International Journal of Mathematics and Mathematical Sciences
Volume 2008 (2008), Article ID 531424, 8 pages
doi:10.1155/2008/531424
An extension of the spectral mapping theorem
A.R. Medghalchi1
and S.M. Tabatabaie2
1Faculty of Mathematical Sciences and Computer Engineering, Tabbiat Moallem University, Tehran 15618, Iran
2Department of Mathematics, The University of Qom, Qom 3716146611, Iran
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.