Linear disjointness preservers of W*-algebras

Abstract

In this paper, we give a complete description of the structure of zero product and orthogonality preserving linear maps between W*-algebras. In particular, two W*-algebras are *-isomorphic if and only if there is a bijective linear map between them preserving their zero product or orthogonality structure in two directions. It is also the case when they have equivalent linear and left (right) ideal structures.

Linear Maps Which are Anti-derivable at Zero

“…Let us simply observe that a linear map T between Banach algebras is a homomorphism at zero if and only if it preserves zero products (i.e., ab = 0 implies T (a)T (b) = 0). We find in this way a natural link with the results on zero products preservers (see, for example, [1,2,8,10,28,29,32,33,[47][48][49][50][51] for additional details and results). Burgos, Cabello-Sánchez and the third author of this note explore in [6] those linear maps between C * -algebras which are * -homomorphisms at certain points of the domain, for example, at the unit element or at zero.…”

Nonsurjective maps between rectangular matrix spaces preserving disjointness, triple products, or norms

“…Adjusting the proof of Theorem 2.1, we can achieve the equality U = V * , at the expenses that the diagonal matrices Q 1 , Q 2 may have negative entries. (7) If the domain is the set M n (C) of n × n complex matrices or the set H n (C) of n × n complex Hermitian matrices, our results can be deduced from the abstract theorems on C * -algebras; e.g., see [4,20,21,28], and also [6,27]. However, the proofs there do not seem to work for rectangular matrix spaces, or real square matrix spaces.…”

“…The result is used to characterize linear maps that preserve the JB * -triple product, or just the zero triple product. Note that there are interesting results on disjointness preserving maps on different kinds of products over general operator spaces or algebras, see, e.g., [16,17,21,27,28]. However, the basic problem on disjointness preservers from a rectangular matrix space to another rectangular matrix space is unknown, and the existing results do not cover this case.…”

Linear maps between C*-algebras that are *-homomorphisms at a fixed point

“…If A is a unital C * -algebra and B is a Banach algebra, J. Alaminos, M. Brešar, J. Extremera, and A. Villena prove in [2, Theorem 4.1] that a bounded linear operator h : A → B is multiplicative at zero if and only if h(1)h(xy) = h(x)h(y) for all x, y ∈ A. Other related results are given in [9,13,14,20].…”