Continuous and discrete flows on operator algebras

Let $(N,\R,\theta)$ be a centrally ergodic W* dynamical system. When $N$ is not a factor, we show that, for each $t\not=0$, the crossed product induced by the time $t$ automorphism $\theta_t$ is not a factor if and only if there exist a rational number $r$ and an eigenvalue $s$ of the restriction of $\theta$ to the center of $N$, such that $rst=2\pi$. In the C* setting, minimality seems to be the notion corresponding to central ergodicity. We show that if $(A,\R,\alpha)$ is a minimal unital C* dynamical system and $A$ is either prime or commutative but not simple, then, for each $t\not=0$, the crossed product induced by the time $t$ automorphism $\alpha_t$ is not simple if and only if there exist a rational number $r$ and an eigenvalue $s$ of the restriction of $\alpha$ to the center of $A$, such that $rst=2\pi$.Comment: 7 page

Matrices similar to centrosymmetric matrices

In this paper we give conditions on a matrix which guarantee that it is similar to a centrosymmetric matrix. We use this conditions to show that some $4 \times 4$ and $6 \times 6$ Toeplitz matrices are similar to centrosymmetric matrices. Furthermore, we give conditions for a matrix to be similar to a matrix which has a centrosymmetric principal submatrix, and conditions under which a matrix can be dilated to a matrix similar to a centrosymmetric matrix.Comment: 15 page

The numerical range of some periodic tridiagonal operators is the convex hull of the numerical ranges of two finite matrices

In this paper we prove a conjecture stated by the first two authors in \cite{IM} establishing the closure of the numerical range of a certain class of $n+1$-periodic tridiagonal operators as the convex hull of the numerical ranges of two tridiagonal $(n+1) \times (n+1)$ matrices. Furthermore, when $n+1$ is odd, we show that the size of such matrices simplifies to $\frac{n}{2}+1$

The numerical range of periodic banded Toeplitz operators

We prove that the closure of the numerical range of a $(n+1)$-periodic and $(2m+1)$-banded Toeplitz operator can be expressed as the closure of the convex hull of the uncountable union of numerical ranges of certain symbol matrices. In contrast to the periodic $3$-banded (or tridiagonal) case, we show an example of a $2$-periodic and $5$-banded Toeplitz operator such that the closure of its numerical range is not equal to the numerical range of a single finite matrix.Comment: 17 pages, 1 figur