Unbounded Induced Representations of 鈭?Algebras

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• 作者：Yurii Savchuk (1)
  Konrad Schm眉dgen (1)
  
• 关键词：Induced representations ; Group graded algebras ; Well ; behaved representations ; Conditional expectation ; Partial action of a group ; Imprimitivity Theorem ; Mackey analysis ; Primary 16G99 ; 47L60 ; Secondary 22D30 ; 57S99 ; 16W50
• 刊名：Algebras and Representation Theory
• 出版年：2013
• 出版时间：April 2013
• 年：2013
• 卷：16
• 期：2
• 页码：309-376
• 全文大小：972KB
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• 作者单位：Yurii Savchuk (1)
  Konrad Schm眉dgen (1)
  
  1. Mathematisches Institut Johannisgasse 26, Universit盲t Leipzig, 04103, Leipzig, Germany
  
• ISSN：1572-9079

文摘

Induced representations of *-algebras by unbounded operators in Hilbert space are investigated. Conditional expectations of a *-algebra ${{\mathcal{A}}}$ onto a unital *-subalgebra ${{\mathcal{B}}}$ are introduced and used to define inner products on the corresponding induced modules. The main part of the paper is concerned with group graded *-algebras ${{\mathcal{A}}}=\oplus_{g\in G}{{\mathcal{A}}}_g$ for which the *-subalgebra ${{\mathcal{B}}}:={{\mathcal{A}}}_e$ is commutative. Then the canonical projection $p:{{\mathcal{A}}}\to{{\mathcal{B}}}$ is a conditional expectation and there is a partial action of the group G on the set ${{\mathcal{B}}}p$ of all characters of ${{\mathcal{B}}}$ which are nonnegative on the cone $\sum{{\mathcal{A}}}^2{{\mathcal{A}}}p{{\mathcal{B}}}.$ The complete Mackey theory is developed for *-representations of ${{\mathcal{A}}}$ which are induced from characters of ${{\widehat{{{\mathcal{B}}}}^+}}.$ Systems of imprimitivity are defined and two versions of the Imprimitivity Theorem are proved in this context. A concept of well-behaved *-representations of such *-algebras ${{\mathcal{A}}}$ is introduced and studied. It is shown that well-behaved representations are direct sums of cyclic well-behaved representations and that induced representations of well-behaved representations are again well-behaved. The theory applies to a large variety of examples. For important examples such as the Weyl algebra, enveloping algebras of the Lie algebras su(2), su(1,1), and of the Virasoro algebra, and *-algebras generated by dynamical systems our theory is carried out in great detail.