Weighted Jordan homomorphisms Brešar, M.; Godoy, M. L. C.
Linear & multilinear algebra,
05/2023, Volume:
71, Issue:
8
Journal Article
Peer reviewed
Open access
Let A and B be unital rings. An additive map
is called a weighted Jordan homomorphism if
is an invertible central element and
for all
. We provide assumptions, which are in particular fulfilled when
...with
and R any unital ring with
, under which every surjective additive map
with the property that
whenever xy = yx = 0 is a weighted Jordan homomorphism. Further, we show that if A is a prime ring with char
, then a bijective additive map
is a weighted Jordan homomorphism provided that there exists an additive map
such that
for all
.
ZERO JORDAN PRODUCT DETERMINED BANACH ALGEBRAS ALAMINOS, J.; BREŠAR, M.; EXTREMERA, J. ...
Journal of the Australian Mathematical Society,
10/2021, Volume:
111, Issue:
2
Journal Article
Peer reviewed
Open access
A Banach algebra
$A$
is said to be a zero Jordan product determined Banach algebra if, for every Banach space
$X$
, every bilinear map
$\unicodeSTIX{x1D711}:A\times A\rightarrow X$
satisfying
...$\unicodeSTIX{x1D711}(a,b)=0$
whenever
$a$
,
$b\in A$
are such that
$ab+ba=0$
, is of the form
$\unicodeSTIX{x1D711}(a,b)=\unicodeSTIX{x1D70E}(ab+ba)$
for some continuous linear map
$\unicodeSTIX{x1D70E}$
. 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.
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.
Zero Lie product determined Banach algebras, II Alaminos, J.; Brešar, M.; Extremera, J. ...
Journal of mathematical analysis and applications,
06/2019, Volume:
474, Issue:
2
Journal Article
Peer reviewed
Open access
A Banach algebra A is said to be zero Lie product determined if every continuous bilinear functional φ:A×A→C satisfying φ(a,b)=0 whenever ab=ba is of the form φ(a,b)=ω(ab−ba) for some ω∈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 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×n matrices, where n≥2, over a cyclically amenable Banach algebra with a bounded approximate identity.
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.
Let A and B be C*-algebras, let X be an essential Banach A-bimodule and let T : A → B and S : A → X be continuous linear maps with T surjective. Suppose that T(a)T(b) + T(b)T(a) = 0 and S(a)b + bS(a) ...+ aS(b) + S(b)a = 0 whenever a, b ε A are such that ab = ba = 0. We prove that then T = wΦ and S = D + Ψ, where w lies in the centre of the multiplier algebra of B, Φ: A → B is a Jordan epimorphism, D: A → X is a derivation and Ψ: A → X is a bimodule homomorphism.
This paper considers Banach algebras with properties A or B, introduced recently by Alaminos et al. The class of Banach algebras satisfying either of these two properties is quite large; in ...particular, it includes C super(*)-algebras and group algebras on locally compact groups. Our first main result states that a continuous orthogonally additive n-homogeneous polynomial on a commutative Banach algebra with property A and having a bounded approximate identity is of a standard form. The other main results describe Banach algebras A with property B and having a bounded approximate identity that admit non-zero continuous symmetric orthosymmetric n-linear maps from A super(n)into C.
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