[論文レビュー] Three Ways to Representations of B^a(E)

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

Advanced Operator Algebra Research参考文献 10被引用数 3

ひとこと要約

本稿では、ヒルベルト $\mathcal{B}$-加群 $E$ 上の随伴可能作用素の代数 $\mathscr{B}^a(E)$ の単位的かつ厳密連続な準同型を、ヒルベルト $\mathcal{C}$-加群上の拡大として表現するための3つの異なる手法を提示する。最も一般的で直接的な証明は、$E \odot E^* \cong \mathscr{K}(E)$ を通じたテンソル積分解を用い、任意の such 表現 $\vartheta$ が $a \mapsto a \otimes \mathrm{id}_{F_\vartheta}$ にユニタリ同値であることを確立する。ここで $F_\vartheta = E^* \odot F$ である。この結果は、$C^*$-代数および $W^*$-代数上のヒルベルト加群への古典的表現理論を拡張するものである。

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.

研究の動機と目的

• 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 $\vartheta(a) = a \otimes \mathrm{id}_{F_\vartheta}$.

提案手法

• 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.

実験結果

リサーチクエスチョン

• 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 $F_\vartheta$?

主な発見

• 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.