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

TL;DR

This paper explores the properties of $MI$-spaces, a class of operator spaces linked to isometries in Hilbert spaces, revealing their extreme non-symmetry, non-multiplicativity, and non-commutativity, with implications for $C^*$-algebras.

Contribution

It introduces new properties of $MI$-spaces, including their structure, non-symmetry, and non-commutativity, and models shifts of arbitrary multiplicity.

Findings

$MI$-spaces containing finite multiplicity isometries are one-dimensional

Each separable subspace can be realized as a shift range

$MI$-spaces are highly non-symmetric and non-commutative

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.