On the Poincaré disc and its higher-dimensional analogs one has a canonical formal star product of Wick type. We define a locally convex topology on a certain class of real-analytic functions on the disc for which the star product is continuous and converges as a series. The resulting Fréchet algebra is characterized explicitly in terms of the set of all holomorphic functions on an extended and doubled disc of twice the dimension endowed with the natural topology of locally uniform convergence. We discuss the holomorphic dependence on the deformation parameter and the positive functionals and their GNS representations of the resulting Fréchet algebra. *

Given a locally convex vector space with a topology induced by Hilbert seminorms and a continuous bilinear form on it we construct a topology on its symmetric algebra such that the usual star product of exponential type becomes continuous. Many properties of the resulting locally convex algebra are explained. We compare this approach to various other discussions of convergent star products in finite and infinite dimensions. We pay special attention to the case of a Hilbert space and to nuclear spaces.

Consider an ordered abelian group G (i.e. an abelian group with a translation invariant partial order) endowed with a biadditive operation µ : G×G → G. We will show that under some additional assumptions concerning the compatibility between the operation µ and the order, the operation µ is automatically both associative and commutative, and all squares µ(g, g) with g ∈ G are positive. These additional assumptions essentially are that the order is (integrally) closed (sometimes also referred to as "Archimedean", especially in the context of lattice-ordered groups) and localizable (the structure of the set of positive elements of G does not obstruct adjoining multiplicative inverses of "strictly" positive elements). Some results of this type have already been obtained by various authors -some of them more than half a century ago -in much more restricitve contexts, especially in the uniformly bounded and the lattice-ordered cases; we discuss how most of these classical results can be obtained from our main theorem. We also prove that a partially ordered skew field, in which there is no element that dominates all natural numbers, is a field.

We develop a general theory of symmetry reduction of states on (possibly non-commutative) * -algebras that are equipped with a Poisson bracket and a Hamiltonian action of a commutative Lie algebra g. The key idea advocated for in this article is that the "correct" notion of positivity on a * -algebra A is not necessarily the algebraic one whose positive elements are the sums of Hermitian squares a * a with a ∈ A, but can be a more general one that depends on the example at hand, like pointwise positivity on * -algebras of functions or positivity in a representation as operators. The notion of states (normalized positive Hermitian linear functionals) on A thus depends on this choice of positivity on A, and the notion of positivity on the reduced algebra A µ-red should be such that states on A µ-red are obtained as reductions of certain states on A. In the special case of the * -algebra of smooth functions on a Poisson manifold M , this reduction scheme reproduces the coisotropic reduction of M , where the reduced manifold M µ-red is just a geometric manifestation of the reduction of the evaluation functionals associated to certain points of M . However, we are mainly interested in applications to non-formal deformation quantization and therefore also discuss the reduction of the Weyl algebra, and of the polynomial algebra whose non-commutative analogs we plan to examine in future projects.

We study formal and non-formal deformation quantizations of a family of manifolds that can be obtained by phase space reduction from [Formula: see text] with the Wick star product in arbitrary signature. Two special cases of such manifolds are the complex projective space [Formula: see text] and the complex hyperbolic disc [Formula: see text]. We generalize several older results to this setting: The construction of formal star products and their explicit description by bidifferential operators, the existence of a convergent subalgebra of “polynomial” functions, and its completion to an algebra of certain analytic functions that allow an easy characterization via their holomorphic extensions. Moreover, we find an isomorphism between the non-formal deformation quantizations for different signatures, linking, e.g., the star products on [Formula: see text] and [Formula: see text]. More precisely, we describe an isomorphism between the (polynomial or analytic) function algebras that is compatible with Poisson brackets and the convergent star products. This isomorphism is essentially given by Wick rotation, i.e. holomorphic extension of analytic functions and restriction to a new domain. It is not compatible with the [Formula: see text]-involution of pointwise complex conjugation.