Three Ways to Representations of B^a(E)

TL;DR

This paper compares three methods for representing the algebra of adjointable operators on Hilbert modules, highlighting a new, simpler proof and discussing the applicability of older, more detailed approaches.

Contribution

It introduces a new, straightforward proof for representations of B^a(E) and analyzes the advantages of older methods in specific contexts.

Findings

The new proof is simple and direct, even for normal representations of B(H).

Older approaches provide more detailed information useful for certain problems.

The paper clarifies when each method is most effective.

Abstract

We describe three methods to determine the structure of (sufficiently continuous) representations of the algebra B^a(E) of all adjointable operators on a Hilbert B-module E by operators on a Hilbert C-module. While the last and latest proof is simple and direct and new even for normal representations of B(H) (H some Hilbert space), the other ones are direct generalizations of the representation theory of B(H) (based on Arveson's and on Bhat's approaches to product systems of Hilbert spaces) and depend on technical conditions (for instance, existence of a unit vector or restriction to von Neumann algebras and von Neumann modules). We explain why for certain problems the more specific information available in the older approaches is more useful for the solution of the problem.