Centralizing additive maps on rank r block triangular matrices
Let F be a field and let k, n1, . . . , nk be positive integers with n1 + · · · + nk = n > 2. We denote by Tn1,...,nk a block triangular matrix algebra over F with unity In and center Z(Tn1,...,nk ). Fixing an integer 1 < r 6 n with r =6 n when |F| = 2, we prove that an additive map ψ: Tn1,......
Elmentve itt :
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| Dokumentumtípus: | Cikk |
| Megjelent: |
2021
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| Sorozat: | Acta scientiarum mathematicarum
87 No. 1-2 |
| Kulcsszavak: | Matematika |
| doi: | 10.14232/actasm-020-586-y |
| Online Access: | http://acta.bibl.u-szeged.hu/73917 |
| Tartalmi kivonat: | Let F be a field and let k, n1, . . . , nk be positive integers with n1 + · · · + nk = n > 2. We denote by Tn1,...,nk a block triangular matrix algebra over F with unity In and center Z(Tn1,...,nk ). Fixing an integer 1 < r 6 n with r =6 n when |F| = 2, we prove that an additive map ψ: Tn1,...,nk → Tn1,...,nk satisfies ψ(A)A−Aψ(A) ∈ Z(Tn1,...,nk ) for every rank r matrices A ∈ Tn1,...,nk if and only if there exist an additive map µ: Tn1,...,nk → F and scalars λ, α ∈ F, in which α 6= 0 only if r = n, n1 = nk = 1 and |F| = 3, such that ψ(A) = λA + µ(A)In + α(a11 + ann)E1n for all A = (aij ) ∈ Tn1,...,nk , where Eij ∈ Tn1,...,nk is the matrix unit whose (i, j)th entry is one and zero elsewhere. Using this result, a complete structural characterization of commuting additive maps on rank s > 1 upper triangular matrices over an arbitrary field is addressed. |
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| Terjedelem/Fizikai jellemzők: | 63-94 |
| ISSN: | 2064-8316 |