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Let A be a unital C⁎-algebra equipped with its natural order. As usual the effect algebra of A is the interval { a ∈ A : 0 ≤ a ≤ I A } , where I A denotes the unit of A. In this paper, we give a complete description of order isomorphisms between effect algebras of C⁎-algebras of type C (X) ⊗ B (H) , where C (X) stands for the algebra of all continuous complex valued functions on a (non pathological) Hausdorff compact space X and B (H) denotes the algebra of all bounded linear operators on a complex Hilbert space H. Our results generalize some works by L. Molnár and P. Šemrl. [ABSTRACT FROM AUTHOR]

Let m, n, p and q be positive integers such that p q ≤ m n and min (p , q) ≤ min (m , n). Let M m , n (C) denote the vector space of matrices of m rows and n columns with complex entries and let M n (C) stand for M n , n (C). For any matrix A ∈ M m (C) let σ (A) denote its spectrum. We prove that a couple of maps ϕ : M m , n (C) → M p , q (C) a n d ψ : M n , m (C) → M q , p (C) satisfy σ ϕ (M) ψ (N) = σ (M N) for all (M , N) ∈ M m , n (C) × M n , m (C) , if and only if there exist invertible matrices P ∈ M p (C) and Q ∈ M q (C) such that one of the following assertions holds: (m , n) = (p , q) , and we have ϕ (M) = P M Q and ψ (N) = Q − 1 N P − 1 for all (M , N) ∈ M m , n (C) × M n , m (C). (m , n) = (q , p) , and we have ϕ (M) = P M t Q and ψ (N) = Q − 1 N t P − 1 for all (M , N) ∈ M m , n (C) × M n , m (C). [ABSTRACT FROM AUTHOR]

Let be the algebra of all bounded linear operators on an infinite-dimensional complex Hilbert space , and let and be two subsets of containing all rank one operators. Fix , and let and denote the ϵ-pseudo spectrum and the ϵ-pseudo spectral radius of any operator. We show that if a surjective map satisfies then either there exists a unitary operator U on such that for all , or there is a conjugate unitary operator U on such that for all. We also characterize surjective maps satisfying [ABSTRACT FROM AUTHOR]

Let be the algebra of all bounded linear operators on a complex Banach space X. For an operator , let be the local spectral radius of T at any vector. For an integer , let be a finite sequence such that and at least one of the terms in appears exactly once. The generalized product of k operators is defined by and includes the usual product TS and the triple product TST. We show that a surjective map ϕ on satisfies for all and all if and only if there exists a map such that for all. [ABSTRACT FROM AUTHOR]

Let Mn(C) denote the algebra of all n × n complex matrices, and fix a nonzero vector x0 in Cn. For any matrix T ∈Mn(C), let σT (x0) denote the local spectrum of T at x0. Given three scalars μ, ν and ξ simultaneously nonzero, we study maps ϕ on Mn(C) satisfying σμSTS+νTS+ξST (x0) = σμϕ(S)ϕ(T)ϕ(S)+νϕ(T)ϕ(S)+ξϕ(S)ϕ(T)(x0) for all S, T ∈Mn(C). Our main result extends and unifies the main results of several papers on maps on Mn(C) preserving the local spectrum of different products. [ABSTRACT FROM AUTHOR]

Let M p (C) be the algebra of all p × p complex matrices, and let σ (T) denote the spectrum of any matrix T ∈ M p (C). For an integer n ≥ 2 , let f (ξ 1 , ... , ξ n) be a ξ 1 -linear polynomial with complex coefficients and n -non commuting indeterminates ξ 1 , ... , ξ n. Under minor natural conditions on f , we show that if a map φ on M p (C) satisfies (0.1) σ (f (φ (T 1) ,... , φ (T n))) = σ (f (T 1 ,... , T n)) for all T 1 ,... , T n ∈ M p (C) , then there exist a nonzero scalar λ and an invertible matrix A ∈ M p (C) such that φ has one of the following forms: (0.2) T ↦ λ A T A − 1 for all T ∈ M p (C) , or (0.3) T ↦ λ A T t A − 1 for all T ∈ M p (C). Here T t denotes the transpose of T. In general, not all of mappings of the form (0.2) or the form (0.3) satisfy (0.1). When n = 2 , we describe the set of all scalars λ for which the mapping (0.2) or (0.3) satisfies (0.1). We also obtain analogue results of spectrum preserving maps on the real linear space H p of all self-adjoint matrices in M p (C). Our results extend and unify several results obtained earlier on maps on M p (C) preserving the spectrum of generalized product or generalized Jordan product of matrices. [ABSTRACT FROM AUTHOR]