Vector-valued spectra of Banach algebra valued continuous functions

2015/10/21 by Mortaza Abtahi, Abtahi, Mortaza, Sara Farhangi +1
Mathematics · #46E40 #46H10 #46J10 #46J20 #FOS: Mathematics #Functional Analysis (math.FA) #math.FA #msc:46E40 #msc:46H10 #msc:46J10 #msc:46J20

paper · pdf · doi:10.48550/arxiv.1510.06641

arxiv created 2015/12/29 · arxiv updated 2015/12/31

Abstract

Given a compact space X, a commutative Banach algebra A, and an A-valued function algebra \mathscrA on X, the notions of vector-valued spectrum of functions f∈\mathscrA are discussed. The A-valued spectrum SP_A(f) of every f∈\mathscrA is defined in such a way that f(X) ⊂ SP_A(f). Utilizing the A-characters introduced in (M. Abtahi, Vector-valued characters on vector-valued function algebras, arXiv:1509.09215 [math.FA]), it is proved that SP_A(f) = \Ψ(f):\textΨ is an A-character of \mathscrA\. For the so-called natural A-valued function algebras, such as C(X,A) and Lip(X,A), we see that SP_A(f)=f(X). When A = ℂ, Banach A-valued function algebras reduce to Banach function algebras, A-characters reduce to characters, and A-valued spectrums reduce to usual spectrums.

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