Seminormed *-subalgebras of l∞(X)

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:455

Article

Alaghmandan, Mahmood^1,2  Ghasemi, Mehdi³ 

[1] Chalmers Univ Technol, Dept Math Sci, SE-41296 Gothenburg, Sweden

[2] Univ Gothenburg, SE-41296 Gothenburg, Sweden

[3] Univ Saskatchewan, Dept Math & Stat, 106 Wiggins Rd, Saskatoon, SK S7N 5E6, Canada

关键词: Measurable functions;    Commutative normed algebras;    Function algebras;    Gelfand spectrum;    Seminormed algebras;    Measures on Boolean rings;   

DOI  :  10.1016/j.jmaa.2017.05.041

来源: Elsevier

【 摘 要 】

Arbitrary representations of an involutive commutative unital F-algebra A as a subalgebra of F-X are considered, where F = C or R and X not equal empty set. The Gelfand spectrum of A is explained as a topological extension of X where a seminorm on the image of A in F-X is present. It is shown that among all seminorms, the sup-norm is of special importance which reduces F-X to l(infinity)(X). The Banach subalgebra of l(infinity) (X) of all Sigma-measurable bounded functions on X, M-b (X, Sigma), is studied for which Sigma is a sigma-algebra of subsets of X. In particular, we study lifting of positive measures from (X, Sigma) to the Gelfand spectrum of M-b (X, Sigma) and observe an unexpected shift in the support of measures. In the case that Sigma is the Borel algebra of a topology, we study the relation of the underlying topology of X and the topology of the Gelfand spectrum of M-b (X, Sigma). (C) 2017 Elsevier Inc. All rights reserved.

【 授权许可】

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