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.
At first conditions are given for existence of a relative integral basis for OK≅Okn−1⊕I with [K;k]=n. Then the constrtiction of the ideal I in OK≅Okn−1⊕I is given for proof of existence of a relative integral basis for OK4(m1,m2)/Ok(​m3). Finally existence and construction of the relative integral basis for OK6(n3,−3)/Ok3(n3),OK6(n3,−3)/Ok2(−3) for some values of n are given
Abstract. We study discontinuous invertibility preserving linear mappings from a Banach algebra into the algebra of n × n matrices and give an explicit representation of such a mapping when n = 2.
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