Rank Preserving Maps on CSL Algebras

Decompositions of Reflexive Bimodules over Maximal Abelian Selfadjoint Algebras

We generalize the notion of `diagonal' from the class of CSL algebras to masa bimodules. We prove that a reflexive masa bimodule decomposes as a sum of two bimodules, the diagonal and a module generalizing the w*-closure of the Jacobson radical of a CSL algebra. The latter module turns out to be reflexive, a result which is new even for CSL algebras. We show that the projection onto the direct summand contained in the diagonal is contractive and preserves compactness and reduces rank of operators. Stronger results are obtained when the module is the reflexive hull of its rank-one subspace. Key…

Rank-preserving module maps

In this paper, we characterize rank one preserving module maps on a Hilbert $C^\ast-$module and study its applications on free probability theory.

Additive preserving rank one maps on Hilbert $C^\ast$-modules

In this paper, we characterize a class of additive maps on Hilbert $C^\ast$-modules which maps a "rank one" adjointable operators to another rank one operators.

On positive definite preserving linear transformations of rank $r$ on real symmetric matrices

We study on what conditions on $B_k,$ \ a linear transformation of rank $r$ \label{form} T(A)=\sum_{k=1}^r\tr(AB_k)U_k where $U_k,\ k=1,2,..., r$ are linear independent and all positive definite; is positive definite preserving. We give some first results for this question. For the case of rank one and two, the necessary and sufficient conditions are given. We also give some sufficient conditions for the case of rank $r.$

Additive mappings in algebras of unbounded operators preserving operators of rank one

Let $ \mathcal D$ be a dense linear manifold in a Hilbert space $\mathcal H$ and let $L^+(\mathcal D)$ be the *-algebra of all linear operators $A$ such that $A \mathcal D \subset \mathcal D, A^* \mathcal D \subset \mathcal D$. Denote by $F(\mathcal D)$ the *-ideal of $L^+(\mathcal D)$ consisting of all finite-rank operators. A characterization of the structure of additive mappings of $F(\mathcal D)$ preserving operators of rank one or projections of rank one is given. The corresponding results for algebras of bounded operators on Banach spaces were given by P. uSemrl.

II$_1$-型因子上的秩度量等距与保行列式映射

Rank metric isometries and determinant-preserving mappings on II$_1$-factors

该研究完整刻画了II$_1$-型因子对应的Murray--von Neumann代数上秩度量等距的一般形式，借此在II$_1$-型因子场景下证明Frobenius定理，证实了1996年的Harris--Kadison猜想。