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Higherdimensional algebra II: 2Hilbert spaces
"... A 2Hilbert space is a category with structures and properties analogous to those of a Hilbert space. More precisely, we define a 2Hilbert space to be an abelian category enriched over Hilb with a ∗structure, conjugatelinear on the homsets, satisfying 〈fg,h 〉 = 〈g,f ∗ h 〉 = 〈f,hg ∗ 〉. We also ..."
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Cited by 43 (13 self)
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A 2Hilbert space is a category with structures and properties analogous to those of a Hilbert space. More precisely, we define a 2Hilbert space to be an abelian category enriched over Hilb with a ∗structure, conjugatelinear on the homsets, satisfying 〈fg,h 〉 = 〈g,f ∗ h 〉 = 〈f,hg ∗ 〉. We also define monoidal, braided monoidal, and symmetric monoidal versions of 2Hilbert spaces, which we call 2H*algebras, braided 2H*algebras, and symmetric 2H*algebras, and we describe the relation between these and tangles in 2, 3, and 4 dimensions, respectively. We prove a generalized DoplicherRoberts theorem stating that every symmetric 2H*algebra is equivalent to the category Rep(G) of continuous unitary finitedimensional representations of some compact supergroupoid G. The equivalence is given by a categorified version of the Gelfand transform; we also construct a categorified version of the Fourier transform when G is a compact abelian group. Finally, we characterize Rep(G) by its universal properties when G is a compact classical group. For example, Rep(U(n)) is the free connected symmetric 2H*algebra on one even object of dimension n. 1
On the PCTTheorem in the Theory of Local Observables
, 2000
"... We review the PCTtheorem and problems connected with its demonstration. We add a new proof of the PCTtheorem in the theory of local observables which is similar to that one of Jost in Wightman quantum field theory. We also look at consequences in case the PCTsymmetry is given on the algebraic le ..."
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Cited by 4 (0 self)
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We review the PCTtheorem and problems connected with its demonstration. We add a new proof of the PCTtheorem in the theory of local observables which is similar to that one of Jost in Wightman quantum field theory. We also look at consequences in case the PCTsymmetry is given on the algebraic level. At the end we present some examples which answer general questions and throw some light on open problems.
Hilbert C*systems for Actions of the Circle Group
, 2000
"... The paper contains constructions of Hilbert systems for the action of the circle group T using subgroups of implementable Bogoljubov unitaries w.r.t. Fock representations of the Fermion algebra for suitable data of the selfdual framework: H is the reference Hilbert space, the conjugation and P a bas ..."
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Cited by 2 (1 self)
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The paper contains constructions of Hilbert systems for the action of the circle group T using subgroups of implementable Bogoljubov unitaries w.r.t. Fock representations of the Fermion algebra for suitable data of the selfdual framework: H is the reference Hilbert space, the conjugation and P a basis projection on H: According to a general result for Hilbert systems of this type, the group C(spec Z ! T ) of T valued functions on spec Z is isomorphic to the stabilizer of A. In particular, examples are presented where the center Z of the xed point algebra A can be calculated explicitly.