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Let H and K be complex complete indefinite inner product spaces, and ℬ(H,K) (ℬ(H) if K = H) the set of all bounded linear operators from H into K. For every T ∈ ℬ(H,K), denote by $T^{†}$ the indefinite conjugate of T. Suppose that Φ: ℬ(H) → ℬ(K) is a bijective linear map. We prove that Φ satisfies $\Phi \left(A\right)\Phi \left(B\right)=\Phi \left(B\right)\Phi {\left(A\right)}^{†}$ for all A, B ∈ ℬ(H) with $AB=B{A}^{†}$ if and only if there exist a nonzero real number c and a generalized indefinite unitary operator U ∈ ℬ(H,K) such that $\Phi \left(A\right)=cUA{U}^{†}$ for all A ∈ ℬ(H).

We show that every spectrally bounded linear map Φ from a Banach algebra onto a standard operator algebra acting on a complex Banach space is square-zero preserving. This result is used to show that if Φ₂ is spectrally bounded, then Φ is a homomorphism multiplied by a nonzero complex number. As another application to the Hilbert space case, a classification theorem is obtained which states that every spectrally bounded linear bijection Φ from ℬ(H) onto ℬ(K), where H and K are infinite-dimensional...

A linear map L on an algebra is said to be Lie derivable at zero if L([A,B]) = [L(A),B] + [A,L(B)] whenever [A,B] = 0. It is shown that, for a 𝒥-subspace lattice ℒ on a Banach space X satisfying dim K ≠ 2 whenever K ∈ 𝒥(ℒ), every linear map on ℱ(ℒ) (the subalgebra of all finite rank operators in the JSL algebra Alg ℒ) Lie derivable at zero is of the standard form A ↦ δ (A) + ϕ(A), where δ is a generalized derivation and ϕ is a center-valued linear map. A characterization of linear maps Lie derivable...

We characterize surjective nonlinear maps Φ between unital C*-algebras 𝒜 and ℬ that satisfy w(Φ(A)-Φ(B))) = w(A-B) for all A,B ∈ 𝒜 under a mild condition that Φ(I) - Φ(0) belongs to the center of ℬ, where w(A) is the numerical radius of A and I is the unit of 𝒜.

Let ₁, ₂ be (not necessarily unital or closed) standard operator algebras on locally convex spaces X₁, X₂, respectively. For k ≥ 2, consider different products $T₁\ast \cdots \ast T_k$ on elements in ${}_i$, which covers the usual product $T₁\ast \cdots \ast T_k=T₁\cdots T_k$ and the Jordan triple product T₁ ∗ T₂ = T₂T₁T₂. Let Φ: ₁ → ₂ be a (not necessarily linear) map satisfying $\sigma \left(\Phi \left(A₁\right)\ast \cdots \ast \Phi \left(A_k\right)\right)=\sigma \left(A₁\ast \cdots \ast A_k\right)$ whenever any one of $A_i$’s has rank at most one. It is shown that if the range of Φ contains all rank one and rank two operators then Φ must be a Jordan isomorphism multiplied by a root...