Continuity of Banach algebra valued functions
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Let K be a compact Hausdorff space, a commutative com- plex Banach algebra with identity and ( ) the set of characters of . In this note, we study the class of functions f : K → such that A ◦ f is continuous, where A denotes the Gelfand representation of . The inclusion relations between this class, the class of continuous functions, the class of bounded functions and the class of weakly contin- uous functions relative to the weak topology σ( , ( )), are discussed. We also provide some results on its completeness under the norm defined by |||f||| = ||Ω ◦f||∞.
Abstract
Let K be a compact Hausdorff space, a commutative com- plex Banach algebra with identity and ( ) the set of characters of . In this note, we study the class of functions f : K → such that A ◦ f is continuous, where A denotes the Gelfand representation of . The inclusion relations between this class, the class of continuous functions, the class of bounded functions and the class of weakly contin- uous functions relative to the weak topology σ( , ( )), are discussed. We also provide some results on its completeness under the norm defined by |||f||| = ||Ω ◦f||∞.
참고문헌 (4)
- R. G. Douglas, "Banach Algebra Techniques in Operator Theory" (1972), Academic Press
- W. Fupinwong, "The fixed point property of unital abelian Banach algebras", Fixed Point Theory Appl, 2010, pp.13~ (2010)
- R. V. Kadison, "Fundamentals of the Theory of Operator Algebras. Vol. I" (1983), Academic Press
- R. V. Kadison, "Fundamentals of the Theory of Operator Algebras. Vol. II" (1986), Academic Press