set of left semi-Fredholm elements in both directions. Lemma 3.2 implies that θ̂ is well defined. By Theorem 3.6, θ̂ is unital and bijective. θ̂ preserves left invertibility in both directions and so preserves invertibility in both directions by [4, Theorem 2.2]. Therefore θ̂(Inv(CR(A))) = Inv(CR(B)). It is easy to see that θ̂ is ∗-preserving. Therefore, θ̂ is a Jordan isomorphism by [9] and is an isometry by [8, Lemma 4.3]. (vi) ⇒ (ii) and (vii) ⇒ (ii) are proved in a similar way. Finally, if A and B are residually finite C∗-algebras, then CR(A) and CR(B) are finite and since a Jordan isomorphism is spectrum preserving it is easy to see that (ii) implies each of the assertions (v) − (vii). Theorem 4.2. Let A be a unital C∗-algebra of real rank zero and B a Banach algebra. Suppose that the linear map θ : A → B is surjective up to the inessential ideal. Consider the following assertions. Then (i) − (v) are equivalent and (vi) − (viii) imply (v). (i) θ is σe-preserving; (ii) θ is ∂σe-preserving; (iii) θ is ησe-preserving; (iv) θ preserves Fredholm elements in both directions and θ(1) = 1(mod I(B)). (v) θ(I(A)) ⊆ I(B) and the induced map θ̂ : CR(A) → CR(B) is a continuous Jordan isomorphism; (vi) θ is σle-preserving; (vii) θ is σre-preserving; (viii) θ is (σle ∩ σre)-preserving; Proof. It is obvious that (i) ⇒ (ii),(i) ⇒ (iii),(i) ⇔ (iv) and (v) implies each of the statements (i) − (iv) . If each of the statements (i) − (iii) holds then re(θ(x)) = re(x) and Lemma 3.1 implies that θ(I(A)) ⊆ I(B). Thus the induced map θ̂ : CR(A) → CR(B) is a spectral isometry and so is injective. By Theorem 3.6, θ̂ is unital and preserves idempotents. Let p and q be orthogonal projections in A. Then, p + q is a projection and θ̂(p) + θ̂(q) = ( θ̂(p) + θ̂(q) )2 = θ̂(p) + θ̂(p)θ̂(q) + θ̂(q)θ̂(p) + θ̂(q). CUBO 17, 1 (2015) Maps preserving Fredholm or semi-Fredholm elements . . . 39 It follows that θ̂(p)θ̂(q)+θ̂(q)θ̂(p) = 0 and hence θ̂(p)θ̂(q) = θ̂(p)θ̂(q)θ̂(p) = θ̂(q)θ̂(p). Therefore, θ̂(p) and θ̂(q) are orthogonal idempotents. Let a = ∑n j=1 λjpj be a linear combination of mutually orthogonal projections p1, ...,pn ∈ A. Then, θ̂(a2) = θ̂ ( n∑ j=1 λ2j pj ) = n∑ j=1 λ2j θ̂pj = ( θ̂(a) )2 for θ̂(p1), ..., θ̂(pn) are mutually orthogonal idempotents. Since every selfadjoint element in A is the norm limit of finite linear combinations of mutually orthogonal projections and θ̂ is continuous, θ̂(a2) = (θ̂(a))2 for every selfadjoint element a in A. Replacing a by a + b in this identity yields θ̂(ab + ba) = θ̂(a)θ̂(b) + θ̂(b)θ̂(a) for all a,b ∈ Asa. Suppose a = a1 + ia2 with a1,a2 ∈ Asa. By the above argument θ̂(a2) = θ̂ ( a21 + i(a1a2 + a2a1) − a 2 2 ) = θ̂(a1) 2 + i ( θ̂(a1)θ̂(a2) + θ̂(a2)θ̂(a1) ) − θ̂(a2) 2 = (θ̂(a))2. This proves that θ̂ is a Jordan isomprphism and (v) holds. (vi) ⇒ (v). Since ∂σe(θ(x)) ⊆ σle(θ(x)) for all x ∈ A, we have re(θ(x)) = max { |λ| : λ ∈ σle(θ(x)) } = max { |λ| : λ ∈ σle(x) } = re(x). Lemma 3.1 implies that θ(I(A)) ⊆ I(B). The induced map θ̂ : CR(A) → CR(B) is σl-preserving and by Theorem 3.6, θ̂ is a bijective unital linear map preserving idempotents and hence is a Jordan isomorphism. (vii) ⇒ (v) and (viii) ⇒ (v) are proved similarly. The following corollary is a direct consequence of Theorem 4.2 and [7, Corollary 5.3]. Corollary 4.3. Let A be a C∗-algebra of real rank zero and B a C∗-algebra. For a surjective up to inessential ideal linear map θ : A → B such that θ(1) = 1(mod I(B)), the conditions (i) − (v) in the above theorem are equivalent to the following conditions. (i) There exist α,β > 0 such that αγe(a) ≤ γe(θ(a)) ≤ βγe(a), for all a ∈ A. (ii) σge(θ(a)) = σge(a), for all a ∈ A. Acknowledgement. The authors would like to thank the referee for valuable comments and suggestions. Received: November 2012. Accepted: March 2013. References [1] B. Aupetit, A primer on spectral theory, Springer-Verlag, New York, 1991. 40 Mohadeseh Rostamani & Shirin Hejazian CUBO 17, 1 (2015) [2] B. A. Barnes, G. J. Murphy, M. R. F. Smyth and T. T. West, Riesz and Fredholm theory in Banach algebras, Research Notes in Math. 67, Pitman, Boston, MA, 1982. [3] M. Bendaoud and A. Bourhim, Essentially spectrally bounded linear maps, Proc. Amer. Math. Soc. 137 (2009) 3329-3334. [4] M. Bendaoud, A. Bourhim, M. Burgos and M. Sarih, Linear maps preserving Fredholm and Atkinson elements of C*-algebras, Linear Multilinear Algebra, 57 (2009) 823-838. [5] M. Bendaoud, A. Bourhim and M. Sarih, Linear maps preserving the essential spectral radius, Linear Alg. Appl. 428 (2008) 1041-1045. [6] B. Blackadar, Operator Algebras: Theory of C*-Algebras and von Neumann Algebras, Springer-Verlag, Berlin Heidelberg, 2006. [7] A. Bourhim and M. Burgos, Linear maps preserving regularity in C*-algebras, Illinois J. Math. 53 (2009), 899-914. [8] A. Bourhim, M. Burgos and V. Shulman, Linear maps preserving the minimum and reduced minimum moduli, J. Funct. Anal. 258 (2010) 50-66 [9] M-D. Choi, D. Hadwin, E. Nordgren, H. Radjavi and P. Rosenthal, On positive linear maps preserving invertibility, J. Funct. Anal. 59 (1984) 462-469 [10] J. L. Cui and J. C. Hou, Linear maps between Banach algebras compressing certain spectral functions, Rocky Mountain J. Math. 34 (2004) 565-585. [11] S. O. Kim and C. Park, Linear maps on C∗-algebras preserving the set of operators that are invertible in A/I. Canad. Math. Bull. 54 (1) (2011) 141-146. [12] M. Mbekhta, Linear maps preserving the set of Fredholm operators, Proc. Amer. Math. Soc. 135 (2007) 3613-3619. [13] M. Mbekhta, Linear maps preserving the minimum and surjectivity moduli of operators, Operators and Matrices 4 (4) (2010) 511-518. [14] M. Mbekhta, L. Rodman, and P. Šemrl, Linear maps preserving generalized invertibility, Integr. Equ. Oper. Theory 55 (2006) 93-109. [15] M. R. F. Smyth, Riesz theory in Banach algebras, Math. Z. 145 (1975) 145-155. [16] M. R. F. Smyth, Fredholm theory in Banach algebras, Banach Center Publications, 8 (1982) 403-414. Introduction Preliminaries Fredholm and semi-Fredholm preservers Essential spectral preservers on C*-algebras