Let $A$ be a unital $C^{*}$-algebra which has a faithful state. If $varphi:Arightarrow A$ is a unital linear map which is bijective and invertibility preserving or surjective and spectral radius preserving, then $varphi$ is a Jordan isomorphism. Also, we discuss other types of linear preserver maps on $A$

Ali Asghar Khalilzadeh

Fatemeh Golfarshchi

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Sahand Communications in Mathematical Analysis (SCMA) Vol. 11, No. 1 (2018), 91-97http://scma.maragheh.ac.irDOI: 10.22130/scma.2017.23702Linear Maps Preserving Invertibility or Spectral Radius onSome C∗-algebrasFatemeh Golfarshchi1∗ and Ali Asghar Khalilzadeh2Abstract. Let A be a unital C∗-algebra which has a faithful state.If φ : A → A is a unital linear map which is bijective and invert-ibility preserving or surjective and spectral radius preserving, thenφ is a Jordan isomorphism. Also, we discuss other types of linearpreserver maps on A.1. IntroductionLet A and B be unital Banach algebras. A linear map φ : A −→ Bis said to be invertibility preserving if a is invertible in A, then φ(a) isinvertible in B, and φ is called Jordan homomorphism if φ(a2) = (φ(a))2for all a ∈ A. Also φ is said to be spectrum preserving if σ(φ(a)) = σ(a)and spectral radius preserving (spectral isometry) if r(φ(a)) = r(a), forall a ∈ A. The map φ said to be unital if φ(1) = 1.A Hilbert C∗-module over a C∗-algebra A is a right A-module Eequipped with an A-valued inner product ⟨., .⟩ which is A-conjugatelinear in the first variable and A-linear in the second variable, suchthat E is a Banach space with the norm ∥x∥ = ∥⟨x, x⟩∥12 (x ∈ E).Every C∗-algebra A is a Hilbert A-module if we define the inner product⟨a, b⟩ = a∗b, for all a, b ∈ A. Let E,F be Hilbert modules over the C∗-algebra A. A map T : E → F is said to be adjointable if there existsa map T ∗ : F → E satisfying ⟨x, T (y)⟩ = ⟨T ∗(x), y⟩, for all x ∈ F andy ∈ E. Such a map T ∗ is called the adjoint of T . We denote the set of alladjointable maps from E to F by B(E,F ), which is the most important2010 Mathematics Subject Classification. 47B48, 46H05, 47C10, 47A75.Key words and phrases. C∗-algebra, Hilbert C∗-module, Invertibility preserving,Spectral radius preserving, Jordan isomorphism.Received: 28 May 2016, Accepted: 12 January 2017.∗ Corresponding author.9192 F. GOLFARSHCHI AND A. A. KHALILZADEHclass of operators on Hilbert C∗-modules. It is well known that with theoperator norm ∥T∥ = sup{∥T (x)∥ : x ∈ E, ∥x∥ ≤ 1}, B(E) = B(E,E)is a C∗-algebra. The topology on B(E,F ) generated by the seminormsT →∥ Tx ∥ (x ∈ E) and T →∥ T ∗y ∥ (y ∈ F ) is called the stricttopology on B(E,F ). A net (Tλ)λ∈Λ converges strictly to an operatorT ∈ B(E,F ) if and only if ∥ (Tλ−T )(x) ∥→ 0 and ∥ (T ∗λ −T ∗)(y) ∥→ 0,for all x ∈ E and y ∈ F , [6].Let A and B be C∗-algebras. An element a ∈ A is said to be positiveand denoted by a ≥ 0, if a is self-adjoint and σ(a) ⊆ R+. Also a linearmap φ : A→ B is called positive if φ(a) ≥ 0 for all a ≥ 0.Let A be a C∗-algebra and let B be a strictly closed C∗-subalgebraof M(A)(the multiplier algebra of A). A retraction from A to B is apositive linear map φ : A→ B satisfying the following conditions:(i) φ(ab) = φ(a)b (a ∈ A, b ∈ B),(ii) φ(A) is strictly dense in B,(iii) There exists an approximate unit (ui) of A such that (φ(ui))converges strictly in M(A) to a projection p ∈ B.If q is a projection in M(A), then the map a→ pap is a retraction fromA to pM(A)p. A retraction φ on A is said to be faithful if a ∈ A+ andφ(a) = 0, then a = 0, see [6]. A state on a C∗-algebra A is a positivelinear functional on A of norm one. Any positive linear unital functionalon A is a state. Similarly a state ψ on A is said to be faithful if a ∈ A+and ψ(a) = 0, then a = 0. Any state on C∗-algebra A is a retractionfrom A to C (identified with the subalgebra C1 of M(A)).Let A be a C∗ − algebra and let p and q be projections in A. Then qis said to be a subprojection of p, if p− q is a projection in A, also p andq is called orthogonal if pq = 0. We know that if q is a subprojection ofp, then pq = qp = q. We say a projection p in a von Neumann algebra iscountably decomposable if any collection of mutually orthogonal nonzerosubprojections of p is countable. A von Neumann algebra A is countablydecomposable if 1A is countably decomposable. Since by [11, Proposition2.1.10] every von Neumann algebra on a separable Hilbert space has aseparable predual and by [11, Proposition 2.1.9] every von Neumannalgebra with separable predual is a countably decomposable, one mayconclude that every von Neumann algebra on a separable Hilbert spaceis countably decomposable.The study of linear preserving maps has attracted the attention ofmany mathematicians in recent decades. For example, Sourour andBresar studied invertibility preserving linear maps, and Jafarian, Semrland Aupetit studied spectrum preserving linear maps [1, 4, 5, 12]. Thespectral radius preserving linear maps has been studied by Bresar andMathieu [3, 8], also Marcus and Moyls studied eigenvalue preservingLINEAR MAPS PRESERVING INVERTIBILITY OR SPECTRAL RADIUS ON ... 93linear maps on matrix algebra [7]. In this paper, we consider linearpreserver maps on the algebra of adjointable operators on a Hilbert C∗-module, and we will show that every unital bijective and inevitabilitypreserving or surjective and spectral radius preserving linear map on aunital C∗-algebra which has a faithful state, is a Jordan isomorphism.2. Main ResultsLet A be a C∗-algebra, B be a strictly closed C∗-subalgebra of M(A)and E be a Hilbert A-module. Suppose that φ is a retraction fromA to B. We define a B-valued semi-inner product ⟨x, y⟩φ on E by⟨x, y⟩φ = φ(⟨x, y⟩), for all x, y ∈ E. Let Nφ = {x ∈ E : ⟨x, x⟩φ = 0} and(x+N, y +N)φ = ⟨x, y⟩φ, for all x, y ∈ E. Then (., .)φ defines an innerproduct on ENφ and Eφ = (ENφ) is a B-Hilbert module (the closure withrespect to the norm defined by ∥x+N∥ = ∥(x+N,x+N)φ∥12 ). If φ isa state on A, then Eφ is a Hilbert space. It is well known that the mapπφ : B(E) → B(Eφ) defined by πφ(T )(x+Nφ) = T (x) +Nφ is a unital∗- homomorphism and is an isomorphism, when φ is faithful, see [6, 13].We need the following lemma to prove our main results.Lemma 2.1. Let A be a unital C∗-algebra which has a faithful state andE be a Hilbert A-module. Then there exists a Hilbert space H such thatB(E) is isomorphic to B(H).Proof. Let ρ be a faithful state on A. Then ρ is a faithful retractionfrom A to C, so H = Eρ is a Hilbert space and πρ : B(E) → B(Eρ) is a∗-isomorphism. □In the following, theorem we characterize the linear map φ which isbijective and invertibility preserving.Theorem 2.2. Let A and B be unital C∗-algebras which have a faithfulstate. Let E and F be Hilbert A-module and B-module respectively. Ifφ : B(E) → B(F ) is a bijective and invertibility preserving unital linearmap, then φ is a Jordan isomorphism.Proof. Let ρ and ζ be faithful states on A andB respectively. Since ρ andζ are retractions, there exist Hilbert spaces Eρ and Fς such that mapsπρ : B(E) → B(Eρ) and πζ : B(F ) → B(Fζ) are unital ∗-isomorphisms,by Lemma 2.1. If ψ = πζφπ−1ρ , then ψ is a linear, bijective and invertibil-ity preserving map, so by [12, Theorem 1.1] ψ is a Jordan isomorphism.Therefore, φ is a Jordan isomorphism. □In the following theorems, we review a map φ with different condi-tions.94 F. GOLFARSHCHI AND A. A. KHALILZADEHTheorem 2.3. Let A be a unital C∗-algebra which has a faithful stateand let E be a Hilbert A-module. If φ : B(E) → B(E) is a unitalsurjective and spectral radius preserving linear map, then either(1) there exists an invertible operator P ∈ B(E) such thatφ(T ) = PTP−1, T ∈ B(E).or(2) there exists an invertible operator Q ∈ B(E) such thatφ(T ) = QT ∗Q−1, T ∈ B(E).Proof. Let ρ be a faithful state on A. Then πρ : B(E) → B(Eρ) is aunital ∗-isomorphism, by Lemma 2.1. Let ψ = πζφπ−1ρ . Since ψ is asurjective and spectral radius preserving linear map, by [3, Theorem 1]there exists c ∈ C such that |c| = 1 and either(1) there exists a bijective V ∈ B(Eρ) such that ψ(S) = cV SV −1,for all S ∈ B(Eρ), or(2) there exists a bijectiveW ∈ B(Eρ) such that ψ(S) = cWS∗W−1,for all S ∈ B(Eρ).Now let P = π−1ρ (V ) and Q = π−1ρ (W ). Then P is an invertible elementof B(E) and we havePπ−1ρ (V−1) = π−1ρ (V )π−1ρ (V−1)= π−1ρ (V V−1)= I,andπ−1ρ (V−1)P = π−1ρ (V−1)π−1ρ (V )= π−1ρ (V−1V )= I,so P−1 = π−1ρ (V−1). Similarly we can show that Q is an invertibleelement of B(E) and Q−1 = π−1ρ (W−1). Also we have φ(T ) = cPTP−1or φ(T ) = cQT ∗Q−1. Since φ is a unital map, c = 1 and thereforeφ(T ) = PTP−1 or φ(T ) = QT ∗Q−1.□Theorem 2.4. Let A be a unital C∗-algebra which has a faithful state,E be an infinite dimensional Hilbert A-module and φ : B(E) → B(E) isa surjective linear map.(i) If σ(φ(T )φ(S)) = σ(TS), for all T, S ∈ B(E), then there existsan invertible C ∈ B(E) such that for all T ∈ B(E) eitherφ(T ) = CTC−1 or φ(T ) = −CT ∗C−1.LINEAR MAPS PRESERVING INVERTIBILITY OR SPECTRAL RADIUS ON ... 95(ii) If σ((φ(T ))∗φ(S)) = σ(φ(T ∗S)), for all T, S ∈ B(E), then thereexist unitary operators U, V ∈ B(E) such that φ(T ) = UTV forall T ∈ B(E).Proof. Let ρ be a faithful state on A. Then πρ : B(E) → B(Eρ) is aunital ∗-isomorphism by Lemma 2.1. Let ψ = πρφπ−1ρ . Then ψ is asurjective linear map on B(Eρ) = B(Eρ). If A,B ∈ B(Eρ), thenσ(ψ(A)ψ(B)) = σ(πρφπ−1ρ (A)πρφπ−1ρ (B))= σ(πρ(φπ−1ρ (A)φπ−1ρ (B)))= σ(φπ−1ρ (A)φπ−1ρ (B))= σ(π−1ρ (A)π−1ρ (B))= σ(π−1ρ (AB))= σ(AB).On the other hand ρ is a faithful state, so Nρ = {0}, thus Eρ = ( ENρ ) isisomorph with E, so Eρ is an infinite dimensional Hilbert space. There-fore, by [9, Theorem 3] there exists an invertible P ∈ B(Eρ) such thatfor all T ∈ B(Eρ) either ψ(T ) = PTP−1 or ψ(T ) = −PT ∗P−1. LetC = π−1ρ (P ). Then C is invertible and C−1 = π−1ρ (P−1). Also we haveφ(T ) = (π−1ρ ψπρ)(T ) = π−1ρ (Pπρ(T )P−1)= π−1ρ (P )Tπ−1ρ (P−1)= CTC−1.The proof of φ(T ) = −CTC−1 is similar.To prove (ii), if A,B ∈ B(Eρ), thenσ((ψ(A))∗ψ(B)) = σ((πρφπ−1ρ )(A))∗(πρφπ−1ρ )(B))= σ(πρ((φπ−1ρ (A))∗φπ−1ρ (B)))= σ((φπ−1ρ (A))∗φπ−1ρ (B)))= σ(φ((π−1ρ (A))∗π−1ρ (B)))= σ((π−1ρ (A∗B))= σ(A∗B).Therefore, by [9, Theorem 4] there exist unitary operators P,Q ∈ B(Eρ)such that ψ(T ) = PTQ, for all T ∈ B(Eρ). Let U = π−1ρ (P ) and96 F. GOLFARSHCHI AND A. A. KHALILZADEHV = π−1ρ (Q), thenUU∗ = π−1ρ (P )(π−1ρ (P ))∗= π−1ρ (P )π−1ρ ((P )∗)= π−1ρ (PP∗)= π−1ρ (I)= I.The proof of V ∗V = V V ∗ = I is similar. Alsoφ(T ) = (π−1ρ ψπρ)(T )= π−1ρ (Pπρ(T )Q)= π−1ρ (P )Tπ−1ρ (Q)= UTV.□Lemma 2.5. Let A be a unital C∗-algebra which has a faithful state andφ : A→ A be a unital linear map. If φ is bijective and invertibility pre-serving or surjective and spectral radius preserving, then φ is a Jordanisomorphism.Proof. SinceA is a unital C∗-algebra, a linear map ψ : K(A) → A definedby ψ(Σλiθaibi) = Σλiaib∗i for all ai, bi ∈ A and λi ∈ C is an isomorphismand by [13, Theorem 15.2.12], M(K(A)) ≃ B(A), so B(A) ≃ A andtherefore φ is a Jordan isomorphism, by Theorems 2.2 and 2.3. □Remark 2.6. If A is a unital finite dimension C∗-algebra, A has a faith-ful state by [10, Exercise 6.2], so by Lemma 2.5 every unital surjectiveand spectral radius preserving linear map on A is a Jordan isomorphismwhich is a result of [8, Corollary 5].Remark 2.7. If A is a unital von Neumann algebra on a separableHilbert space, then A is countably decomposable, so by [2, PropositionIII.2.2.27] A has a faithful state and by Lemma 2.5 every surjective andspectrum preserving linear map on A is a Jordan isomorphism which isa result of [1, Theorem 1.3].Theorem 2.8. Let A be an infinite dimension unital C∗-algebra whichhas a faithful state. If φ : A→ A is a surjective linear map, then(i) if σ(φ(a)φ(b)) = σ(ab), for all a, b ∈ A, then there exists c ∈ Asuch that for all a ∈ A, φ(a) = c−1ac or φ(a) = −cac−1,(ii) if σ((φ(a))∗φ(b)) = σ(a∗b), for all a, b ∈ A, then there existunitary elements u, v ∈ A such that for all a ∈ A, φ(a) = uav.LINEAR MAPS PRESERVING INVERTIBILITY OR SPECTRAL RADIUS ON ... 97Proof. Since A is a unital C∗-algebra, B(A) ≃ A and the result followsby Theorem 2.4. □References1. B. Aupetit, Spectrum-preserving linear mappings between Banach algebras orJordan-Banach algebras, J. Lond. Math Soc., 62 (2000) 917-924.2. B. Blackadar, Theory of C∗-algebras and Von Neumann algebras, Springer-Verlag, New York, 2006.3. M. Bresar and P. Semrl, Linear maps preserving the spectral radius, J. Funct.Anal., 142 (1996) 360-368.4. M. Bresar and P. Semrl, Invertibility preserving maps preserve idempotents,Michigan Math. J., 45 (1998) 483-488.5. A. Jafarian and A.R. Sourour, Spectrum preserving linear maps, J. Funct. Anal.,66 (1986) 255-261.6. E.C. Lance, Hilbert C∗-modules A toolkit for operator algebraists, Lond. Math.Soc., Lecture Notes Ser 210, Cambridge University Press, Cambridge, 1995.7. M. Marcus and B.N. Moyls, Linear transformations on algebras of matrices,Canad. J. Math., 11 (1959) 61-66.8. M. Mathieu and A.R. Sourour, Hereditary properties of spectral isometries, Arch.Math., 82 (2004) 222-229.9. L. Molnar, Some characterizations of the automorphisms of B(H) and C(X),Proc. Amer. Math. Soc., 130 (2001) 111-120.10. J. Murphy, C∗-algebras and Operator Theory, Academic Press, Boston, 1990.11. S. Sakai, C∗-algebras and W ∗-algebras, Springer-Verlag, New York, 1971.12. A.R. Sourour, Invertibility preserving linear maps on L(X), Trans. Amer. Math.Soc., 348 (1996) 13-30.13. N.E. Wegge-Olsen, K-theory and C∗-algebras, Oxford University Press, NewYork, 1993.1 Department of Multimedia, Tabriz Islamic Art University, Tabriz,Iran.E-mail address: f.golfarshchi@tabriziau.ac.ir2 Department of Mathematics, Sahand University of Technology, Sa-hand Street, Tabriz, Iran.E-mail address: khalilzadeh@sut.ac.ir

Linear Maps Preserving Invertibility or Spectral Radius on Some $C^{*}$-algebras

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Linear Maps Preserving Invertibility or Spectral Radius on Some $C^{*}$ -algebras

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Linear Maps Preserving Invertibility or Spectral Radius on Some C∗C^{*}C∗-algebras

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Linear Maps Preserving Invertibility or Spectral Radius on Some $C^{*}$ -algebras