M. Brešar’s research while affiliated with University of Ljubljana and other places

We call an algebra A A commutator-simple if [ A , A ] [A,A] does not contain nonzero ideals of A A . After providing several examples, we show that in these algebras derivations are determined by a condition that is applicable to the study of local derivations. This enables us to prove that every continuous local derivation D : L 1 ( G ) → L 1 ( G ) D\colon L^1(G)\to L^1(G) , where G G is a unimodular locally compact group, is a derivation. We also give some remarks on homomorphism-like maps in commutator-simple algebras.

We call an algebra A commutator-simple if [A,A] does not contain nonzero ideals of A. After providing several examples, we show that in these algebras derivations are determined by a condition that is applicable to the study of local derivations. This enables us to prove that every continuous local derivation $D\colon L^1(G)\to L^1(G)$, where G is a unimodular locally compact group, is a derivation. We also give some remarks on homomorphism-like maps in commutator-simple algebras.

Let A and B be Banach algebras with bounded approximate identities and let Φ:A→B be a surjective continuous linear map which preserves two-sided zero products (i.e., Φ(a)Φ(b)=Φ(b)Φ(a)=0 whenever ab=ba=0). We show that Φ 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 Φ:A→B that satisfy Φ(a)Φ(b)+Φ(b)Φ(a)=0 whenever ab=ba=0. We show in particular that if Φ is surjective and A is a C⁎-algebra, then Φ is a weighted Jordan homomorphism.

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 12, 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)≠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.

Let A and B be Banach algebras with bounded approximate identities and let $\Phi:A\to B$ be a surjective continuous linear map which preserves two-sided zero products (i.e., $\Phi(a)\Phi(b)=\Phi(b)\Phi(a)=0$ whenever ab=ba=0). We show that $\Phi$ 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 $L^1(G)$ with G any locally compact group. We also study a more general type of continuous linear maps $\Phi:A\to B$ that satisfy $\Phi(a)\Phi(b)+\Phi(b)\Phi(a)=0$ whenever ab=ba=0. We show in particular that if $\Phi$ is surjective and A is a $C^*$-algebra, then $\Phi$ is a weighted Jordan homomorphism.

A Banach algebra A is said to be a zero Jordan product determined Banach algebra if, for every Banach space X , every bilinear map \unicode[STIX]{x1D711}:A\times A\rightarrow X satisfying \unicode[STIX]{x1D711}(a,b)=0 whenever a , $b\in A$ are such that ab+ba=0 , is of the form \unicode[STIX]{x1D711}(a,b)=\unicode[STIX]{x1D70E}(ab+ba) for some continuous linear map \unicode[STIX]{x1D70E} . We show that all $C^{\ast }$ -algebras and all group algebras $L^{1}(G)$ of amenable locally compact groups have this property and also discuss some applications.

A Banach algebra A is said to be a zero Jordan product determined Banach algebra if every continuous bilinear map $\varphi\colon A\times A\to X$, where X is an arbitrary Banach space, which satisfies $\varphi(a,b)=0$ whenever a, $b\in A$ are such that ab+ba=0, is of the form $\varphi(a,b)=\sigma(ab+ba)$ for some continuous linear map $\sigma$. We show that all $C^*$-algebras and all group algebras $L^1(G)$ of amenable locally compact groups have this property, and also discuss some applications.

A Banach algebra A is said to be zero Lie product determined if every continuous bilinear functional $\varphi \colon A\times A\to \mathbb{C}$ satisfying $\varphi(a,b)=0$ whenever ab=ba is of the form $\varphi(a,b)=\omega(ab-ba)$ for some $\omega\in A^*$. We prove that A has this property provided that any of the following three conditions holds: (i) A is a weakly amenable Banach algebra with property $\mathbb{B}$ and having a bounded approximate identity, (ii) every continuous cyclic Jordan derivation from A into $A^*$ is an inner derivation, (iii) A is the algebra of all $n\times n$ matrices, where $n\ge 2$, over a cyclically amenable Banach algebra with a bounded approximate identity.

A Banach algebra A is said to be zero Lie product determined if every continuous bilinear functional $\varphi \colon A\times A\to \mathbb{C}$ satisfying $\varphi(a,b)=0$ whenever ab=ba is of the form $\varphi(a,b)=\omega(ab-ba)$ for some $\omega\in A^*$. We prove that A has this property provided that any of the following three conditions holds: (i) A is a weakly amenable Banach algebra with property $\mathbb{B}$ and having a bounded approximate identity, (ii) every continuous cyclic Jordan derivation from A into $A^*$ is an inner derivation, (iii) A is the algebra of all $n\times n$ matrices, where $n\ge 2$, over a cyclically amenable Banach algebra with a bounded approximate identity.

A Banach algebra A is said to be zero Lie product determined if every continuous bilinear functional $\varphi \colon A\times A\to \mathbb{C}$ with the property that $\varphi(a,b)=0$ whenever a and b commute is of the form $\varphi(a,b)=\tau(ab-ba)$ for some $\tau\in A^*$. In the first part of the paper we give some general remarks on this class of algebras. In the second part we consider amenable Banach algebras and show that all group algebras $L^1(G)$ with G an amenable locally compact group are zero Lie product determined.