Abstract
Let (Formula presented.) be a locally compact abelian group. By modifying a theorem of Pedersen, it follows that actions of (Formula presented.) on (Formula presented.)-algebras (Formula presented.) and (Formula presented.) are outer conjugate if and only if there is an isomorphism of the crossed products that is equivariant for the dual actions and preserves the images of (Formula presented.) and (Formula presented.) in the multiplier algebras of the crossed products. The rigidity problem discussed in this paper deals with the necessity of the last condition concerning the images of (Formula presented.) and (Formula presented.). There is an alternative formulation of the problem: an action of the dual group (Formula presented.) together with a suitably equivariant unitary homomorphism of (Formula presented.) give rise to a generalized fixed-point algebra via Landstad’s theorem, and a problem related to the above is to produce an action of (Formula presented.) and two such equivariant unitary homomorphisms of (Formula presented.) that give distinct generalized fixed-point algebras. We present several situations where the condition on the images of (Formula presented.) and (Formula presented.) is redundant, and where having distinct generalized fixed-point algebras is impossible. For example, if (Formula presented.) is discrete, this will be the case for all actions of (Formula presented.).
| Original language | English (US) |
|---|---|
| Journal | International Journal of Mathematics |
| DOIs | |
| State | Accepted/In press - Feb 28 2018 |
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Keywords
- Action
- crossed-product
- exterior equivalence
- generalized fixed-point algebra
- outer conjugacy
ASJC Scopus subject areas
- Mathematics(all)
Cite this
Rigidity theory for C∗-dynamical systems and the “Pedersen rigidity problem”. / Kaliszewski, Steven; Omland, Tron; Quigg, John.
In: International Journal of Mathematics, 28.02.2018.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Rigidity theory for C∗-dynamical systems and the “Pedersen rigidity problem”
AU - Kaliszewski, Steven
AU - Omland, Tron
AU - Quigg, John
PY - 2018/2/28
Y1 - 2018/2/28
N2 - Let (Formula presented.) be a locally compact abelian group. By modifying a theorem of Pedersen, it follows that actions of (Formula presented.) on (Formula presented.)-algebras (Formula presented.) and (Formula presented.) are outer conjugate if and only if there is an isomorphism of the crossed products that is equivariant for the dual actions and preserves the images of (Formula presented.) and (Formula presented.) in the multiplier algebras of the crossed products. The rigidity problem discussed in this paper deals with the necessity of the last condition concerning the images of (Formula presented.) and (Formula presented.). There is an alternative formulation of the problem: an action of the dual group (Formula presented.) together with a suitably equivariant unitary homomorphism of (Formula presented.) give rise to a generalized fixed-point algebra via Landstad’s theorem, and a problem related to the above is to produce an action of (Formula presented.) and two such equivariant unitary homomorphisms of (Formula presented.) that give distinct generalized fixed-point algebras. We present several situations where the condition on the images of (Formula presented.) and (Formula presented.) is redundant, and where having distinct generalized fixed-point algebras is impossible. For example, if (Formula presented.) is discrete, this will be the case for all actions of (Formula presented.).
AB - Let (Formula presented.) be a locally compact abelian group. By modifying a theorem of Pedersen, it follows that actions of (Formula presented.) on (Formula presented.)-algebras (Formula presented.) and (Formula presented.) are outer conjugate if and only if there is an isomorphism of the crossed products that is equivariant for the dual actions and preserves the images of (Formula presented.) and (Formula presented.) in the multiplier algebras of the crossed products. The rigidity problem discussed in this paper deals with the necessity of the last condition concerning the images of (Formula presented.) and (Formula presented.). There is an alternative formulation of the problem: an action of the dual group (Formula presented.) together with a suitably equivariant unitary homomorphism of (Formula presented.) give rise to a generalized fixed-point algebra via Landstad’s theorem, and a problem related to the above is to produce an action of (Formula presented.) and two such equivariant unitary homomorphisms of (Formula presented.) that give distinct generalized fixed-point algebras. We present several situations where the condition on the images of (Formula presented.) and (Formula presented.) is redundant, and where having distinct generalized fixed-point algebras is impossible. For example, if (Formula presented.) is discrete, this will be the case for all actions of (Formula presented.).
KW - Action
KW - crossed-product
KW - exterior equivalence
KW - generalized fixed-point algebra
KW - outer conjugacy
UR - http://www.scopus.com/inward/record.url?scp=85042748294&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85042748294&partnerID=8YFLogxK
U2 - 10.1142/S0129167X18500167
DO - 10.1142/S0129167X18500167
M3 - Article
AN - SCOPUS:85042748294
JO - International Journal of Mathematics
JF - International Journal of Mathematics
SN - 0129-167X
ER -