On a Class of Subalgebras of C(x) and the Intersection of Their Free Maximal Ideals
نویسندگان
چکیده
Let X be a Tychonoff space and A a subalgebra of C(X) containing C∗(X). Suppose that CK(X) is the set of all functions in C(X) with compact support. Kohls has shown that CK(X) is precisely the intersection of all the free ideals in C(X) or in C∗(X). In this paper we have proved the validity of this result for the algebra A. Gillman and Jerison have proved that for a realcompact space X, CK(X) is the intersection of all the free maximal ideals in C(X). In this paper we have proved that this result does not hold for the algebra A, in general. However we have furnished a characterisation of the elements that belong to all the free maximal ideals in A. The paper terminates by showing that for any realcompact space X, there exists in some sense a minimal algebra Am for which X becomes Am-compact. This answers a question raised by Redlin and Watson in 1987. But it is still unsettled whether such a minimal algebra exists with respect to set inclusion.
منابع مشابه
ON MAXIMAL IDEALS OF R∞L
Let $L$ be a completely regular frame and $mathcal{R}L$ be the ring of real-valued continuous functions on $L$. We consider the set $$mathcal{R}_{infty}L = {varphi in mathcal{R} L : uparrow varphi( dfrac{-1}{n}, dfrac{1}{n}) mbox{ is a compact frame for any $n in mathbb{N}$}}.$$ Suppose that $C_{infty} (X)$ is the family of all functions $f in C(X)$ for which the set ${x in X: |f(x)|geq dfrac{1...
متن کاملASSOCIATED PRIME IDEALS IN C(X)
The minimal prime decomposition for semiprime ideals is defined and studied on z-ideals of C(X). The necessary and sufficient condition for existence of the minimal prime decomposition of a z-ideal / is given, when / satisfies one of the following conditions: (i) / is an intersection of maximal ideals. (ii) I is an intersection of O , s, when X is basically disconnected. (iii) I=O , when x X h...
متن کاملCertain subalgebras of Lipschitz algebras of infinitely differentiable functions and their maximal ideal spaces
We study an interesting class of Banach function algebras of innitely dierentiable functions onperfect, compact plane sets. These algebras were introduced by Honary and Mahyar in 1999, calledLipschitz algebras of innitely dierentiable functions and denoted by Lip(X;M; ), where X is aperfect, compact plane set, M = fMng1n=0 is a sequence of positive numbers such that M0 = 1 and(m+n)!Mm+n ( m!Mm)...
متن کاملOn the reducible $M$-ideals in Banach spaces
The object of the investigation is to study reducible $M$-ideals in Banach spaces. It is shown that if the number of $M$-ideals in a Banach space $X$ is $n(<infty)$, then the number of reducible $M$-ideals does not exceed of $frac{(n-2)(n-3)}{2}$. Moreover, given a compact metric space $X$, we obtain a general form of a reducible $M$-ideal in the space $C(X)$ of continuous functions on $X$. The...
متن کاملMaximal Ideals in Subalgebras of C(x)
Let X be a completely regular space, and let A(X) be a subalgebra of C(X) containing C*{X). We study the maximal ideals in A(X) by associating a filter Z(f) to each / 6 A(X). This association extends to a oneto-one correspondence between M(A) (the set of maximal ideals of A(X)) and ßX. We use the filters Z(f) to characterize the maximal ideals and to describe the intersection of the free maxima...
متن کامل