Completely bounded norms of right module maps

It is well-known that if T is a $D_m-D_n$ bimodule map on the m × n complex matrices, then T is a Schur multiplier and $6T{6}_{\mathit{cb}}=6T6$. If n = 2 and T is merely assumed to be a right D₂-module map, then we show that $6T{6}_{\mathit{cb}}=6T6$. However, this property fails if m ? 2 and n ? 3. For m ? 2 and n = 3, 4 or n ? m² we give examples of maps T attaining the supremum$C(m\text{,}n)=\sup T{6}_{\mathit{cb}}:T\text{a right}{D}_n\text{-module map on}{M}_{m\text{,}n}\text{with}6T6\le 1\}\text{,}$we show that $C(m\text{,}{m}^2)=\sqrt{m}$ and succeed in finding sharp results for C(m, n) in certain other cases. As a consequence, if H is an infinite-dimensional Hilbert space and D is a masa in B(H), then there is a bounded right D-module map on K(H) which is not completely bounded.     

Dynamical systems method and surjectivity of nonlinear maps

If F : H → H is a $C_{\mathrm{loc}}^2$-map in a real Hilbert space, sup_{u ∈ B(u0,R)}∥[F′(u)]⁻¹∥ ⩽ m(R), and ${\mathrm{limsup}}_{R>0}\frac{R}{m(R)}=\infty$, then F is surjective.     

Dimension group of a non-negative integer irreducible matrix

In this paper we are concerned with a non-negative integer and irreducible matrix $A:{\mathbb{Z}}^d\to {\mathbb{Z}}^d$. The main contribution is to prove that if the matrix satisfies certain spectral and algebraic constraints, the cone:$C=\{v\in {\mathbb{Z}}^d/\exists n⩾0\mathrm{and}{A}^{n}v⩾0\}\subset {\mathbb{Z}}^d$is defined by linear maps ${\varphi }_0\text{,}\dots \text{,}{\varphi }_{k-1}:{\mathbb{Z}}^d\to \mathbb{R}$, in the sense that v ∈ C is equivalent to, ϕ_l(v) ⩾ 0 for all l = 0, … , k − 1 (where k is the index of cyclicity of the irreducible matrix). This result allows us to characterize the dimension group generated by the matrix, it is a subgroup of ${\mathbb{R}}^k$ endowed with an order induced by the positive cone of ${\mathbb{R}}^k$.     

On the trivectors of a 6-dimensional symplectic vector space. II

Let V be a 6-dimensional vector space over a field $\mathbb{F}$, let f be a nondegenerate alternating bilinear form on V and let $\mathit{Sp}(V\text{,}f)?{\mathit{Sp}}_6(\mathbb{F})$ denote the symplectic group associated with $(V\text{,}f)$. The group $\mathit{GL}(V)$ has a natural action on the third exterior power ${?}^{3}V$ of V and this action defines five families of nonzero trivectors of V. Four of these families are orbits for any choice of the field $\mathbb{F}$. The orbits of the fifth family are in one-to-one correspondence with the quadratic extensions of $\mathbb{F}$ that are contained in a fixed algebraic closure $\overline{\mathbb{F}}$ of $\mathbb{F}$. In this paper, we divide the orbits corresponding to the separable quadratic extensions into suborbits for the action of $\mathit{Sp}(V\text{,}f)?\mathit{GL}(V)$ on ${?}^{3}V$.