[Paper Review] Three Ways to Representations of B^a(E)

Michael Skeide|arXiv (Cornell University)|Apr 30, 2004

Advanced Operator Algebra Research10 references3 citations

TL;DR

This paper presents three distinct methods to represent unital, strictly continuous homomorphisms of the algebra $\mathscr{B}^a(E)$ of adjointable operators on a Hilbert $\mathcal{B}$-module $E$ as amplifications on a Hilbert $\mathcal{C}$-module. The most general and direct proof uses tensor product decomposition via $E \odot E^* \cong \mathscr{K}(E)$, establishing that any such representation $\vartheta$ is unitarily equivalent to $a \mapsto a \otimes \mathrm{id}_{F_\vartheta}$, with $F_\vartheta = E^* \odot F$. This result extends the classical representation theory of $\mathscr{B}(H)$ to Hilbert modules over $C^*$- and $W^*$-algebras.

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.

Motivation & Objective

• To generalize the classical representation theory of $\mathscr{B}(H)$ to representations of $\mathscr{B}^a(E)$ on Hilbert $\mathcal{C}$-modules.
• To provide three distinct proofs of the structure theorem for unital, strictly continuous homomorphisms $\vartheta: \mathscr{B}^a(E) \to \mathscr{B}^a(F)$.
• To clarify the interrelations and relative strengths of older approaches (Arveson, Bhat) versus a new, direct proof based on tensor product decomposition.
• To establish the canonical identification $F \cong E \odot F_\vartheta$ with $F_\vartheta = E^* \odot F$, showing $\vartheta(a) = a \otimes \mathrm{id}_{F_\vartheta}$.

Proposed method

• The primary method uses the canonical isomorphism $E \odot E^* \cong \mathscr{K}(E)$, where $E^*$ is the dual module, to decompose $F$ as $E \odot (E^* \odot F)$, identifying $F_\vartheta = E^* \odot F$.
• The representation $\vartheta$ is realized as amplification: $\vartheta(a) = a \otimes \mathrm{id}_{F_\vartheta}$, with unitary equivalence via $u: E \odot F_\vartheta \to F$.
• The Arveson approach generalizes the intertwiner space construction from $E_0$-semigroups on $\mathscr{B}(H)$ to Hilbert modules, relying on the existence of a unit vector and $C^*$-algebraic conditions.
• The Bhat approach generalizes the rank-one operator method, requiring restriction to von Neumann algebras and $W^*$-modules.
• The commutant method uses duality and commutant algebras: $F_{\vartheta} \cong (E^* \odot F)^{\prime}$, with $F \cong E \odot F_{\vartheta}$ via canonical isomorphisms.
• All constructions are shown to yield the same result: $\vartheta(a) = a \otimes \mathrm{id}_{F_\vartheta}$, with $F_\vartheta$ uniquely determined up to canonical isomorphism.

Experimental results

Research questions

• RQ1 How can the classical representation of $\mathscr{B}(H)$ as $\mathrm{id} \otimes a$ be generalized to representations of $\mathscr{B}^a(E)$ on Hilbert $\mathcal{C}$-modules?
• RQ2 What are the structural differences and relative advantages of the Arveson, Bhat, and tensor product-based approaches in the context of Hilbert modules?
• RQ3 Under what conditions do the older approaches (Arveson/Bhat) remain more useful than the new, direct method?
• RQ4 How does the tensor product decomposition $E \odot E^* \cong \mathscr{K}(E)$ enable the construction of $F_\vartheta$ and the identification $F \cong E \odot F_\vartheta$?
• RQ5 What is the role of Morita equivalence and commutant algebras in characterizing the module $F_\vartheta$?

Key findings

• The representation $\vartheta: \mathscr{B}^a(E) \to \mathscr{B}^a(F)$ is unitarily equivalent to $a \mapsto a \otimes \mathrm{id}_{F_\vartheta}$, with $F_\vartheta = E^* \odot F$, via a canonical isomorphism $u: E \odot F_\vartheta \to F$.
• The new proof is direct, general, and applies to $C^*$- and $W^*$-algebras, even for normal representations of $\mathscr{B}(H)$, and is novel in this context.
• The Arveson and Bhat approaches generalize classical product system constructions but require technical conditions such as existence of a unit vector or restriction to von Neumann algebras.
• The module $F_\vartheta$ is uniquely determined up to canonical isomorphism, and the identification $F \cong E \odot F_\vartheta$ is universal and canonical.
• The result holds for both $C^*$- and $W^*$-algebras, with the $W^*$-version involving commutant algebras and normal homomorphisms.
• The construction reveals that $\vartheta$ is an isomorphism if and only if $F_\vartheta$ is a Morita equivalence bimodule.