The spectral theory of commutative C*-algebras: The constructive spectrum
The spectral theory of commutative C*-algebras: The constructive spectrum
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DOI：
10.2989/16073600009485989
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发表时间：
2000-12
影响因子：
0.7
通讯作者：
B. Banaschewski;C. Mulvey
中科院分区：
文献类型：
--
作者：
B. Banaschewski;C. Mulvey
本文在Grothendieck拓扑E中引入了交换C*-代数的概念，并由此引入了A的谱MFN A的概念，它是由适当的命题理论在Topos E中确定的，它描述了A上乘法线性泛函的基本性质。此外，以类似的方式定义了Topos E中复数的Locale CE，并建立了它的一些基本性质，如它的完全正则性和CE中单位平方的紧性。最后证明了Locale MFN A是紧的且完全正则的，推广了交换C*-代数上的乘性线性泛函构成弱*拓扑中紧Hausdorff空间的经典结果.
This paper introduces the notion of a commutative C*-algebra in a Grothendieck topos E and subsequently that of the spectrum MFn A of A, presented as the locale determined by an appropriate propositional theory in the topos E which describes the basic properties of a multplicative linear functional on A. Further, the locale CE of complex numbers in the topos E is defined in a similar manner and some of its basic properties are established, such as its complete regularity and the compactness of the unit square in CE. Finally, it is shown that the locale MFn A is compact and completely regular, extending the classical result that the multiplicative linear functionals on a commutative C*-algebra form a compact Hausdorff space in the weak* topology.