因子von Neumann代数上导子的等价刻画
Equivalent Characterization of Derivations on Factor von Neumann Algebras
设R是含非平凡幂等元P的素环,C∈R,C=PC.本文证明可加映射△:R→R在C可导,即△(AB)=△(A)B+A△(B),∀ A,B∈R,AB=C当且仅当存在导子δ:{R}→{R},使得△(A)=δ(A)+△(I)A,∀ A∈R.没有I1型中心直和项的von Neumann代数上的可导映射也有类似结论.利用该结论证明了,若非零算子C∈B(X),使得ran(C)或ker(C)在X中可补,则可加映射△:B(X)→B(X)在C可导当且仅当它是导子.特别地,证明了因子von Neumann代数上的可加映射在任意但固定的非零算子可导当且仅当它是导子.
Let R be a prime ring containing a nontrivial idempotent P. Suppose C ∈ R satisfies C=PC. It is shown that an additive map Δ:R → R is derivable at C, that is, Δ(AB)=Δ(A)B + AΔ(B) for every A, B ∈ R with AB=C if and only if there exists a derivation δ:R → R such that Δ(A)=δ(A) + Δ(I)A for all A ∈ R. Similar results are obtained for von Neumann algebras with no central abelian projections. As its application, we obtain that, if nonzero operator C ∈ B(X) such that ran(C) or ker(C) is complementary in X, then an additive map Δ:B(X) → B(X) is derivable at C if and only if it is a derivation. In particular, we show that an additive map from a factor von Neumann algebra into itself is derivable at an arbitrary but fixed nonzero operator if and only if it is a derivation.
素环 / 导子 / 可补子空间 / von Neumann代数 / 中心覆盖 {{custom_keyword}} /
prime rings / derivations / complementary subspaces / von Neumann algebras / central carrier {{custom_keyword}} /
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国家自然科学基金资助项目(11001194)
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