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Dual Group Actions on C*‐Algebras and Their Description by Hilbert Extensions
Author(s) -
Baumgärtel Hellmut,
Lledó Fernando
Publication year - 2002
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/1522-2616(200206)239:1<11::aid-mana11>3.0.co;2-b
Subject(s) - mathematics , bijection , abelian group , homomorphism , hilbert space , superselection , pure mathematics , von neumann algebra , algebra over a field , group (periodic table) , hilbert's fourteenth problem , group algebra , regular representation , discrete mathematics , von neumann architecture , rigged hilbert space , reproducing kernel hilbert space , observable , chemistry , organic chemistry , physics , quantum mechanics
Given a C*‐algebra , a discrete abelian group and a homomorphism Θ : → Out , defining the dual action group Γ ⊂ aut , the paper contains results on existence and characterization of Hilbert extensions of {, Γ}, where the action is given by $ \hat {\cal X} $ . They are stated at the (abstract) C*‐level and can therefore be considered as a refinement of the extension results given for von Neumann algebras for example by V. F. R. Jones [18] or C. E. Sutherland [22, 23]. A Hilbert extension exists iff there is a generalized 2‐cocycle. These results generalize those in [12], which are formulated in the context of superselection theory, where it is assumed that the algebra has a trivial center, i.e. = ℂ 1 . In particular the well‐known “outer characterization” of the second cohomology H 2 (,(), α ) can be reformulated: there is a bijection to the set of all ‐module isomorphy classes of Hilbert extensions. Finally, a Hilbert space representation (due to C. E. Sutherland [22, 23] in the von Neumann case) is mentioned. The C*‐norm of the Hilbert extension is expressed in terms of the norm of this representation and it is linked to the so‐called regular representation appearing in superselection theory.