Abstract
After an appropriate restatement of the Gelfand–Naimark–Segal construction for topological *-algebras we prove that there exists an isomorphism among the set Cycl(A) of weakly continuous strongly cyclic *-representations of a barreled dual-separable *-algebra with unit A, the space HilbA(A*) of the Hilbert spaces that are continuously embedded in A* and are * -invariant under the dual left regular action of A, and the set of the corresponding reproducing kernels. We show that these isomorphisms are cone morphisms and we prove many interesting results that follow from this fact. We discuss how these results can be used to describe cyclic representations on more general inner product spaces.
Highlights
Quantum statistical mechanics and quantum field theories are believed to be fully described in purely algebraic terms, the so-called C*-algebraic approachsee Refs. 1–3 for textbooks and Refs. 4 and 5 for recent reviews on the subjectbeing the most appealing one
If quantum gauge theories are assumed to be described in algebraic terms, the appropriate representation spaces would be more general inner product spaces than Hilbert spaces10 and in that case there is no compelling reason to believe that the *-algebra describing the observable content of the theory should be a normable one
If A is a barreled dualseparable *-algebra with unit we will prove that the set CyclAof weakly continuous strongly cyclic *-representations of A is isomorphic to the set HilbAA*͒ of the Hilbert subspaces of A* that are *-invariant under the left dual regular action of A on A*
Summary
Quantum statistical mechanics and quantum field theories are believed to be fully described in purely algebraic terms, the so-called C*-algebraic approachsee Refs. 1–3 for textbooks and Refs. 4 and 5 for recent reviews on the subjectbeing the most appealing one. If A is a barreled dualseparable *-algebra with unit we will prove that the set CyclAof weakly continuous strongly cyclic *-representations of A is isomorphic to the set HilbAA*͒ of the Hilbert subspaces of A* that are *-invariant under the left dual regular action of A on A*. In turn this bijection can be extended to a multiple isomorphism among these spaces, the space of continuous positive functionals over.
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