Some unitary similarity invariant sets preservers of skew Lie products

Linear Algebra and its Applications 457 (2014) 76–92

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Some unitary similarity invariant sets preservers of skew Lie products ✩ Jianlian Cui a,∗ , Qiting Li a , Jinchuan Hou b,c , Xiaofei Qi c a Department of Mathematical Sciences, Tsinghua University, Beijing 100084, PR China b Department of Mathematics, Taiyuan University of Technology, Taiyuan 030024, PR China c Department of Mathematics, Shanxi University, Taiyuan 030006, PR China

a r t i c l e

i n f o

Article history: Received 20 November 2013 Accepted 6 May 2014 Available online xxxx Submitted by P. Semrl MSC: 47B48 46L10 Keywords: Numerical range Numerical radius Pseudo-spectrum Skew Lie products

a b s t r a c t Let H and K be complex separable Hilbert spaces with dimensions at least three, and B(H) the Banach algebra of all bounded linear operators on H. Let Δ(·) denote W (·) or σε (·), where, for A, W (A) stands for the numerical range of A ∈ B(H) and σε (A) the ε-pseudospectrum of A. It is shown that a bijective map (no algebraic structure assumed) Φ : B(H) → B(K) satisfies that Δ(AB − BA∗ ) = Δ(Φ(A)Φ(B) − Φ(B)Φ(A)∗ ) for all A, B ∈ B(H) if and only if there exists a unitary operator U ∈ B(H, K) such that Φ(A) = μU AU ∗ for all A ∈ B(H), where μ ∈ {−1, 1}. If Δ(·) = W (·), then the injectivity assumption on Φ can be omitted. © 2014 Elsevier Inc. All rights reserved.

✩ This project was supported by National Natural Science Foundation of China (No. 11271217, No. 11171249, No. 11101250). * Corresponding author. E-mail addresses: [email protected] (J. Cui), [email protected] (J. Hou), [email protected] (X. Qi).

http://dx.doi.org/10.1016/j.laa.2014.05.009 0024-3795/© 2014 Elsevier Inc. All rights reserved.

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1. Introduction Let A be a ∗-ring. For any A, B ∈ A, the product AB − BA∗ is called the skew Lie product of A and B. This product is playing a more and more important role in some research topics, and its study has recently attracted many authors’ attention (see, for example, [2–6,12,14–16]). Denote by B(H) the Banach algebra of all bounded linear operators on a real or complex Hilbert space H of dimension greater than 1. Motivated by the theory of rings (and algebras) equipped with the Lie product [A, B] = AB −BA or the Jordan product A ◦ B = AB + BA, Molnár in [12] initiated a systematic study of the skew Lie product, and showed that if N ⊆ B(H) is an ideal, then N = span{AB − BA∗ | A ∈ N , B ∈ B(H)} = span{AB − BA∗ | A ∈ B(H), B ∈ N }; in particular, every element of B(H) is a finite sum of T S − ST ∗ type operators. Later, Brešar and Fošner [2] generalized the above results in [12] to rings with involution in different directions. Let A and B be two von Neumann algebras and Φ : A → B be a map. If Φ(AB − BA∗ ) = Φ(A)Φ(B) − Φ(B)Φ(A)∗ for all A, B ∈ A, we say that Φ preserves skew Lie products; if Φ(A)Φ(B) − Φ(B)Φ(A)∗ = AB − BA∗ for all A, B ∈ A, we say that Φ preserves strong skew Lie products. For A, B ∈ A, if AB = BA∗ implies that Φ(A)Φ(B) = Φ(B)Φ(A)∗ , we say that Φ preserves zero skew Lie products. Clearly, every skew Lie product preserving map is zero skew Lie product preserving. Cui and Li [5] proved that, if A and B are factor Von Neumann algebras and Φ is a bijective map preserving skew Lie products, then Φ is a ∗-ring isomorphism. Cui and Park [6] proved that, if A and B are factor Von Neumann algebras and Φ is a surjective map preserving strong skew Lie products, then Φ is of the form Φ(A) = Ψ (A) + h(A)I for all A ∈ A, where Ψ : A → B is a linear bijective map preserving strong skew Lie products and h : A → R is a real functional satisfying h(0) = 0; in particular, when A = B = B(H), the above map Ψ is the identity map on B(H). Qi and Hou [14] generalized and improved the result in [6] to von Neumann algebras without central summands of type I1 ; they characterized also the corresponding maps on prime involution rings and prime involution algebras. When A = B(H) and B = B(K) with H and K being two complex Hilbert spaces, as a special case of results in [4], Cui and Hou characterized the linear bijective maps Φ : B(H) → B(K) preserving zero skew Lie products, and proved that such a Φ is of the form Φ(A) = cU AU ∗ for all A ∈ B(H), where U ∈ B(H, K) is unitary and c is a nonzero real number. For the maps on von Neumann algebras, even if under the assumption that the involved maps are additive, it seems considerably difficult to characterize maps preserving zero skew Lie products. The purpose of this paper is to discuss the maps that preserve some properties of skew Lie products on factor von Neumann algebras. More exactly, we will respectively characterize the maps preserving numerical range and the maps preserving pseudo-spectrum of skew Lie products of operators on B(H) (see Theorem 2.2 and Theorem 3.3), and characterize the additive maps preserving numerical radius of skew Lie products on factor von Neumann algebras (see Theorem 4.1). Our results reveal that, for a map

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Φ : B(H) → B(K), where H and K are two complex separable Hilbert spaces with the dimension at least three, the following statements are equivalent: (1) Φ is surjective and preserves the numerical range of skew Lie products; (2) Φ is bijective and preserves the ε-pseudo-spectrum of skew Lie products; (3) there exists a unitary operator U ∈ B(H, K) such that Φ(A) = μU AU ∗ for all A ∈ B(H), where μ ∈ {−1, 1}. Let us fix some notations. In this paper, H is a complex separable Hilbert space, and Bs (H) the set of all self-adjoint operators in B(H). For A ∈ B(H), ker A and ran A denote the kernel and range of A, respectively. For a subset M of H, A|M denotes the restriction of A to M , and M ⊥ denotes the orthogonal complementary subspace of M in H. For x ∈ H, [x] denotes the linear subspace spanned by x. For any nonzero x, f ∈ H, denote by x ⊗ f the rank one operator z → z, f x. Note that all rank one operators in B(H) can be written in this form. As usual, I stands for the identity operator in B(H) and, for a finite rank operator A, Tr(A) stands for the trace of A. Denote by respectively C and R the complex field and the real field. 2. Numerical range preservers In this section, we discuss the maps preserving numerical range of the skew Lie product AB − BA∗ on type I factor von Neumann algebras. For A ∈ B(H), the numerical range and numerical radius of A are respectively defined by W (A) = { Ax, x : x ∈ H, x = 1} and w(A) = sup{|λ| : λ ∈ W (A)}. Let us recall some basic results on numerical ranges, which will be used often in this paper. For other properties, see [10]. (1) (2) (3) (4)

W (A) = W (U AU ∗ ) for any unitary U ∈ B(H). W (λA) = λW (A) for any λ ∈ C. W (λI + A) = λ + W (A) for any λ ∈ C. W (A) = {λ} if and only if A = λI.

The following lemma is a result in [13], which will be used to prove our main result. Lemma 2.1. (See [13, Corollary 2].) Let H and K be two complex separable Hilbert spaces with the dimension at least three and let φ : Bs (H) → Bs (K) be a bijective transformation which preserves commutativity in both directions. Then there exists either a unitary or an anti-unitary operator U from H onto K and for every operator A ∈ Bs (H), there is a real valued bounded Borel function fA on σ(A) (the spectrum of A) such that φ(A) = U fA (A)U ∗ . Now we are in a position to state and prove the main result of this section.

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Theorem 2.2. Let H and K be two complex separable Hilbert spaces with the dimension at least three and Φ : B(H) → B(K) be a surjective map. Then Φ satisfies that W (AB − BA∗ ) = W (Φ(A)Φ(B) − Φ(B)Φ(A)∗ ) for all A, B ∈ B(H) if and only if there exist a unitary operator U ∈ B(H, K) and a scalar μ ∈ {1, −1} such that Φ(A) = μU AU ∗ for all A ∈ B(H). Proof. The “if” part is obvious. We will complete the proof of the “only if” part after proving several claims. Claim 1. Φ(RI) = RI, Φ(CI) = CI and Φ preserves self-adjoint elements in both directions. Since W (A) = {0} if and only if A = 0, it follows from W (Φ(A)Φ(B) − Φ(B)Φ(A)∗ ) = W (AB − BA∗ ) that Φ(A)Φ(B) = Φ(B)Φ(A)∗

⇔

AB = BA∗

for A, B ∈ B(H).

(2.1)

Now a similar proof of Claim 2 in [5] implies that Claim 1 holds. Claim 2. There is μ ∈ {−1, 1} such that W (Φ(A)) = μW (A) for all A ∈ B(H). By Claim 1, for every nonzero real number λ, there are a, b ∈ R with b = 0 such that Φ(iλI) = (a + ib)I, where i denotes the imaginary unit. For every A ∈ B(H), we have     W Φ(A)2 − Φ(A)Φ(A)∗ = W A2 − AA∗ . Taking A = iλI in the above equality gives b = ±λ and a = 0, and hence Φ(iλI) = ±iλI. Assume that there exist nonzero λ1 , λ2 ∈ R such that Φ(iλ1 I) = iλ1 I and Φ(iλ2 I) = −iλ2 I. Then   {−2λ1 λ2 } = W iλ1 I(iλ2 I) − iλ2 I(iλ1 I)∗   = W iλ1 I(−iλ2 I) − (−iλ2 I)(iλ1 I)∗ = {2λ1 λ2 } implies that λ1 λ2 = 0, a contradiction. So Φ(iλI) = iλI for every λ ∈ R or Φ(iλI) = −iλI for every λ ∈ R. Take μ ∈ {−1, 1} so that Φ(iλI) = μλI holds for all λ ∈ R. Let A ∈ B(H) be arbitrary. Since 2iλA = (iλI)A − A(iλI)∗ , it follows that     W (2iλA) = W 2Φ(iλI)Φ(A) = W 2μiΦ(A) , and hence, W (Φ(A)) = μW (A) holds for every A ∈ B(H). Claim 3. Φ is injective.

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For A, B ∈ B(H), assume that Φ(A) = Φ(B). Then, for every C ∈ B(H), we have     W CA − AC ∗ = W Φ(C)Φ(A) − Φ(A)Φ(C)∗   = W Φ(C)Φ(B) − Φ(B)Φ(C)∗   = W CB − BC ∗ . For any nonzero vector x ∈ H, letting C = ix ⊗ x in the above equality gives W (x ⊗ xA + Ax ⊗ x) = W (x ⊗ xB + Bx ⊗ x). Note that the center of the rectangular box from the vertical and horizontal support lines of W (x ⊗ xA + Ax ⊗ x) is equal to Ax, x (Ref. the proof of [8, Assertion 2.2.2]). It follows from the above equality that Ax, x = Bx, x for all x ∈ H. Since H is complex, we have A = B. So Φ is injective. Now, by Claims 1 and 3, Eq. (2.1) implies that Φ|Bs (H) : Bs (H) → Bs (K) is a bijective map preserving commutativity in both directions and thus Lemma 2.1 is applicable. Then, there exists a unitary or a conjugate unitary operator U from H onto K, and for every self-adjoint operator S ∈ Bs (H), there exists a real valued bounded Borel function fS on σ(S) such that Φ(S) = U fS (S)U ∗ . Assume firstly that U is unitary. Let μ be as in Claim 2 and define a map Ψ : B(H) → B(H) by Ψ (A) = μU ∗ Φ(A)U for every A ∈ B(H), where μ ∈ {1, −1}. Then Ψ is a bijective map satisfying     W Ψ (A)Ψ (B) − Ψ (B)Ψ (A)∗ = W AB − BA∗ for every A, B ∈ B(H),   W Ψ (A) = W (A) for every A ∈ B(H). Moreover, for every self-adjoint operator S ∈ Bs (H), there exists a real valued bounded Borel function fS on σ(S) such that Ψ (S) = fS (S). Particularly, it follows from the spectral theorem that, for every projection P on H, there exist α and β ∈ R such that Ψ (P ) = αP +β(I −P ). Since W (Ψ (P )) = W (P ) = [0, 1], one gets further that Ψ (P ) = P or I − P . Claim 4. For any rank one projection P ∈ B(H), we have Ψ (iP ) = iP . Let A ∈ B(H) be arbitrary. For every unit vector x ∈ H, since Ψ (x ⊗ x) = x ⊗ x or Ψ (x ⊗ x) = I − x ⊗ x, we have   W (Ax ⊗ x − x ⊗ Ax) = W Ψ (A)x ⊗ x − x ⊗ Ψ (A)x ,

(2.2)

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or    W (Ax ⊗ x − x ⊗ Ax) = W Ψ (A) − Ψ (A)∗ − Ψ (A)x ⊗ x − x ⊗ Ψ (A)x .

(2.3)

Put   X1 = x ∈ H : x = 1, Ψ (x ⊗ x) = x ⊗ x and   X2 = x ∈ H : x = 1, Ψ (x ⊗ x) = I − x ⊗ x . Then, for any unit vector x ∈ H, either x ∈ X1 or x ∈ X2 . As αx ⊗ αx = x ⊗ x for any scalar α with |α| = 1, x ∈ Xi ⇔ αx ∈ Xi , i = 1, 2. So we may identify x and αx in Xi with |α| = 1. Let P = y ⊗ y be an arbitrary rank one projection in B(H); we have to show that Ψ (iy ⊗ y) = iy ⊗ y. It follows from W (Ψ (iP )) = W (iP ) = iW (P ) and W (P ) = [0, 1] that there exists a positive operator B ∈ B(H) with B = 1 such that Ψ (iP ) = iB. We assert that X2 ∩ [y]⊥ = ∅. Assume on the contrary that there is x ∈ X2 ∩ [y]⊥ . Then, Eq. (2.3) implies that 2B = Bx ⊗ x + x ⊗ Bx. Thus we have Bx = x, Bx x and hence, B = x, Bx x ⊗ x. This, together with B = 1, entails that B = x ⊗ x. So X2 ∩ [y]⊥ = {x} has only one point, and Ψ (iy ⊗ y) = ix ⊗ x. Since dim H ≥ 3, one may take z ∈ X1 ∩ [y]⊥ so that z⊥x. Let Q = 12 (y + z) ⊗ (y + z). Then Q is a rank one projection and Ψ (Q) = Q or Ψ (Q) = I − Q. If Ψ (Q) = Q, then  iW

y ⊗ (y + z) + (y + z) ⊗ y 2



  = W iy ⊗ yQ − Q(iy ⊗ y)∗   = W ix ⊗ xQ − Q(ix ⊗ x)∗ = {0},

a contradiction. So Ψ (Q) = I − Q, and thus we have 

y ⊗ (y + z) + (y + z) ⊗ y iW 2



  = W iy ⊗ yQ − Q(iy ⊗ y)∗   = W ix ⊗ x(I − Q) − (I − Q)(ix ⊗ x)∗ = iW (2x ⊗ x). √

√

However, this leads to a contradiction [ 1−2 2 , 1+2 2 ] = W ( y⊗(y+z)+(y+z)⊗y ) = [0, 2]. 2 Hence we must have X2 ∩ [y]⊥ = ∅, and X1 ∩ [y]⊥ = {x ∈ [y]⊥ : x = 1}.

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Now, for any x ∈ X1 ∩ [y]⊥ , Eq. (2.2) implies that   {0} = W Ψ (iP )x ⊗ x − x ⊗ xΨ (iP )∗ = iW (Bx ⊗ x + x ⊗ Bx) and consequently, Bx ⊗ x + x ⊗ Bx = 0. This implies that Bx = 0 as B is positive, that is, B|[y]⊥ ∩X1 = 0. As [y]⊥ = span{[y]⊥ ∩ X1 }, we see that B|[y]⊥ = 0, which implies that B = αy ⊗ y for some α > 0. Note that B is positive and B = 1, we have B = y ⊗ y and hence Ψ (iy ⊗ y) = iy ⊗ y, as desired. Claim 5. For any rank one projection P ∈ B(H), we have Ψ (P ) = P . Assume, on the contrary, that there is a rank one projection P = y ⊗ y such that Ψ (y ⊗ y) = I − y ⊗ y. For any unit vector x ∈ H with x, y = 0, by Claim 4, we have     {0} = W (ix ⊗ x)P − P (ix ⊗ x)∗ = W 2ix ⊗ x − ix ⊗ xP + P (ix ⊗ x)∗ = W (2ix ⊗ x). This implies that 2ix ⊗ x = 0, a contradiction. Hence Ψ (P ) = P for every rank one projection P ∈ B(H). Claim 6. Ψ (A) = A for every A ∈ B(H). Let A ∈ B(H) be arbitrary. For any unit vector x ∈ H, it follows from Claim 5 that   iW (x ⊗ xA + Ax ⊗ x) = W (ix ⊗ x)A − A(ix ⊗ x)∗   = W Ψ (ix ⊗ x)Ψ (A) − Ψ (A)Ψ (ix ⊗ x)∗   = iW x ⊗ xΨ (A) + Ψ (A)x ⊗ x . So,   W (x ⊗ xA + Ax ⊗ x) = W x ⊗ xΨ (A) + Ψ (A)x ⊗ x , which implies that 

Ψ (A)x, x = Ax, x

holds for every unit vector x ∈ H. Hence we must have Ψ (A) = A for every A ∈ B(H). Consequently, Φ(A) = μU AU ∗ for every A ∈ B(H), where μ ∈ {−1, 1} and U is unitary. Finally, let us prove that the case U is conjugate unitary does not occur. Assume, on the contrary, that U is conjugate unitary. Similar to the discussion of Claims 4–5, for every rank one projection P , we have Φ(P ) = μU P U ∗ and Φ(iP ) = iμU P U ∗ , where μ ∈ {−1, 1}. Let A ∈ B(H) be arbitrary. For any unit vector x ∈ H, it follows from

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  iW (x ⊗ xA + Ax ⊗ x) = W (ix ⊗ x)A − A(ix ⊗ x)∗   = W Φ(ix ⊗ x)Φ(A) − Φ(A)Φ(ix ⊗ x)∗   = iμW U x ⊗ xU ∗ Φ(A) + Φ(A)U x ⊗ xU ∗ that   W (x ⊗ xA + Ax ⊗ x) = μW U x ⊗ xU ∗ Φ(A) + Φ(A)U x ⊗ xU ∗ .

(2.4)



¯ Take an orthonormal basis {ei }i∈Λ of H and define J by J( i∈Λ ξi ei ) = i∈Λ ξi ei . ∗ −1 ∗ t Then J : H → H is conjugate unitary and J = J = J , also JA J = A for every A ∈ B(H), where At is the transpose of A with respect to the given orthonormal basis of H above. Let U = V J; then V ∈ B(H, K) is unitary. Note that the center of the rectangular box from the vertical and horizontal support lines of W (U x ⊗ xU ∗ Φ(A) + Φ(A)U x ⊗ xU ∗ ) is equal to Φ(A)U x, U x = V Jx, Φ(A)∗ V Jx = Jx, V ∗ Φ(A)∗ V Jx = JV ∗ Φ(A)∗ V Jx, x = U ∗ Φ(A)∗ U x, x , and hence,  μ U ∗ Φ(A)∗ U x, x = Ax, x holds for every x ∈ H. This implies that Φ(A) = μU A∗ U ∗ for every A ∈ B(H). Take A = x ⊗ f and B = x ⊗ x, where x and f are orthogonal unit vectors. Then it follows from W (Φ(A)Φ(B) − Φ(B)Φ(A)∗ ) = W (AB − BA∗ ) that x ⊗ f = f ⊗ x, a contradiction. This completes the proof. 2 Note that, if a map Φ : B(H) → B(K) has the form Φ(A) = γU AU ∗ for every A ∈ B(H), where U is unitary or conjugate unitary and γ ∈ C with |γ| = 1, then w(Φ(A)Φ(B) − Φ(B)Φ(A)∗ ) = w(AB − BA∗ ) for all A, B ∈ B(H), that is, Φ preserves numerical radius of the skew Lie products. Thus the following problem is natural and interesting. Problem. Let H and K be complex Hilbert spaces and Φ : B(H) → B(K) a bijective map. Assume that w(Φ(A)Φ(B) − Φ(B)Φ(A)∗ ) = w(AB − BA∗ ) for all A, B ∈ B(H). Whether or not there exist a unitary or conjugate unitary operator U from H into K and a γ ∈ C with |γ| = 1 such that Φ(A) = γU AU ∗ for every A ∈ B(H)? At present we cannot solve this problem. But, in the case when the map under consideration is additive, the answer is true. In fact, a characterization of the additive maps on general factor von Neumann algebras that preserve the numerical radius of skew Lie products will be given in Section 4. 3. Pseudo-spectrum preservers Eigenvalues, more generally, spectra, have been a standard tool of the mathematical sciences for a century and a half. In many applications, for example, in structural

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mechanics, in aeronautics, in quantum mechanics, in ecology, in probability theory, in electrical engineering and in numerical analysis, and so on, eigenvalue analysis has been proved highly successful. It is well known that in applied mathematics, it is not always enough to ask whether an operator is singular or nonsingular; the norm A−1  may be important as well as the fact that it is finite. It should not be surprising that for applications, it is sometimes desirable to extend these definitions too by asking not just whether (zI − A)−1 exists but whether it is large or small. For every ε > 0, the ε-pseudospectrum σε (A) of A ∈ B(H), a new subset of the complex plane, is defined as follows:     σε (A) = z ∈ C : (zI − A)−1  > ε−1 . For convention, we write (zI − A)−1  = ∞ if z ∈ σ(A). From the definition, it follows that the collection of ε-pseudospectra of A is a family of strictly nested closed subsets of C, which grow to fill the whole complex plane as ε → ∞, and it follows from the upper-semicontinuity of the spectrum that the intersection of all the pseudospectra is the spectrum, σε (A) = σ(A). ε>0

Pseudospectra can also be defined in other equivalent ways:     σε (A) = z ∈ C : (zI − A)x < ε for some unit vector x ∈ X ,   σε (A) = z ∈ σ(A + E) : E ∈ B(H), E < ε and

  σε (A) = z ∈ C : smin (zI − A) < ε ,

where smin denotes the minimal singular value in the matrix case or the infimum of s-numbers in operator case [9]. Now let us recall some properties of the pseudospectrum (Ref. [17]), which will be frequently used in our proof. Let ε > 0 be arbitrary and D(a, ε) = {μ ∈ C : |μ − a| < ε}, where a ∈ C. Proposition 3.1. Let ε > 0 and let A ∈ B(H). Then the following statements are true. σ(A) + D(0, ε) ⊆ σε (A). If A is normal, then σε (A) = σ(A) + D(0, ε). For any c ∈ C, σε (A + cI) = c + σε (A). ε (A). For any nonzero c ∈ C, σε (cA) = cσ |c| σε (A) = σε (At ), where At denotes the transpose of A relative to an arbitrary but fixed orthonormal basis of H. (6) σε (U AU ∗ ) = σε (A), where U ∈ B(H) is unitary.

(1) (2) (3) (4) (5)

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The following proposition was proved in [7]. Proposition 3.2. Let ε > 0 and A ∈ B(H). Then the following statements hold. (1) A = aI if and only if σε (A) = D(a, ε), where a ∈ C. (2) Let a ∈ C be nonzero. There exists a nontrivial projection P ∈ B(H) such that A = aP if and only if σε (A) = D(0, ε) ∪ D(a, ε). (3) A is self-adjoint if and only if σε (A) ⊂ {z ∈ C : | Im z| < ε}. (4) A is skew self-adjoint if and only if σε (A) ⊂ {z ∈ C : | Re z| < ε}. The following is our main result in this section. Theorem 3.3. Let H and K be two complex separable Hilbert spaces with the dimension at least three, and ε > 0. Then a bijective map Φ : B(H) → B(K) satisfies that σε (Φ(A)Φ(B) − Φ(B)Φ(A)∗ ) = σε (AB − BA∗ ) for all A, B ∈ B(H) if and only if there exist a unitary operator U ∈ B(H, K) and μ ∈ {−1, 1} such that Φ(A) = μU AU ∗ for every A ∈ B(H). Proof. Only the “only if” part should be proved. Assume that Φ preserves the ε-pseudospectrum. Then Proposition 3.2(1) implies that AB = BA∗

⇔

Φ(A)Φ(B) = Φ(B)Φ(A)∗

for A, B ∈ B(H).

(3.1)

Now, by [5, proof of Claim 2] we see that Φ(RI) = RI, Φ(CI) = CI and Φ preserves self-adjoint operators in both directions. Claim 1. σε (Φ(A)) = σε (A) for every A ∈ B(H). Take A = I and B = iI. Then     D(2i, ε) = σε BA − AB ∗ = σε Φ(iI)Φ(I) − Φ(I)Φ(iI)∗ and     D(−2, ε) = σε B 2 − BB ∗ = σε Φ(iI)2 − Φ(iI)Φ(iI)∗ . Note that Φ(iI) = (a + ib)I and Φ(I) = cI for some a, b, c ∈ R. Substituting them into the above two equalities gives D(2i, ε) = σε (2ibcI),

   D(−2, ε) = σε 2iab − 2b2 I ,

which, together with Proposition 3.2(1), implies that 2bci = 2i and 2abi − 2b2 = −2. Hence either (a, b, c) = (0, 1, 1) or (a, b, c) = (1, −1, −1). Thus either (Φ(I), Φ(iI)) =

86

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(I, iI) or (Φ(I), Φ(iI)) = (−I, −iI). If Φ(I) = −I, then −Φ satisfies the conditions of the theorem. So, without loss of generality we may assume that Φ(I) = I. In this case it is easily checked that Φ( 12 iI) = 12 iI. For any A ∈ B(H), it follows from 

 ∗  1 1 iA − A iI 2 2   ∗    1 1 = σε iΦ(A) iIΦ(A) − Φ(A) iI = σε 2 2   = iσε Φ(A)

iσε (A) = σε (iA) = σε

that σε (A) = σε (Φ(A)). Claim 2. There exists a unitary or a conjugate unitary operator U from H onto K such that for every projection P ∈ B(H), Φ(P ) = U P U ∗

or

Φ(P ) = U (I − P )U ∗ .

Since Φ preserves self-adjoint elements in both directions, it follows from Eq. (3.1) that Φ|Bs (H) : Bs (H) → Bs (K) is a bijective map preserving commutativity in both directions. Applying Lemma 2.1, there exists a unitary or conjugate unitary operator U from H onto K, and for every self-adjoint operator S ∈ Bs (H), there exists a real valued bounded Borel function fS on σ(S) such that Φ(S) = U fS (S)U ∗ . Thus, for every projection P , there exist real numbers αP and βP such that Φ(P ) = αP U P U ∗ + βP U (I − P )U ∗ . Note that, Proposition 3.2 and Claim 1 imply that Φ(P ) is a projection if and only if P is a projection. Thus, for every projection P on H, we have Φ(P ) = U P U ∗ or Φ(P ) = U (I − P )U ∗ . Claim 3. For any rank one projection P ∈ B(H), Φ(iP ) = iU P U ∗ . Let P ∈ B(H) be an arbitrary rank one projection. Since σε (Φ(iP )) = σε (iP ) = D(0, ε) ∪ D(i, ε), by Proposition 3.2(2), there exists a projection Q ∈ B(H) such that Φ(iP ) = iQ. We assert that Φ(I − P ) = U (I − P )U ∗ . Otherwise, assume that Φ(I − P ) = U P U ∗ . As dim H ≥ 3, there exist two rank one projections R1 , R2 ≤ I − P satisfying R1 R2 = 0. If Φ(Rj ) = U (I − Rj )U ∗ , j = 1, 2, then,   D(0, ε) = σε Φ(iP )U (I − Rj )U ∗ − U (I − Rj )U ∗ Φ(iP )∗   = σε i 2Q − QU Rj U ∗ − U Rj U ∗ Q . This, together with Proposition 3.2(1), ensures that 2Q − QU Rj U ∗ − U Rj U ∗ Q = 0, j = 1, 2, and hence, Q = QU R1 U ∗ = QU R2 U ∗ . Since R1 and R2 are of rank one, it follows that Q = U R1 U ∗ = U R2 U ∗ , a contradiction. So there exists at most one rank one

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projection R1 ≤ I − P such that Φ(R1 ) = U (I − R1 )U ∗ and for any rank one projection R2 ≤ I − P satisfying R1 R2 = 0, we have Φ(R2 ) = U R2 U ∗ . Assume that there is some rank one projection R1 ≤ I − P so that Φ(R1 ) = U (I − R1 )U ∗ . Then   D(0, ε) = σε Φ(iP )U (I − R1 )U ∗ − U (I − R1 )U ∗ Φ(iP )∗   = σε i 2Q − QU R1 U ∗ − U R1 U ∗ Q ensures Q = QU R1 U ∗ . Hence Q = U R1 U ∗ and Φ(iP ) = iU R1 U ∗ . In this case, if Φ(P ) = U P U ∗ , then   D(0, ε) = σε iU R1 U ∗ U P U ∗ + iU P U ∗ U R1 U ∗   = σε Φ(iP )Φ(P ) − Φ(P )Φ(iP )∗ = σε (2iP ) implies that P = 0, a contradiction. So we must have Φ(P ) = U (I − P )U ∗ . However this will lead to a contradiction. In fact, for any rank one projections R1 , R2 ≤ I − P satisfying R1 R2 = 0, since there exist projections Hj (j = 1, 2) such that Φ(iRj ) = iHj (j = 1, 2), we have   D(0, ε) = σε (iRj P + iP Rj ) = σε Φ(iRj )U (I − P )U ∗ − U (I − P )U ∗ Φ(iRj )∗   = σε i 2Hj − Hj U P U ∗ − U P U ∗ Hj , which ensures that H1 = H2 = U P U ∗ , and Φ(iR1 ) = iU P U ∗ = Φ(iR2 ), contradicting to the injectivity of Φ. Thus we have proved that Φ(I − P ) = U (I − P )U ∗ . Since Φ(iP ) = iQ for some projection Q, it follows from     D(0, ε) = σε iP (I − P ) + i(I − P )P = σε Φ(iP )U (I − P )U ∗ − U (I − P )U ∗ Φ(iP )∗   = σε i 2Q − QU P U ∗ − U P U ∗ Q that Q = U P U ∗ , so Φ(iP ) = iU P U ∗ . The claim holds. Claim 4. There exists a unitary operator U on H such that Φ(iA) = iU AU ∗ for every self-adjoint operator A or Φ(iA) = U (iA)t U ∗ for every self-adjoint operator A, where At is the transpose of A for an arbitrarily but fixed orthonormal basis of H. For any A ∈ B(H) and for any unit vector x ∈ H, Claim 3 and the fact     σε iP A − A(iP )∗ = σε Φ(iP )Φ(A) − Φ(A)Φ(iP )∗

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imply that   σε (x ⊗ xA + Ax ⊗ x) = σε U x ⊗ xU ∗ Φ(A) + Φ(A)U x ⊗ xU ∗ .

(3.2)

Let A ∈ Bs (H) be arbitrary. Since iAx ⊗ x + x ⊗ x(iA) is skew self-adjoint, Eq. (3.2), Proposition 3.1(2) and Proposition 3.2(4) imply that U x ⊗ xU ∗ Φ(iA) + Φ(iA)U x ⊗ xU ∗ is also skew self-adjoint. Hence we have σ(U x ⊗ xU ∗ Φ(iA) + Φ(iA)U x ⊗ xU ∗ ) = σ(ix ⊗ xA + iAx ⊗ x), and   Tr Φ(iA)U x ⊗ xU ∗ = i Ax, x

(3.3)

hold for all x ∈ H. So, if U is unitary, it follows that Φ(iA) = U (iA)U ∗ . Now assume that U is conjugate unitary. Let U = V J, where J is defined in the proof of Claim 6 in Theorem 2.2. Then V is unitary, and Eq. (3.3) implies that Φ(iA) = V (iA)t V ∗ . This completes the proof of Claim 4. Assume U is unitary. Consider the map A → U ∗ Φ(A)U . Then this map satisfies the conditions of the theorem. So, by Claim 4, we might as well assume that Φ(iA) = iA for every self-adjoint operator A or Φ(iA) = iAt for every self-adjoint operator A. We assume first that Φ(iA) = iA for every self-adjoint operator A. We will prove that Φ(A) = A for every A ∈ B(H). For any nonzero vector x ∈ H and any λ > 0, we have Φ(iλx ⊗ x) = iλx ⊗ x. Let A ∈ B(H) be arbitrary. It follows that     iλσ λε Φ(A)x ⊗ x + x ⊗ xΦ(A) = σε iλx ⊗ xΦ(A) + iλΦ(A)x ⊗ x = σε (iλx ⊗ xA + iAλx ⊗ x) = iλσ λε (Ax ⊗ x + x ⊗ xA).

Notice that σ(Φ(A)x ⊗ x + x ⊗ xΦ(A)) = λ>0 σ λε (Φ(A)x ⊗ x + x ⊗ xΦ(A)). Thus we get σ(Φ(A)x ⊗ x + x ⊗ xΦ(A)) = σ(Ax ⊗ x + x ⊗ xA), which entails that Φ(A)x, x = Ax, x

for any x ∈ H, and therefore, Φ(A) = A. Next we show that the case that Φ(iA) = iAt for every self-adjoint operator A does not occur. Assume on the contrary that Φ(iA) = iAt for every self-adjoint operator A. Then for any x ∈ H and any λ > 0, Φ(iλx ⊗ x) = iλJx ⊗ Jx, where J is defined as in Claim 6 in the proof of Theorem 2.2. Let A ∈ B(H) be arbitrary. Similar to the previous discussion we obtain that σ(Φ(A)Jx ⊗ Jx + Jx ⊗ JxΦ(A)) = σ(Ax ⊗ x + x ⊗ xA) for any nonzero x ∈ H, and hence JΦ(A)∗ Jx, x = Jx, Φ(A)∗ Jx = Φ(A)Jx, Jx = Ax, x

for any nonzero vector x ∈ H. It follows that JΦ(A)∗ J = A, that is, Φ(A) = JA∗ J = At . Now, for any orthogonal unit vectors x, f ∈ H, let A = x ⊗ f and B = x ⊗ x; then AB − BA∗ = 0 and BA − A∗ B = x ⊗ f − f ⊗ x. Thus we have D(0, ε) = σε (AB − BA∗ ) = σε (Φ(A)Φ(B)−Φ(B)Φ(A)∗ ) = σε (At B t −B t (At )∗ ) = σε (BA−A∗ B) = σε (x⊗f −f ⊗x), which implies that x ⊗ f = f ⊗ x, a contradiction. So the case Φ(A) = At does not occur. The proof is complete. 2

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4. Numerical radius preservers Let A be a unital C*-algebra with the unit I. Recall that a state τ of A is a positive linear functional of A with τ (I) = 1. For A ∈ A, the numerical range and the numerical radius of A are defined by   W (A) = τ (A) : τ runs over all states on A and   w(A) = sup |λ| : λ ∈ W (A) , respectively. It is well known that numerical radius, w(·), is a norm (but not a C*-norm) and is equivalent to the original norm on A. Recall that a factor von Neumann algebra A is a von Neumann algebra with the center Z(A) = CI. Every factor von Neumann algebra is prime. Recall that a ring R is called prime if, for any A, B ∈ R, ARB = {0} implies that either A = 0 or B = 0. In this section we answer the problem posed in the end of Section 2 on the factor von Neumann algebras when the considered maps are additive. Theorem 4.1. Let A and B be two factor von Neumann algebras. Assume that Φ : A → B is an additive surjective map. Then Φ satisfies w(Φ(A)Φ(B) − Φ(B)Φ(A)∗ ) = w(AB − BA∗ ) for all A, B ∈ A if and only if Φ is a ∗-isomorphism, or a conjugate ∗-isomorphism, or their negative multiple. To prove the theorem, we need the following lemma. Lemma 4.2. (See [1, Lemmas 4–5].) Let A and B be two unital C*-algebras with units I and I  , respectively. Assume that Φ : A → B is an additive surjective map. If w(Φ(A)) = w(A) for every A ∈ A, then the following statements hold. (1) If Φ(λI) = λI  for all scalars λ, then Φ is a ∗-Jordan isomorphism. (2) If Φ(λI) = λI  for all scalars λ, then Φ is a conjugate ∗-Jordan isomorphism. Proof of Theorem 4.1. The proof of the theorem will be divided into several steps. Step 1. Φ(RI) = RI, Φ(CI) = CI and Φ preserves self-adjoint elements in both directions. For A ∈ A, since w(A) = 0 if and only if A = 0, the result follows from the proof of Claim 1 in Theorem 2.2. Step 2. Φ(iλI) = ±iλI for every real number λ, and w(Φ(A)) = w(A) for all A ∈ A.

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By Step 1, for every nonzero real number λ, there are a, b ∈ R with b = 0 such that Φ(iλI) = (a + ib)I. Let A ∈ A be arbitrary. We have     w Φ(A)2 − Φ(A)Φ(A)∗ = w A2 − AA∗ .

(4.1)

Taking A = iλI in the above equality gives   b2 a2 + b2 = λ4 .

(4.2)

By Step 1, there exist c, d ∈ R with d = 0 such that Φ((c + id)I) = iλI. Substituting this into Eq. (4.1), we get   d2 c2 + d2 = λ4 .

(4.3)

Take A = iλI and B = (c + id)I; then w(AB − BA∗ ) = w(Φ(A)Φ(B) − Φ(B)Φ(A)∗ ) implies that c2 + d2 = b2 , which, together with Eqs. (4.2) and (4.3) and b = 0, implies that d2 = a2 + b2 . Hence a2 + c2 = 0, and therefore, a = c = 0. Now Eq. (4.2) entails that b = ±λ, and Φ(iλI) = ±iλI. For any A ∈ A and every λ ∈ R, it follows from 2iλA = iλA − A(iλ)∗ that w(A) = w(Φ(A)). ¯ for all λ ∈ C, where μ ∈ {−1, 1}. Step 3. Φ(λI) = μλI for all λ ∈ C, or Φ(λI) = μλI For any a, b ∈ R, let λ = a + ib. Steps 1 and 2 imply that there exist c, d ∈ R with a + b2 = c2 + d2 such that Φ((a + ib)I) = (c + id)I. By Step 2, we only need to consider the case a = 0 and b = 0, and consequently, c = 0 and d = 0. Denote [A, B]∗ = AB −BA∗ for all A, B ∈ A. Since 2

        2|ab| = w (a + ib)I, aI ∗ = w Φ (a + ib)I , Φ(aI) ∗ = 2|ad|, ¯ Since Φ is additive, it one has d = ±b, and so c = ±a. Hence Φ(λI) = ±λI or ±λI. is easily checked that Φ(λI) ≡ λI for all λ ∈ C, or Φ(λI) ≡ −λI for all λ ∈ C, or ¯ for all λ ∈ C, or Φ(λI) ≡ −λI ¯ for all λ ∈ C. Φ(λI) ≡ λI ¯ for every λ ∈ C. By Step 2 Assume that Φ(λI) ≡ λI for every λ ∈ C or Φ(λI) ≡ λI and Lemma 4.2, Φ is a ∗-Jordan isomorphism or a conjugate ∗-Jordan isomorphism. Since every factor von Neumann algebra is prime, it follows from [11] that Φ is a ∗-isomorphism or a ∗-anti-isomorphism or a conjugate ∗-isomorphism or a conjugate ∗-anti-isomorphism. If Φ is a ∗-anti-isomorphism or a conjugate ∗-anti-isomorphism, then we have     w AB − BA∗ = w Φ(A)Φ(B) − Φ(B)Φ(A)∗    = w Φ BA − A∗ B   = w BA − A∗ B

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for all A, B ∈ A. However, this implies that AB = BA∗ if and only if BA = A∗ B for A, B ∈ A, which is impossible. In fact, take an arbitrary nontrivial projection P ∈ A; then A = P AP + P A(I − P ) + (I − P )AP + (I − P )A(I − P ). Pick any nonzero element A ∈ P A(I − P ). It is clear that AP = P A∗ = 0. Note that A = P A ∈ P A(I − P ) and A∗ = A∗ P ∈ (I − P )AP . So P A = A∗ P , a contradiction. Hence Φ is a ∗-isomorphism or a conjugate ∗-isomorphism. ¯ for every λ ∈ C, then the same discusIf Φ(λI) ≡ −λI for every λ ∈ C or Φ(λI) ≡ −λI sion is applied to −Φ, and we have −Φ is a ∗-isomorphism or a conjugate ∗-isomorphism. This completes the proof of the theorem. 2 Remark. (1) In Theorem 4.1, if Φ is assumed that Φ(RI) = RI and Φ(CI) = CI, then the condition that A and B are factors can be deleted. (2) If the condition of Theorem 4.1 is relaxed so that A and B are unital factor C*-algebras, then we may prove that Φ or −Φ is either a ∗-Jordan isomorphism or a conjugate ∗-Jordan isomorphism. However, we do not know whether the converse is true or not. Acknowledgements The authors wish to express their thanks to the referees. They read the original manuscript carefully and gave many helpful comments to improve the paper. References [1] Z.-F. Bai, J.-C. Hou, Z.-B. Xu, Maps preserving numerical radius on C*-algebras, Studia Math. 162 (2004) 97–104. [2] M. Brešar, M. Fošner, On rings with involution equipped with some new product, Publ. Math. Debrecen 57 (2000) 121–134. [3] M.A. Chebotar, Y. Fong, P.-H. Lee, On maps preserving zeros of the polynomial xy − yx∗ , Linear Algebra Appl. 408 (2005) 230–243. [4] J.-L. Cui, J.-C. Hou, Linear maps preserving elements annihilated by a polynomial XY − Y X † , Studia Math. 174 (2) (2006) 183–199. [5] J.-L. Cui, C.-K. Li, Maps preserving product XY − Y X ∗ on factor von Neumann algebras, Linear Algebra Appl. 431 (2009) 833–842. [6] J.-L. Cui, C. Park, Strong Lie skew-products preserving maps on factor von Neumann algebras, Acta Math. Sci. (2012) 531–538. [7] J.-L. Cui, C.-K. Li, Y.-T. Poon, Pseudospectra of special operators and pseudosectrum preservers, J. Math. Anal. Appl. (2014), http://dx.doi.org/10.1016/j.jmaa.2014.05.041, in press. [8] H. Gau, C.-K. Li, C*-isomorphisms, Jordan isomorphisms, and numerical range preserving maps, Proc. Amer. Math. Soc. 135 (2007) 2907–2914. [9] I.C. Gohberg, M.G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators, Amer. Math. Soc., Providence, RI, 1969. [10] P.R. Halmos, A Hilbert Space Problem Book, 2nd ed., Springer-Verlag, New York, 1982. [11] I.N. Herstein, Topics in Ring Theory, Springer, Berlin, 1991. [12] L. Molnár, A condition for a subspace of B(H) to be an ideal, Linear Algebra Appl. 235 (1996) 229–234.

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[13] L. Molnár, P. Šemrl, Non-linear commutativity preserving maps on self-adjoint operators, Quart. J. Math. 56 (2005) 589–595. [14] X. Qi, J. Hou, Strong skew commutativity preserving maps on von Neumann algebras, J. Math. Anal. Appl. 397 (2013) 362–370. [15] P. Šemrl, Quadratic functionals and Jordan *-derivations, Studia Math. 97 (1991) 157–165. [16] P. Šemrl, On Jordan *-derivations and an application, Colloq. Math. 59 (1990) 241–251. [17] L.N. Trefethen, M. Embree, Spectra and Pseudospectra, the Behavior of Nonormal Matrices and Operators, Princeton University Press, Princeton, 2005.