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

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 Esemigroups 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. 1. Genesis and Justification B(H) is the algebra of all linear, bounded operators in a separable Hilbert space H over the complex numbers C, with the identity I. A subspace of H is always closed. A shift is always unilateral. An operator space is a linear subspace of B(H). For A1, ..., An ∈ B(H) span(A1, ..., An) is the set of all linear combinations of A1, ..., An. Operator spaces contained in the set MI of all scalar multiples of all isometries in a Hilbert space are the subject of investigation in this paper. For brevity, an operator space contained in MI will be called an MI − space. Another possible name: ”a subspace of MI” is misleading because MI is not a linear space. As will be shown, these are strange spaces, indeed. Just for a start, the shift of multiplicity one and its square cannot belong to one MI-space, because their sum does not belong to MI. This example will be further explained after Proposition 3.2.. Even though the set MI is a semigroup with operator multiplication, it turns out cf. Proposition 3.2.that on MI-spaces this multiplicative structure trivializes. My interest in MI-spaces came from an attempt to extend the following result of H. Radjavi and P. Rosenthal [R-R]: Each linear space of operators contained in the set of all normal operators is commutative. With John B. Conway [Con-Sz] we replaced ”normal” by ”hyponormal” in that theorem and showed that such result is false. Trying to understand