Test spaces (or manuals) provide a simple, elegant and very general mathematical framework for the study of probability theory – classical, quantum and otherwise. In many particular cases of interest, test spaces carry significant topological structure. This paper inaugurates a general study of topological test spaces. Among other results, we show that any topological test space with a compact ...

1. In [12] and [13] Selberg introduced a trace formula for a compact, locally symmetric space of negative curvature. There is a natural algebra of operators on any such space which commute with the Laplacian. The Selberg trace formula gives the trace of these operators. Selberg also pointed out the importance of deriving such a formula when the symmetric space is assumed only to have finite vol...

The algebraic approach to bundles in non-commutative geometry and the definition of quantum real weighted projective spaces are reviewed. Principal U(1)-bundles over quantum real weighted projective spaces are constructed. As the spaces in question fall into two separate classes, the negative or odd class that generalises quantum real projective planes and the positive or even class that genera...

The notion of quantum group equivariant homogeneous vector bundles on quantum homogeneous spaces is introduced. The category of such quantum vector bundles is shown to be exact, and its Grothendieck group is determined. It is also shown that the algebras of functions on quantum homogeneous spaces are noetherian.

We introduce, and investigate the properties of, the family of quantum Brègman distances, based on embeddings into suitable vector spaces (with the reflexive noncommutative Orlicz spaces over semi-finite W-algebras and noncommutative Lp spaces over any W-algebras providing two important examples). This allows us to define geometric categories for nonlinear quantum inference theory, with morphis...

Quantum groups lead to an algebraic structure that can be realized on quantum spaces. These are noncommutative spaces that inherit a well deened mathematical structure from the quantum group symmetry. In turn such quantum spaces can be interpreted as noncom-mutative connguration spaces for physical systems which carry a symmetry like structure. These connguration spaces will be generalized to n...