<abstract xmlns="http://www.w3.org/1999/xhtml">Let 𝒜 be a Banach *-algebra. By 𝒮𝒜 we denote the set of all self-adjoint elements of 𝒜 and by 𝒪𝒜 we denote the set of those elements in 𝒜 which can be represented as finite real-linear combinations of mutually orthogonal projections. The main purpose of this paper is to prove the following result:Suppose that 
<inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ausm-2017-0011_eq_001.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mover accent="true"><mrow><msub><mi>𝒪</mi><mi>𝒜</mi></msub></mrow><mo stretchy="true">¯</mo></mover><mo>=</mo><msub><mi>𝒮</mi><mi>𝒜</mi></msub></math><tex-math>$\overline {{\cal O}_{\cal A} } = {\cal S}_{\cal A }$</tex-math></alternatives></inline-formula>
 and {dn} is a sequence of uniformly bounded linear mappings satisfying 
<inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ausm-2017-0011_eq_002.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mtext>d</mtext><mtext>n</mtext></msub><mo stretchy="false">(</mo><mtext>p</mtext><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true"><msubsup><mo>∑</mo><mrow><mtext>k</mtext><mo>=</mo><mn>0</mn></mrow><mtext>n</mtext></msubsup><mrow><msub><mtext>d</mtext><mrow><mtext>n</mtext><mo>−</mo><mtext>k</mtext></mrow></msub><mo stretchy="false">(</mo><mtext>p</mtext><mo stretchy="false">)</mo><msub><mtext>d</mtext><mtext>k</mtext></msub><mo stretchy="false">(</mo><mtext>p</mtext><mo stretchy="false">)</mo></mrow></mstyle></math><tex-math>${\rm{d}}_{\rm{n}} ({\rm{p}}) = \sum\nolimits_{{\rm{k}} = 0}^{\rm{n}} {{\rm{d}}_{{\rm{n}} - {\rm{k}}} ({\rm{p}}){\rm{d}}_{\rm{k}} ({\rm{p}})} $</tex-math></alternatives></inline-formula>
, where p is an arbitrary projection in 𝒜. Then dn(𝒜) ⊆ ∪ϕ∈Φ𝒜 ker ϕ for each n ≥ 1. In particular, if 𝒜 is semi-prime and further, dim(∪ϕ∈Φ𝒜 ker ϕ) ≤ 1, then dn = 0 for each n ≥ 1.</abstract>

Let 𝒜 be a Banach *-algebra. By 𝒮𝒜 we denote the set of all self-adjoint elements of 𝒜 and by 𝒪𝒜 we denote the set of those elements in 𝒜 which can be represented as finite real-linear combinations of mutually orthogonal projections. The main purpose of this paper is to prove the following result:Suppose that 
𝒪𝒜¯=𝒮𝒜$\overline {{\cal O}_{\cal A} }  = {\cal S}_{\cal A }$
 and {dn} is a sequence of uniformly bounded linear mappings satisfying 
dn(p)=∑k=0ndn−k(p)dk(p)${\rm{d}}_{\rm{n}} ({\rm{p}}) = \sum\nolimits_{{\rm{k}} = 0}^{\rm{n}} {{\rm{d}}_{{\rm{n}} - {\rm{k}}} ({\rm{p}}){\rm{d}}_{\rm{k}} ({\rm{p}})} $
, where p is an arbitrary projection in 𝒜. Then dn(𝒜) ⊆ ∪ϕ∈Φ𝒜 ker ϕ for each n ≥ 1. In particular, if 𝒜 is semi-prime and further, dim(∪ϕ∈Φ𝒜 ker ϕ) ≤ 1, then dn = 0 for each n ≥ 1.

Some conditions under which derivations are zero on Banach *-algebras

Acta Universitatis Sapientiae, Mathematica

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

{"article-title":"Some conditions under which derivations are zero on Banach *-algebras"}

Let 𝒜 be a Banach *-algebra. By 𝒮𝒜 we denote the set of all self-adjoint elements of 𝒜 and by 𝒪𝒜 we denote the set of those elements in 𝒜 which can be...

Some conditions under which derivations are zero on Banach *-algebras

Published Online: Aug 05, 2017

Page range: 176 - 184

Received: Apr 26, 2017

DOI: https://doi.org/10.1515/ausm-2017-0011

Keywords
(Jordan) derivation, pre-higher derivation, Banach *-algebra

© 2017 Amin Hosseini, published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Some conditions under which derivations are zero on Banach *-algebras

Published Online: Aug 05, 2017

Page range: 176 - 184

Received: Apr 26, 2017

DOI: https://doi.org/10.1515/ausm-2017-0011

Keywords(Jordan) derivation, pre-higher derivation, Banach *-algebra

© 2017 Amin Hosseini, published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
(Jordan) derivation, pre-higher derivation, Banach *-algebra