Maps preserving zero products

Abstract

Abstract. A linear map T from a Banach algebra A into another B preserves zero products if T (a)T (b) = 0 whenever a, b ∈ A are such that ab = 0. This paper is mainly concerned with the question of whether every continuous linear surjective map T : A → B that preserves zero products is a weighted homomorphism. We show that this is indeed the case for a large class of Banach algebras which includes group algebras.Our method involves continuous bilinear maps φ : A × A → X (for some Banach space X) with the property that φ(a, b) = 0 whenever a, b ∈ A are such that ab = 0. We prove that such a map necessarily satisfies φ(aµ, b) = φ(a, µb) for all a, b ∈ A and for all µ from the closure with respect to the strong operator topology of the subalgebra of M(A) (the multiplier algebra of A) generated by doubly power-bounded elements of M(A). This method is also shown to be useful for characterizing derivations through the zero products.Introduction. S. Banach [4] was the first to describe isometries on L p ([0, 1]) with p = 2. Although Banach did not give the full proof for this case (this was provided by J. Lamperti [26]), he made the key observation that isometries on L p ([0, 1]) must take functions with disjoint support into functions with disjoint support. This property arises in a variety of situations and has been considered by several authors. For example, in the theory of Banach lattices there is an extensive literature about linear maps T : X → Y , where X and Y are Banach lattices, with the property that |T (x)| ∧ |T (y)| = 0 whenever x, y ∈ X are such that |x| ∧ |y| = 0. Such maps are usually called disjointness preserving operators or d-homomorphisms. The reader interested in this setting is referred to the monograph [1]. The concept of a disjointness preserving operator was exported to function algebras by E. Beckenstein and L. Narici (see [5] for general information).

Reflexivity and hyperreflexivity of bounded $N$-cocycles from group algebras

“…The author would like to thank Armando R. Villena for providing the preprints of the manuscripts [1], [2], and [3]. The author also would like to thank the referee for carefully reading this article and providing helpful comments.…”

“…In [1], it is shown that L 1 (G) always has property (B). In Section 3, we combine their result together with our results in Section 2 and the structure of groups with polynomial growth to show that the space of bounded n-cocycles from L 1 (G)…”

“…In Section 2, we formulate the definition for both algebraical and topological reflexivity of subspaces of bounded n-linear maps. We then recall the class of Banach algebras with property (B), which was introduced in [1] and shown to be useful in studying maps preserving zero products (Section 2.3). We show that we can characterize bounded n-cocycles for Banach algebras with property (B) provided that the algebra has sufficiently many "local identities".…”

Identifying Derivations Through the Spectra of Their Values

“…Our first result is an easy consequence of Lemma 3.5 and a deeper result from [1]. From now on we will consider the norm inequality condition (N), which, as observed in the introduction, follows immediately from the spectral inclusion condition (S) if a and b are selfadjoint.…”

Characterizing Jordan Derivations of Matrix Rings Through Zero Products

“…For more studies concerning Jordan derivations we refer the reader to [5,8,10,12,16,17,18,19] and the references therein. Also, there have been a number of papers concerning the study of conditions under which (generalized or Jordan) derivations of rings can be completely determined by the action on some sets of points [1,2,3,7,9,13,14,15,21,22].…”