Multiplicative linear functionals in a commutative Banach algebra

Ershad, F.; Seddighi, K.

doi:10.1007/bf01196583

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Example 41 ([11]) Let1

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2001

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“…Let P and Q be arbitrary points of B with Q in the interior. Consider the three continuous linear functionals ν 1 , ν 2 and ν 3 defined by ν 1 (f ) = f (P ) + f x (Q), ν 2 (f ) = f (Q), and ν 3 (f ) = f x (Q) + if y (Q) for all f ∈ A, where x, y, z denote the coordinates of a point in R 3 . We show that each element of the subspace…”

Section: Example 41 ([11]) Letmentioning

confidence: 99%

Sufficient conditions for a linear functional to be multiplicative

Seddighi

Haghighi

2001

Proc. Amer. Math. Soc.

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Abstract. A commutative Banach algebra A is said to have the P (k, n) property if the following holds: Let M be a closed subspace of finite codimension n such that, for every x ∈ M , the Gelfand transformx has at least k distinct zeros in ∆(A), the maximal ideal space of A. Then there exists a subset Z of ∆(A) of cardinality k such thatM vanishes on Z, the set of common zeros of M . In this paper we show that if X ⊂ C is compact and nowhere dense, then R(X), the uniform closure of the space of rational functions with poles off X, has the P (k, n) property for all k, n ∈ N. We also investigate the P (k, n) property for the algebra of real continuous functions on a compact Hausdorff space.

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Section: Example 41 ([11]) Letmentioning

confidence: 99%