Weighted Jordan homomorphisms

The first author was supported by the Slovenian Research Agency (ARRS) Grant P1- 0288. The second author was suported by MCIU/AEI/FEDER Grant PGC2018-093794- B-I00, Junta de Andalucía Grant FQM-185, MIU Grant FPU18/00419 and MIU Grant EST19/00466.Let A and B be unital rings. An additive map T : A → B is called a weighted Jordan homomorphism if c = T (1) is an invertible central element and cT (x2) = T (x)2 for all x ∈ A. We provide assumptions, which are in particular fulfilled when A = B = Mn(R) with n ≥ 2 and R any unital ring with 1 2 , under which every surjective additive map T : A → B with the property that T (x)T (y) + T (y)T (x) = 0 whenever xy = yx = 0 is a weighted Jordan homomorphism. Further, we show that if A is a prime ring with char(A) 6= 2, 3, 5, then a bijective additive map T : A → A is a weighted Jordan homomorphism provided that there exists an additive map S : A → A such that S(x2) = T (x)2 for all x ∈ A.Slovenian Research Agency (ARRS) Grant P1- 0288MCIU/AEI/FEDER Grant PGC2018-093794- B-I00Junta de Andalucía Grant FQM-185MIU Grant FPU18/00419 MIU Grant EST19/0046

Maps preserving two-sided zero products on Banach algebras

Let A and B be Banach algebras with bounded approximate identities and let F: A→B be a surjective continuous linear map which preserves twosided zero products (i.e., F(a)F(b) = F(b)F(a) = 0 whenever ab = ba = 0). We show that F is a weighted Jordan homomorphism provided that A is zero product determined and weakly amenable. These conditions are in particular fulfilled when A is the group algebra L1(G) with G any locally compact group. We also study a more general type of continuous linear maps F : A → B that satisfy F(a)F(b)+F(b)F(a) = 0 whenever ab = ba = 0. We show in particular that if F is surjective and A is a C∗-algebra, then F is a weighted Jordan homomorphism.Slovenian Research Agency - Slovenia P1-0288MCIU/AEI/FEDER Grant PGC2018-093794-B-I00Junta de Andalucia FQM-185 P20_00255 MIU FPU18/00419 EST19/00466Proyectos I+D+i del programa operativo FEDER-Andalucia Grant A-FQM-484-UGR1

SIMULTANEOUS EMBEDDINGS OF GRAPHS AS MEDIAN AND ANTIMEDIAN SUBGRAPHS

The distance DG(v) of a vertex v in an undirected graph G is the sum of the distances between v and all other vertices of G. The set of vertices in G with maximum (minimum) distance is the antimedian (median) set of a graph G. It is proved that for arbitrary graphs G and J and a positive integer r ≥ 2, there exists a connected graph H such that G is the antimedian and J the median subgraphs of H, respectively, and that dH(G, J) = r. When both G and J are connected, G and J can in addition be made convex subgraphs of H