arXiv subjects

Jean Roydor

Publications and source records attributed to Jean Roydor.

8 recordsLinked to original sources

arXiv2026

Operator space theory from the maximal viewpoint

We initiate a theory of ``operator spaces" (in other words subspaces of $C^*$-algebras) with an emphasis on the maximal tensor product. The natural morphisms become the maximally bounded maps instead of the completely bounded ones. With this new looking glass we first examine various issues : lifting and extension properties, local reflexivity, inner derivations, similarity problems.This first part is semi-expository while the second one focuses on the class of von Neumann algebras $M$ such that any completely bounded unital homomorphism on a $C^*$-algebra with values in $M$ is maximally bounded (or equivalently is similar to a $*$-homomorphism via a similarity belonging to $M$). We show that this holds if and only if $M$ is the direct sum of a finite von Neumann algebra and an injective one. This also characterizes the $M$'s such that all completely bounded derivations into $M$ are inner. Our main tool for this is a new characterization of injective von Neumann algebras.

math.OA↗

arXiv2017

Homomorphisms with small bound between Fourier algebras

Inspired by Kalton and Wood's work on group algebras, we describe almost completely contractive algebra homomorphisms from Fourier algebras into Fourier-Stieltjes algebras (endowed with their canonical operator space structure). We also prove that two locally compact groups are isomorphic if and only if there exists an algebra isomorphism $T$ between the associated Fourier algebras (resp. Fourier-Stieltjes algebras) with completely bounded norm $\| T \|_{cb} < \sqrt {3/2}$ (resp. $ \| T \|_{cb} < \sqrt {5}/2$). We show similar results involving the norm distortion $\| T \| \| T ^{-1} \|$ with universal but non-explicit bound. Our results subsume Walter's well-known structural theorems and also Lau's theorem on second conjugate of Fourier algebras.

math.FA↗

arXiv2013

A non-commutative Amir-Cambern theorem for von Neumann algebras and nuclear $C^*$-algebras

We prove that von Neumann algebras and separable nuclear $C^*$-algebras are stable for the Banach-Mazur cb-distance. A technical step is to show that unital almost completely isometric maps between $C^*$-algebras are almost multiplicative and almost selfadjoint. Also as an intermediate result, we compare the Banach-Mazur cb-distance and the Kadison-Kastler distance. Finally, we show that if two $C^*$-algebras are close enough for the cb-distance, then they have at most the same length.

math.OA↗

arXiv2011

Dual operator algebras close to injective von Neumann algebras

We prove that if a non-selfadjoint dual operator algebra admitting a normal virtual diagonal and an injective von Neumann algebra are close enough for the Kadison-Kastler's metric, then they are similar. The bound explicitly depends on the norm of the normal virtual diagonal. This is inspired from E. Christensen's work on perturbation of operator algebras.

math.OA↗

arXiv2011

Near inclusions of amenable operator algebras

We prove that if an amenable operator algebra is nearly contained in a complemented dual operator algebra, then it can be embedded inside this dual operator algebra via a similarity. The proof relies on a B.E. Johnson Theorem on approximately multiplicative maps.

math.OA↗

arXiv2010

Subalgebras of $C(Ω,M_n)$ and their modules

We give an operator space characterization of subalgebras of $C(Ω,M_n)$. We also describe injective subspaces of $C(Ω,M_n)$ and then give applications to sub-TROs of $C(Ω,M_n)$. Finally, we prove an `$n$-minimal version' of the Christensen-Effros-Sinclair representation theorem.

math.OA↗

arXiv2010

Isomorphisms of tensor algebras of topological graphs

We show that if two tensor algebras of topological graphs are algebraically isomorphic, then the graphs are locally conjugate. Conversely, if the base space is at most one dimensional and the edge space is compact, then locally conjugate topological graphs yield completely isometrically isomorphic tensor algebras.

math.OA↗

arXiv2009

Completely 1-complemented subspaces of Schatten spaces

We consider the Schatten spaces S^p in the framework of operator space theory and for any $1\leq p\not=2<\infty$, we characterize the completely 1-complemented subspaces of S^p. They turn out to be the direct sums of spaces of the form S^p(H,K), where H,K are Hilbert spaces. This result is related to some previous work of Arazy-Friedman giving a description of all 1-complemented subspaces of S^p in terms of the Cartan factors of types 1-4. We use operator space structures on these Cartan factors regarded as subspaces of appropriate noncommutative L^p-spaces. Also we show that for any $n\geq 2$, there is a triple isomorphism on some Cartan factor of type 4 and of dimension 2n which is not completely isometric, and we investigate L^p-versions of such isomorphisms.

math.OA↗