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- Title
On the Takai duality for Lp operator crossed products.
- Authors
Wang, Zhen; Zhu, Sen
- Abstract
The aim of this paper is to study a problem raised by Phillips concerning the existence of Takai duality for L p operator crossed products F p (G , A , α) , where G is a locally compact Abelian group, A is an L p operator algebra and α is an isometric action of G on A. Inspired by Williams' proof for the Takai duality theorem for crossed products of C ∗ -algebras, we construct a homomorphism Φ from F p (G ^ , F p (G , A , α) , α ^) to K (l p (G)) ⊗ p A which is a natural L p -analog of Williams' map. For countable discrete Abelian groups G and separable unital L p operator algebras A which have unique L p operator matrix norms, we show that Φ is an isomorphism if and only if either G is finite or p = 2 ; in particular, Φ is an isometric isomorphism in the case that p = 2 . Moreover, it is proved that Φ is equivariant for the double dual action α ^ ^ of G on F p (G ^ , F p (G , A , α) , α ^) and the action Ad ρ ⊗ α of G on K (l p (G)) ⊗ p A .
- Subjects
OPERATOR algebras; COMPACT groups; ISOMORPHISM (Mathematics); MATRIX norms; HOMOMORPHISMS; ABELIAN groups
- Publication
Mathematische Zeitschrift, 2023, Vol 304, Issue 4, p1
- ISSN
0025-5874
- Publication type
Article
- DOI
10.1007/s00209-023-03316-4