Extremely Non-symmetric, Non-multiplicative, Non-commutative Operator Spaces

2008/05/22 by Wacław Szymański, Waclaw Szymanski, Szymanski, Waclaw
Mathematics · #Advanced Operator Algebra Research #Advanced Topics in Algebra #Holomorphic and Operator Theory #math.FA #math.OA #msc:46L07

paper · pdf · doi:10.48550/arxiv.0805.3513

arxiv created 2008/05/22 · arxiv updated 2009/12/01

Abstract

Motivated by importance of operator spaces contained in the set of all scalar multiples of isometries (MI-spaces) in a separable Hilbert space for C^*-algebras and E-semigroups we exhibit more properties of such spaces. For example, if an MI-space contains an isometry with shift part of finite multiplicity, then it is one-dimensional. We propose a simple model of a unilateral shift of arbitrary multiplicity and show that each separable subspace of a Hilbert space is the range of a shift. Also, we show that MI-spaces are non-symmetric, very unfriendly to multiplication, and prove a Commutator Identity which elucidates the extreme non-commutativity of these spaces.

Related

• Isomorphisms of operator algebras
• A double commutant theorem for operator algebras
• Isometries on extremely non-complex Banach spaces
• Operator space characterizations of C*-algebras and ternary rings
• On the unitary part of isometries in commuting, completely non doubly commuting pairs