On the minimal rank in non-reflexive operator spaces over finite fields

Pazzis, Clément de Seguins

doi:10.1016/j.laa.2014.08.017

Cited by 2 publications

(1 citation statement)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This result is in sharp contrast with classical sufficient conditions for algebraic reflexivity (see [6,7,9]), which are based upon comparing the minimal rank for the non-zero operators in S with the dimension of S.…”

Section: ∀S ∈ S F(s) ∈ Im Smentioning

confidence: 71%

Range-compatible homomorphisms on matrix spaces

Pazzis

2015

Linear Algebra and its Applications

Self Cite

View full text Add to dashboard Cite

Let K be a (commutative) field, and U and V be finite-dimensional vector spaces over K. Let S be a linear subspace of the space L(U,V) of all linear operators from U to V. A map F from S to V is called range-compatible when F(s) belongs to the range of s for all s in S. Obvious examples of such maps are the evaluation maps s -> s(x), with x in U. In this article, we classify all the range-compatible group homomorphisms on S provided that the codimension of S in L(U,V) is less than or equal to 2 dim(V)-3, unless this codimension equals 2 dim(V)-3 and K has only two elements. Under those assumptions, it is shown that the linear range-compatible maps are the evaluation maps, and the above upper-bound on the codimension of S is optimal for this result to hold. As an application, we obtain new sufficient conditions for the algebraic reflexivity of an operator space and, with the above conditions on the codimension of S, we give an explicit description of the range-restricting and range-preserving homomorphisms on S.Comment: 60 page

show abstract

Section: ∀S ∈ S F(s) ∈ Im Smentioning

confidence: 71%

Range-compatible homomorphisms on matrix spaces

Pazzis

2015

Linear Algebra and its Applications

Self Cite

View full text Add to dashboard Cite

show abstract

LDB division algebras

Pazzis

2015

Journal of Algebra

Self Cite

View full text Add to dashboard Cite

An LDB division algebra is a triple $(A,\star,\bullet)$ in which $\star$ and $\bullet$ are regular bilinear laws on the finite-dimensional non-zero vector space $A$ such that $x \star (x \bullet y)$ is a scalar multiple of $y$ for all vectors $x$ and $y$ of $A$. This algebraic structure has been recently discovered in the study of the critical case in Meshulam and \v Semrl's estimate of the minimal rank in non-reflexive operator spaces. In this article, we obtain a constructive description of all LDB division algebras over an arbitrary field together with a reduction of the isotopy problem to the similarity problem for specific types of quadratic forms over the given field. In particular, it is shown that the dimension of an LDB division algebra is always a power of $2$, and that it belongs to $\{1,2,4,8\}$ if the characteristic of the underlying field is not $2$.Comment: 35 page

show abstract

On the minimal rank in non-reflexive operator spaces over finite fields

Cited by 2 publications

References 9 publications

Range-compatible homomorphisms on matrix spaces

Range-compatible homomorphisms on matrix spaces

LDB division algebras

Contact Info

Product

Resources

About