International Journal of Mathematics and Mathematical Sciences
Volume 2008 (2008), Article ID 531424, 8 pages
doi:10.1155/2008/531424
Research Article
An Extension of the Spectral Mapping Theorem
1Faculty of Mathematical Sciences and Computer Engineering, Tabbiat Moallem University, Tehran 15618, Iran
2Department of Mathematics, The University of Qom, Qom 3716146611, Iran
Received 8 January 2008; Accepted 4 May 2008
Academic Editor: Ingo Witt
Copyright © 2008 A. R. Medghalchi and S. M. Tabatabaie. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
We give an extension of the spectral mapping theorem
on hypergroups and prove that if K is a commutative strong hypergroup with K^=Xb(K) and κ is a weakly continuous representation of M(K) on a W∗-algebra such that for every t∈K, κt is an ∗-automorphism, spκ is a synthesis
set for L1(K) and κ(L1(K)) is without order, then for any μ in a closed regular
subalgebra of M(K) containing L1(K), σ(κ(μ))=μ^(spκ)¯, where spκ is the
Arveson spectrum of κ.