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Let M n (F) be the algebra of n × n matrices over an arbitrary field F. We consider linear maps Φ : M n (F) → M r (F) preserving matrices annihilated by a fixed polynomial f (x) = (x − a 1) ⋯ (x − a m) with m ≥ 2 distinct zeroes a 1 , a 2 , ... , a m ∈ F ; namely, f (Φ (A)) = 0 whenever f (A) = 0. Suppose that f (0) = 0 , and the zero set Z (f) = { a 1 , ... , a m } is not an additive group. Then Φ assumes the form (†) A ↦ S ( A ⊗ D 1 A t ⊗ D 2 0 s ) S − 1 , for some invertible matrix S ∈ M r (F) , invertible diagonal matrices D 1 ∈ M p (F) and D 2 ∈ M q (F) , where s = r − n p − n q ≥ 0. The diagonal entries λ in D 1 and D 2 , as well as 0 in the zero matrix 0 s , are zero multipliers of f (x) in the sense that λ Z (f) ⊆ Z (f). In general, assume that Z (f) − a 1 is not an additive group. If Φ (I n) commutes with Φ (A) for all A ∈ M n (F) , or if f (x) has a unique zero multiplier λ = 1 , then Φ assumes the form (†). The above assertions follow from the special case when f (x) = x (x − 1) = x 2 − x , for which the problem reduces to the study of linear idempotent preservers. It is shown that a linear map Φ : M n (F) → M r (F) sending disjoint rank one idempotents to disjoint idempotents always assume the above form (†) with D 1 = I p and D 2 = I q , unless M n (F) = M 2 (Z 2). [ABSTRACT FROM AUTHOR]

Let 1 ≤ p ≤ + ∞ . We show that the positive part of the closed unit ball of a non-commutative L p -space, as a metric space, is a complete Jordan ⁎ -invariant for the underlying von Neumann algebra. [ABSTRACT FROM AUTHOR]

Abstract: Let be standard operator algebras on complex Banach spaces , respectively. For , let be a sequence with terms chosen from , and define the generalized Jordan product on elements in . This includes the usual Jordan product , and the triple . Assume that at least one of the terms in appears exactly once. Let a map satisfy that whenever any one of has rank at most one. It is shown in this paper that if the range of contains all operators of rank at most three, then must be a Jordan isomorphism multiplied by an th root of unity. Similar results for maps between self-adjoint operators acting on Hilbert spaces are also obtained. [Copyright &y& Elsevier]

In this paper, we study bijective linear maps θ : A → B between A W ⁎ -algebras preserving either zero products or range orthogonality. Such a map is automatically continuous, and provides an algebra or a ⁎-algebra isomorphism π ( ⋅ ) = θ ( ⋅ ) θ ⁎ ⁎ ( 1 ) − 1 from A onto B . Our results extend previous works for the case of W ⁎ -algebras, and also cover such maps between C ⁎ -algebras satisfying an extra assumption on preserving abelian C ⁎ -subalgebras. [ABSTRACT FROM AUTHOR]

We show that a compact operator A is a multiple of a positive semi-definite operator if and only if σ ( A B ) ⊆ W ( A ) W ( B ) ¯ , for all (rank one) operators B . An example of a normal operator is given to show that the equivalence conditions may fail in general. We then obtain conditions to identify other classes of operators A so that equivalence conditions hold. [ABSTRACT FROM AUTHOR]