Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1206.0694

Derivations and Projections on Jordan Triples. An introduction to nonassociative algebra, continuous cohomology, and quantum functional analysis

Bernard Russo
View original source ↗Read source document ↗Export metadata ↓

Abstract

This paper is an elaborated version of the material presented by the author in a three hour minicourse at "V International Course of Mathematical Analysis in Andalusia," Almeria, Spain, September 12-16, 2011. Part I is devoted to an exposition of the properties of derivations on various algebras and triple systems in finite and infinite dimensions, the primary questions addressed being whether the derivation is automatically continuous and to what extent it is an inner derivation. Part II discusses cohomology theory of algebras and triple systems, in both finite and infinite dimensions. Although the cohomology of associative and Lie algebras is substantially developed, in both finite and infinite dimensions, the same could not be said for Jordan algebras. Moreover, the cohomology of triple systems has a rather sparse literature which is essentially non-existent in infinite dimensions. Thus, one of the goals of this paper is to encourage the study of continuous cohomology of some Banach triple systems. Part III discusses three topics, two very recent, which involve the interplay between Jordan theory and operator space theory (quantum functional analysis). The first one, a joint work of the author, discusses the structure theory of contractively complemented Hilbertian operator spaces, and is instrumental to the third topic, which is concerned with some recent work on enveloping TROs and K-theory for JB*-triples. The second topic presents some very recent joint work by the author concerning quantum operator algebras. One section in Part III is devoted to the subject of contractive projections, which play an important role in the structure theory of Jordan triples.

Explore related subjects

math.OA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Bernard Russo. 2015-12-10. Derivations and Projections on Jordan Triples. An introduction to nonassociative algebra, continuous cohomology, and quantum functional analysis. https://arxiv.org/abs/1206.0694

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

$C^*$-extreme maps and $*$-homomorphisms from $C(X)$ to finite von Neumann algebras

Given a unital inclusion of $C^*$-algebras $\mathcal C \subset \mathcal A$ and a unital inclusion $\mathcal C \subset \mathcal B$ into a von Neumann algebra $\mathcal B$, we investigate the extreme points of unital completely positive maps from $\mathcal A$ to $\mathcal B$ that fix $\mathcal C$ denoted by $UCP_\mathcal C(\mathcal A, \mathcal B)$. This space is obviously convex and $C^*$-convex with respect to the $C^*$-algebra $\mathcal C' \cap \mathcal B$. In this article we show that the $\mathcal C' \cap \mathcal B$-extreme points are exactly the $*$-homomorphisms that fix $\mathcal C$ when $\mathcal A$ is commutative and $\mathcal B$ has a normal faithful center valued trace. This generalizes a result due to Farenick and Morenz where $\mathcal C = \mathbb C 1$ and $\mathcal B = M_n(\mathbb C)$.

math.OA↗
arXiv2026

Quantum Cheeger Inequalities for KMS-Symmetric Quantum Markov Semigroups

In this paper, we establish a quantum Cheeger inequality for primitive KMS-symmetric quantum Markov semigroups in terms of projection conductance. We discuss both projection conductance and classical conductance for graph-based KMS-symmetric quantum Markov semigroups. We show that hypercontractivity and the logarithmic Sobolev inequality hold for primitive KMS-symmetric quantum Markov semigroups. We also present applications of the quantum Cheeger inequality to logarithmic Sobolev inequalities, hypercontractivity, and complete modified logarithmic Sobolev inequalities.

math.OA↗
arXiv2026

A characterization of simplicity of reduced groupoid C*-algebras

We show that, for a second-countable locally compact Hausdorff étale minimal groupoid with compact unit space, simplicity of the reduced groupoid C*-algebra implies the existence of a comeager set of unit points with C*-simple isotropy group. Combining this result with work of Christensen and Neshveyev on exotic completions of isotropy group algebras, we show that the converse implication is also true. Finally, we construct a Hausdorff étale minimal groupoid with an isotropy group whose induced exotic completion differs from its reduced group C*-algebra, answering a question of Christensen and Neshveyev.

math.OA↗