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Revista Colombiana de Matemáticas
Print version ISSN 0034-7426
Rev.colomb.mat. vol.53 supl.1 Bogotá Dec. 2019 Epub Mar 24, 2020
https://doi.org/10.15446/recolma.v53nsupl.84095
Artículos originales
The Pedersen Rigidity Problem
1 Arizona State University, USA
If is an action of a locally compact abelian group G on a C*-algebra A, Takesaki-Takai duality recovers (A, α) up to Morita equivalence from the dual action of Ĝ on the crossed product A αG. Given a bit more information, Landstad duality recovers (A, α) up to isomorphism. In between these, by modifying a theorem of Pedersen, (A, α) is recovered up to outer conjugacy from the dual action and the position of A in M(A αG). Our search (still unsuccessful, somehow irritating) for examples showing the necessity of this latter condition has led us to formulate the "Pedersen Rigidity problem". We present numerous situations where the condition is redundant, including G discrete or A stable or commutative. The most interesting of these "no-go theorems" is for locally unitary actions on continuous-trace algebras.
Keywords: action; crossed-product; exterior equivalence; outer conjugacy; generalized fixed-point algebra
Si α es una acción de un grupo abeliano localmente compacto G sobre una C*-álgebra A, la dualidad de Takesaki-Takai recupera (A, α), salvo equivalencia de Morita, de la acción dual de Ĝ sobre el producto cruzado A αG. Mediante un poco más de información, la dualidad de Landstad recupera (A, α) salvo isomorfismo. De manera intermedia, mediante la modificación de un teorema de Pedersen, (A α) es recuperado, salvo conjugación externa, de la acción dual y de la posición de A en M(A αG). Nuestra búsqueda (todavía sin éxito, de alguna manera irritante) de ejemplos que prueben la necesidad de esta última condición, nos ha conducido a a formular el "problema de rigidez de Pedersen". Presentamos numerosas situaciones donde la condición es redundante, incluídos los casos en que G es discreto, o bien A es estable o conmutativo. Lo más interesante de estos "teoremas de no usar" es para acciones localmente unitarias sobre álgebras trazo-continuas.
Palabras clave: Acción; producto cruzado; equivalencia exterior; conjugación externa; álgebra generalizada de punto fijo
References
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Received: June 05, 2018; Accepted: November 11, 2018