Commuting additive maps on tensor products of matrix algebras / Wong Jian Yong
Let k ⩾ 1 and n1, . . . , nk ⩾ 2 be integers. Let F be a field and letMni be the algebra of ni × ni matrices over F for i = 1, . . . , k. Let ⊗ki=1Mni be the tensor product of Mn1 , . . . ,Mnk . In this dissertation, we obtain a complete structural characterization of additive maps ψ : ⊗k i=1 M...
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| Format: | Thesis |
| Published: |
2021
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| Subjects: | |
| Online Access: | http://studentsrepo.um.edu.my/12911/ http://studentsrepo.um.edu.my/12911/2/Wong_Jian_Yong.pdf http://studentsrepo.um.edu.my/12911/1/Wong_Jian_Yong.pdf |
| _version_ | 1848774752290734080 |
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| author | Wong , Jian Yong |
| author_facet | Wong , Jian Yong |
| author_sort | Wong , Jian Yong |
| building | UM Research Repository |
| collection | Online Access |
| description | Let k ⩾ 1 and n1, . . . , nk ⩾ 2 be integers. Let F be a field and letMni be the algebra of ni × ni matrices over F for i = 1, . . . , k. Let
⊗ki=1Mni be the tensor product of Mn1 , . . . ,Mnk . In this dissertation, we obtain a complete structural characterization of additive maps ψ :
⊗k
i=1
Mni
→
⊗k
i=1
Mni satisfying
ψ(⊗k
i=1Ai)(⊗ki
=1Ai) = (⊗ki
=1Ai) ψ(⊗ki
=1Ai)
for all A1 ∈ S1,n1 , . . . ,Ak ∈ Sk,nk , where
Si,ni =
{
E(ni)
st + αE(ni)
pq : α ∈ F and 1 ⩽ p, q, s, t ⩽ ni are not all distinct integers
}
and E(ni)
st is the standard matrix unit inMni for i = 1, . . . , k. In particular, we show that
ψ :Mn1
→Mn1 is an additive map commuting on S1,n1 if and only if there exist a scalar
λ ∈ F and an additive map μ :Mn1
→ F such that
ψ(A) = λA + μ(A)In1
for all A ∈ Mn1 , where In1
∈ Mn1 is the identity matrix. As an application, we
classify additive maps ψ :
⊗k
i=1
Mni
→
⊗k
i=1
Mni satisfying ψ(⊗ki
=1Ai)(⊗ki
=1Ai) =
(⊗ki
=1Ai) ψ(⊗ki=1Ai) for all A1 ∈ Rn1
r1 , . . . ,Ak ∈ Rnk
rk . Here, Rni
ri denotes the set of rank
ri matrices inMni and 1 < ri ⩽ ni is a fixed integer such that ri ̸= ni when ni = 2 and
|F| = 2 for i = 1, . . . , k.
|
| first_indexed | 2025-11-14T14:03:18Z |
| format | Thesis |
| id | um-12911 |
| institution | University Malaya |
| institution_category | Local University |
| last_indexed | 2025-11-14T14:03:18Z |
| publishDate | 2021 |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | um-129112022-02-28T23:55:46Z Commuting additive maps on tensor products of matrix algebras / Wong Jian Yong Wong , Jian Yong QA Mathematics Let k ⩾ 1 and n1, . . . , nk ⩾ 2 be integers. Let F be a field and letMni be the algebra of ni × ni matrices over F for i = 1, . . . , k. Let ⊗ki=1Mni be the tensor product of Mn1 , . . . ,Mnk . In this dissertation, we obtain a complete structural characterization of additive maps ψ : ⊗k i=1 Mni → ⊗k i=1 Mni satisfying ψ(⊗k i=1Ai)(⊗ki =1Ai) = (⊗ki =1Ai) ψ(⊗ki =1Ai) for all A1 ∈ S1,n1 , . . . ,Ak ∈ Sk,nk , where Si,ni = { E(ni) st + αE(ni) pq : α ∈ F and 1 ⩽ p, q, s, t ⩽ ni are not all distinct integers } and E(ni) st is the standard matrix unit inMni for i = 1, . . . , k. In particular, we show that ψ :Mn1 →Mn1 is an additive map commuting on S1,n1 if and only if there exist a scalar λ ∈ F and an additive map μ :Mn1 → F such that ψ(A) = λA + μ(A)In1 for all A ∈ Mn1 , where In1 ∈ Mn1 is the identity matrix. As an application, we classify additive maps ψ : ⊗k i=1 Mni → ⊗k i=1 Mni satisfying ψ(⊗ki =1Ai)(⊗ki =1Ai) = (⊗ki =1Ai) ψ(⊗ki=1Ai) for all A1 ∈ Rn1 r1 , . . . ,Ak ∈ Rnk rk . Here, Rni ri denotes the set of rank ri matrices inMni and 1 < ri ⩽ ni is a fixed integer such that ri ̸= ni when ni = 2 and |F| = 2 for i = 1, . . . , k. 2021-05 Thesis NonPeerReviewed application/pdf http://studentsrepo.um.edu.my/12911/2/Wong_Jian_Yong.pdf application/pdf http://studentsrepo.um.edu.my/12911/1/Wong_Jian_Yong.pdf Wong , Jian Yong (2021) Commuting additive maps on tensor products of matrix algebras / Wong Jian Yong. Masters thesis, Universiti Malaya. http://studentsrepo.um.edu.my/12911/ |
| spellingShingle | QA Mathematics Wong , Jian Yong Commuting additive maps on tensor products of matrix algebras / Wong Jian Yong |
| title | Commuting additive maps on tensor products of matrix algebras / Wong Jian Yong |
| title_full | Commuting additive maps on tensor products of matrix algebras / Wong Jian Yong |
| title_fullStr | Commuting additive maps on tensor products of matrix algebras / Wong Jian Yong |
| title_full_unstemmed | Commuting additive maps on tensor products of matrix algebras / Wong Jian Yong |
| title_short | Commuting additive maps on tensor products of matrix algebras / Wong Jian Yong |
| title_sort | commuting additive maps on tensor products of matrix algebras / wong jian yong |
| topic | QA Mathematics |
| url | http://studentsrepo.um.edu.my/12911/ http://studentsrepo.um.edu.my/12911/2/Wong_Jian_Yong.pdf http://studentsrepo.um.edu.my/12911/1/Wong_Jian_Yong.pdf |