AN INTRODUCTION TO THE THEORY OF C*-ALGEBRAS IN CONSTRUCTIVE MATHEMATICS
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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