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1. CJM 2016 (vol 69 pp. 373)

Kaftal, Victor; Ng, Ping Wong; Zhang, Shuang
 Strict Comparison of Positive Elements in Multiplier Algebras Main result: If a C*-algebra $\mathcal{A}$ is simple, $\sigma$-unital, has finitely many extremal traces, and has strict comparison of positive elements by traces, then its multiplier algebra $\operatorname{\mathcal{M}}(\mathcal{A})$ also has strict comparison of positive elements by traces. The same results holds if finitely many extremal traces" is replaced by quasicontinuous scale". A key ingredient in the proof is that every positive element in the multiplier algebra of an arbitrary $\sigma$-unital C*-algebra can be approximated by a bi-diagonal series. An application of strict comparison: If $\mathcal{A}$ is a simple separable stable C*-algebra with real rank zero, stable rank one, and strict comparison of positive elements by traces, then whether a positive element is a positive linear combination of projections is determined by the trace values of its range projection. Keywords:strict comparison, bi-diagonal form, positive combinationsCategories:46L05, 46L35, 46L45, 47C15

2. CJM 2002 (vol 54 pp. 571)

Li, Chi-Kwong; Poon, Yiu-Tung
 Diagonals and Partial Diagonals of Sum of Matrices Given a matrix $A$, let $\mathcal{O}(A)$ denote the orbit of $A$ under a certain group action such as \begin{enumerate}[(4)] \item[(1)] $U(m) \otimes U(n)$ acting on $m \times n$ complex matrices $A$ by $(U,V)*A = UAV^t$, \item[(2)] $O(m) \otimes O(n)$ or $\SO(m) \otimes \SO(n)$ acting on $m \times n$ real matrices $A$ by $(U,V)*A = UAV^t$, \item[(3)] $U(n)$ acting on $n \times n$ complex symmetric or skew-symmetric matrices $A$ by $U*A = UAU^t$, \item[(4)] $O(n)$ or $\SO(n)$ acting on $n \times n$ real symmetric or skew-symmetric matrices $A$ by $U*A = UAU^t$. \end{enumerate} Denote by $$\mathcal{O}(A_1,\dots,A_k) = \{X_1 + \cdots + X_k : X_i \in \mathcal{O}(A_i), i = 1,\dots,k\}$$ the joint orbit of the matrices $A_1,\dots,A_k$. We study the set of diagonals or partial diagonals of matrices in $\mathcal{O}(A_1,\dots,A_k)$, {\it i.e.}, the set of vectors $(d_1,\dots,d_r)$ whose entries lie in the $(1,j_1),\dots,(r,j_r)$ positions of a matrix in $\mathcal{O}(A_1, \dots,A_k)$ for some distinct column indices $j_1,\dots,j_r$. In many cases, complete description of these sets is given in terms of the inequalities involving the singular values of $A_1,\dots,A_k$. We also characterize those extreme matrices for which the equality cases hold. Furthermore, some convexity properties of the joint orbits are considered. These extend many classical results on matrix inequalities, and answer some questions by Miranda. Related results on the joint orbit $\mathcal{O}(A_1,\dots,A_k)$ of complex Hermitian matrices under the action of unitary similarities are also discussed. Keywords:orbit, group actions, unitary, orthogonal, Hermitian, (skew-)symmetric matrices, diagonal, singular valuesCategories:15A42, 15A18

3. CJM 2000 (vol 52 pp. 613)

Ou, Zhiming M.; Williams, Kenneth S.
 Small Solutions of $\phi_1 x_1^2 + \cdots + \phi_n x_n^2 = 0$ Let $\phi_1,\dots,\phi_n$ $(n\geq 2)$ be nonzero integers such that the equation $$\sum_{i=1}^n \phi_i x_i^2 = 0$$ is solvable in integers $x_1,\dots,x_n$ not all zero. It is shown that there exists a solution satisfying $$0 < \sum_{i=1}^n |\phi_i| x_i^2 \leq 2 |\phi_1 \cdots \phi_n|,$$ and that the constant 2 is best possible. Keywords:small solutions, diagonal quadratic formsCategory:11E25
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