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Register a new user$C^*$-fermi systems and detailed balance
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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.
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We consider nonlinear reaction systems satisfying mass-action kinetics with slow and fast reactions. It is known that the fast-reaction-rate limit can be described by an ODE with Lagrange multipliers and a set of nonlinear constraints that ask the fast reactions to be in equilibrium. Our aim is to study the limiting gradient structure which is available if the reaction system satisfies the detailed-balance condition. The gradient structure on the set of concentration vectors is given in terms of the relative Boltzmann entropy and a cosh-type dissipation potential. We show that a limiting or effective gradient structure can be rigorously derived via EDP convergence, i.e. convergence in the sense of the Energy-Dissipation Principle for gradient flows. In general, the effective entropy will no longer be of Boltzmann type and the reactions will no longer satisfy mass-action kinetics.
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.
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.
We analyse and interpret the effects of breaking detailed balance on the convergence to equilibrium of conservative interacting particle systems and their hydrodynamic scaling limits. For finite systems of interacting particles, we review existing results showing that irreversible processes converge faster to their steady state than reversible ones. We show how this behaviour appears in the hydrodynamic limit of such processes, as described by macroscopic fluctuation theory, and we provide a quantitative expression for the acceleration of convergence in this setting. We give a geometrical interpretation of this acceleration, in terms of currents that are emph{antisymmetric} under time-reversal and orthogonal to the free energy gradient, which act to drive the system away from states where (reversible) gradient-descent dynamics result in slow convergence to equilibrium.
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.
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