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Register a new userA duality theorem for ergodic actions of compact quantum groups on C*-algebras
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The spectral functor of an ergodic action of a compact quantum group G on a unital C*-algebra is quasitensor, in the sense that the tensor product of two spectral subspaces is isometrically contained in the spectral subspace of the tensor product representation, and the inclusion maps satisfy natural properties. We show that any quasitensor *-functor from Rep(G) to the category of Hilbert spaces is the spectral functor of an ergodic action of G on a unital C*-algebra. As an application, we associate an ergodic G-action on a unital C*-algebra to an inclusion of Rep(G) into an abstract tensor C*-category. If the inclusion arises from a quantum subgroup of G, the associated G-system is just the quantum quotient space. If G is a group and the category has permutation symmetry, the associated system is commutative, and therefore isomorphic to the classical quotient space by a closed subgroup of $G$. If a tensor C*-category has a Hecke symmetry making an object of dimension d and q-quantum determinant one then there is an ergodic action of S_qU(d) on a unital C*-algebra, having the spaces of intertwiners from the tensor unit to powers of the object as its spectral subspaces. The special case od S_qU(2) is discussed.
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We use a tensor C*-category with conjugates and two quasitensor functors into the category of Hilbert spaces to define a *-algebra depending functorially on this data. If one of them is tensorial, we can complete in the maximal C*-norm. A particular case of this construction allows us to begin with solutions of the conjugate equations and associate ergodic actions of quantum groups on the C*-algebra in question. The quantum groups involved are A_u(Q) and B_u(Q).
To a proper inclusion Nsubset M of II_1 factors of finite Jones index [M:N], we associate an ergodic C*-action of the quantum group S_mu U(2). The deformation parameter is determined by -1<mu<0 and [M:N]=|mu+mu^{-1}|. The higher relative commutants can be identified with the spectral spaces of the tensor powers of the defining representation of the quantum group. This ergodic action may be thought of as a virtual subgroup of S_mu U(2) in the sense of Mackey arising from the tensor category generated by M regarded as a bimodule over N. mu is negative as M is a real bimodule.
We give some sufficient conditions for the injectivity of actions of compact quantum groups on $C^{ast}$-algebra. As an application, we prove that any faithful smooth action by a compact quantum group on a compact smooth (not necessarily connected) manifold is injective. A similar result is proved for actions on $C^{ast}$- algebras obtained by Rieffel-deformation of compact, smooth manifolds.
Let G be a classical compact Lie group and G_mu the associated compact matrix quantum group deformed by a positive parameter mu (or a nonzero and real mu in the type A case). It is well known that the category Rep(G_mu) of unitary f.d. representations of G_mu is a braided tensor C*-category. We show that any braided tensor *-functor from Rep(G_mu) to another braided tensor C*-category with irreducible tensor unit is full if |mu| eq 1. In particular, the functor of restriction to the representation category of a proper compact quantum subgroup, cannot be made into a braided functor. Our result also shows that the Temperley--Lieb category generated by an object of dimension >2 can not be embedded properly into a larger category with the same objects as a braided tensor C*-subcategory.
A general form of contractive idempotent functionals on coamenable locally compact quantum groups is obtained, generalising the result of Greenleaf on contractive measures on locally compact groups. The image of a convolution operator associated to a contractive idempotent is shown to be a ternary ring of operators. As a consequence a one-to-one correspondence between contractive idempotents and a certain class of ternary rings of operators is established.
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