Commuting additive maps on tensor products of matrix algebras / Wong Jian Yong - Wong , Jian Yong (2021)
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.