J. Extremera

Characterizing Jordan maps on C*-algebras through zero products

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 e A are such that ab = ba = 0. We prove that then T = w Φ 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.

Uniqueness of norm on Lp(G) and C(G) when G is a compact group

Let G be a compact group. If the trivial representation of G is not weakly contained in the left regular representation of G on L 2 ðGÞ and X is either L p ðGÞ for 1oppN or CðGÞ; then we show that every complete norm jj on X that makes translations from ðX ; jj Þ into itself continuous is equivalent to jjjj p or jjjj N respectively. If 1oppN and every left invariant linear functional on L p ðGÞ is a constant multiple of the Haar integral, then we show that every complete norm jj on L p ðGÞ that makes translations from ðL p ðGÞ; jj Þ into itself continuous and that makes the map t/Lt from G into LðL p ðGÞ; jj Þ bounded is equivalent to jjjj p:

Metric versions of Herstein's theorems on Jordan maps

Let A be a Banach algebra and let B be an ultraprime Banach algebra. If Φ: A → B is a surjective continuous linear map which tends to satisfy the Jordan multiplicativity condition, then we show that Φ comes near to satisfy either the multiplicativity or the anti-multiplicativity condition. In fact, we give a quantitative estimate of this phenomenon. Furthermore, we estimate how much a continuous linear map Δ: B → B approaches to satisfy the derivation identity in the case when Δ tends to satisfy the Jordan derivation identity

Zero Lie product determined Banach algebras

A Banach algebra

Operators Splitting the Arveson Spectrum

A

Maps preserving zero products

is said to be zero Lie product determined if every continuous bilinear functional

Characterizing homomorphisms and derivations on C * -algebras

\varphi \colon A\times A\to \mathbb{C}

Determining elements in Banach algebras through spectral properties

satisfying

On bilinear maps determined by rank one idempotents

\varphi(a,b)=0

Approximately spectrum-preserving maps☆

whenever