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Abstract Abstract
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18.1 Introduction 18.1 Introduction
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18.2 Constructive C*-Algebras 18.2 Constructive C*-Algebras
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18.3 Positive Elements 18.3 Positive Elements
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18.4 Positive Linear Functional and States 18.4 Positive Linear Functional and States
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18.5 Representations 18.5 Representations
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18.6 The GNS Construction Theorem 18.6 The GNS Construction Theorem
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Acknowledgements Acknowledgements
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References References
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18 AN INTRODUCTION TO THE THEORY OF C*-ALGEBRAS IN CONSTRUCTIVE MATHEMATICS
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Published:October 2005
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Abstract
This chapter introduces an elementary theory of C*-algebras in the context of Bishop-style constructive mathematics. It givens proof of the Gelfand-Naĭmark-Segal (GNS) construction theorem in Bishop's constructive mathematics. This important theorem in the theory of operator algebras says that for each C*-algebra and every state, there exists a cyclic representation on some Hilbert space. This chapter's contribution is of particular interest in view of the Bridges-Hellman debate on whether constructive mathematics is able to cope with quantum mechanics. Since quantum mechanics is bound up with the theory of operator algebras on Hilbert spaces, a constructive treatment of the latter has been a challenge for constructive mathematics from the very beginning.
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