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Register a new user$C^*$-non-linear second quantization
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Recently, we have constructed a non{linear (polynomial) extension of the 1-mode Heisenberg group and the corresponding Fock and Weyl representations. The transition from the 1-mode case to the current algebra level, in which the operators are indexed by elements of an appropriate test function space (second quantization), can be done at Lie algebra level. A way to bypass the difficulties of constructing a (non trivial) Hilbert space representation is to try and construct directly a $C^*$-algebra rep- resentation and then to look for its Hilbert space representations. In usual (linear) quantization, this corresponds to the construction of the Weyl $C^*$-algebra. In this paper, we produce such a construction for the above mentioned polynomial extension of the Weyl $C^*$-algebra. The result of this construction is a factorizable system of local alge- bras localized on bounded Borel subsets of $mathbb{R}$ and obtained as induc- tive limit of tensor products of finite sets of copies of the one mode $C^*$-algebra. The $C^*$-embeddings of the inductive system require some non{trivial re{scaling of the generators of the algebras involved. These re{scalings are responsible of a $C^*$-analogue of the no-go theorems, first met at the level of Fock second quantization, namely the proof that the family of Fock states defined on the inductive family of $C^*$-algebras is projective only in the linear case (i.e. the case of the usual Weyl algebra). Thus the solution of the representa- tion problem at $C^*$-level does not automatically imply its solution at Hilbert space level.
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We construct a new class of finite-dimensional C^*-quantum groupoids at roots of unity q=e^{ipi/ell}, with limit the discrete dual of the classical SU(N) for large orders. The representation category of our groupoid turns out to be tensor equivalent to the well known quotient C^*-category of the category of tilting modules of the non-semisimple quantum group U_q({mathfrak sl}_N) of Drinfeld, Jimbo and Lusztig. As an algebra, the C^*-groupoid is a quotient of U_q({mathfrak sl}_N). As a coalgebra, it naturally reflects the categorical quotient construction. In particular, it is not coassociative, but satisfies axioms of the weak quasi-Hopf C^*-algebras: quasi-coassociativity and non-unitality of the coproduct. There are also a multiplicative counit, an antipode, and an R-matrix. For this, we give a general construction of quantum groupoids for complex simple Lie algebras {mathfrak g} eq E_8 and certain roots of unity. Our main tools here are Drinfelds coboundary associated to the R-matrix, related to the algebra involution, and certain canonical projections introduced by Wenzl, which yield the coproduct and Drinfelds associator in an explicit way. Tensorial properties of the negligible modules reflect in a rather special nature of the associator. We next reduce the proof of the categorical equivalence to the problems of establishing semisimplicity and computing dimension of the groupoid. In the case {mathfrak g}={mathfrak sl}_N we construct a (non-positive) Haar-type functional on an associative version of the dual groupoid satisfying key non-degeneracy properties. This enables us to complete the proof.
A systematic theory of product and diagonal states is developed for tensor products of $mathbb Z_2$-graded $*$-algebras, as well as $mathbb Z_2$-graded $C^*$-algebras. As a preliminary step to achieve this goal, we provide the construction of a {it fermionic $C^*$-tensor product} of $mathbb Z_2$-graded $C^*$-algebras. Twisted duals of positive linear maps between von Neumann algebras are then studied, and applied to solve a positivity problem on the infinite Fermi lattice. Lastly, these results are used to define fermionic detailed balance (which includes the definition for the usual tensor product as a particular case) in general $C^*$-systems with gradation of type $mathbb Z_2$, by viewing such a system as part of a compound system and making use of a diagonal state.
A Banach involutive algebra is called a Krein C*-algebra if there is a fundamental symmetry (an involutive automorphism of period 2) such that the C*-property is satisfied when the original involution is replaced with the new one obtained by composing the automorphism with the old involution. For a given fundamental symmetry, a Krein C*-algebra decomposes as a direct sum of an even part (a C*-algebra) and an odd part (a Hilbert C*-bimodule on the even part). Our goal here is to develop a spectral theory for commutative unital Krein C*-algebras when the odd part is a symmetric imprimitivity C*-bimodule over the even part and there exists an additional suitable exchange symmetry between the odd and even parts.
We consider the construction of twisted tensor products in the category of C*-algebras equipped with orthogonal filtrations and under certain assumptions on the form of the twist compute the corresponding quantum symmetry group, which turns out to be the generalised Drinfeld double of the quantum symmetry groups of the original filtrations. We show how these results apply to a wide class of crossed products of C*-algebras by actions of discrete groups. We also discuss an example where the hypothesis of our main theorem is not satisfied and the quantum symmetry group is not a generalised Drinfeld double.
We present a general approach to a modular frame theory in C*-algebras and Hilbert C*-modules. The investigations rely on the idea of geometric dilation to standard Hilbert C*-modules over unital C*-algebras that possess orthonormal Hilbert bases, and of reconstruction of the frames by projections and by other bounded modular operators with suitable ranges. We obtain frame representations and decomposition theorems, as well as similarity and equivalence results for frames. Hilbert space frames and quasi-bases for conditional expectations of finite index on C*-algebras appear as special cases. Using a canonical categorical equivalence of Hilbert C*-modules over commutative C*-algebras and (F)Hilbert bundles the results find a reintepretation for frames in vector and (F)Hilbert bundles. Fields of applications are investigations on Cuntz-Krieger-Pimsner algebras, on conditional expectations of finite index, on various ranks of C*-algebras, on classical frame theory of Hilbert spaces (wavelet and Gabor frames), and others. 2001: In the introduction we refer to related publications in detail.
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