arXiv subjects
Publications and source records attributed to Qingying Bu.
We characterize the positive Schur property in the positive projective tensor products of Banach lattices, we establish the connection with the weak operator topology and we give necessary and sufficient conditions for the space of regular multilinear operators/homogeneous polynomials taking values in a Dedekind complete Banach lattice to have the positive Schur property.
Let $A,B$ be C*-algebras, $B_A(0;r)$ the open ball in $A$ centered at $0$ with radius $r>0$, and $H:B_A(0;r)\to B$ an orthogonally additive holomorphic map. If $H$ is zero product preserving on positive elements in $B_A(0;r)$, we show, in the commutative case when $A=C_0(X)$ and $B=C_0(Y)$, that there exist weight functions $h_n$'s and a symbol map $φ: Y\to X$ such that $$ H(f)=\sum_{n\geq1} h_n (f\circφ)^n, \quad\forall f\in B_{C_0(X)}(0;r). $$ In the general case, we show that if $H$ is also conformal then there exist central multipliers $h_n$'s of $B$ and a surjective Jordan isomorphism $J: A\to B$ such that $$ H(a) = \sum_{n\geq1} h_n J(a)^n, \quad\forall a\in B_A(0;r). $$ If, in addition, $H$ is zero product preserving on the whole $B_A(0;r)$, then $J$ is an algebra isomorphism. %Similar conclusions hold for orthogonally additive $n$-homogeneous polynomials which are $n$-isometries.
In this paper, we show that four main diagonal spaces of injective tensor products are pairwise isometrically isomorphic. When E is a Banach lattice, we show that the tensor diagonal of E is a 1-unconditional basic sequence in both the n-fold injective tensor product of E and the n-fold symmetric injective tensor product of E.
Let $H:M_m\to M_m$ be a holomorphic function of the algebra $M_m$ of complex $m\times m$ matrices. Suppose that $H$ is orthogonally additive and orthogonally multiplicative on self-adjoint elements. We show that either the range of $H$ consists of zero trace elements, or there is a scalar sequence $\{λ_n\}$ and an invertible $S$ in $M_m$ such that $$ H(x) =\sum_{n\geq 1} λ_n S^{-1}x^nS, \quad\forall x \in M_m,%\eqno{(‡)} $$ or $$ H(x) =\sum_{n\geq 1} λ_n S^{-1}(x^t)^nS, \quad\forall x \in M_m. $$ Here, $x^t$ is the transpose of the matrix $x$. In the latter case, we always have the first representation form when $H$ also preserves zero products. We also discuss the cases where the domain and the range carry different dimensions.
We investigate the relationship between the diagonal of the Fremlin projective tensor product of a Banach lattice $E$ with itself and the 2-concavification of $E$.