Quantum locally compact metric spaces

We consider a coamenable compact quantum group \mathbb{G} as a compact quantum metric space if its function algebra \mathrm{C}(\mathbb{G}) is equipped with a Lip-norm. By using a projection P onto direct summands of the Peter–Weyl decomposition, the \mathrm{C}^{*} -algebra \mathrm{C}(\mathbb{G}) can be compressed to an operator system P\mathrm{C}(\mathbb{G})P , and there are induced left and right coactions on this operator system. Assuming that the Lip-norm on \mathrm{C}(\mathbb{G}) is bi-invariant in the sense of Li, there is an induced bi-invariant Lip-norm on the operator system P\mathrm{C}(\mathbb{G})P turning it into a compact quantum metric space. Given an appropriate net of such projections which converges strongly to the identity map on the Hilbert space \mathrm{L}^{2}(\mathbb{G}) , we obtain a net of compact quantum metric spaces. We prove convergence of such nets in terms of Kerr’s complete Gromov–Hausdorff distance. An important tool is the choice of an appropriate state whose induced slice map gives an approximate inverse of the compression map \mathrm{C}(\mathbb{G}) \ni a \mapsto PaP in Lip-norm.

We show how to equip the crossed product between a group of polynomial growth and a compact quantum metric space with a compact quantum metric space structure. When the quantum metric on the base space arises from a spectral triple, which is compatible with the action of the group, we furthermore show that the crossed product becomes a spectral metric space. Lastly, we analyse the spectral triple at the crossed product level from the point of view of unbounded K K KK -theory and show that it arises as an internal Kasparov product of unbounded Kasparov modules.

Order-unit quantum Gromov–Hausdorff distance

We prove that some well known compact quantum spaces such as quantum tori and some quantum two-spheres do not admit a compact quantum group structure. This is achieved by considering existence of traces, characters and nuclearity of the corresponding <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper C Superscript asterisk"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">C</mml:mi> </mml:mrow> <mml:mo> ∗ </mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">\mathrm {C}^*</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -algebras.

We give an elementary introduction to the theory of algebraic and topological quantum groups (in the spirit of S. L. Woronowicz). In particular, we recall the basic facts from Hopf (*-) algebra theory, theory of compact (matrix) quantum groups and the theory of their actions on compact quantum spaces. We also provide the most important examples, including the classification of quantum SL(2)-groups, their real forms and quantum spheres. We also consider quantum SLq(N)-groups and quantum Lorentz groups.

Non-supersymmetric multi-wall configurations are generically unstable. It is proposed that the stabilization in compact space can be achieved by introducing a winding number into the model. A BPS-like bound is studied for the energy of configuration with nonvanishing winding number. Winding number is implemented in an ${\cal N}=1$ supersymmetric nonlinear sigma model with two chiral scalar fields and a bound states of BPS and anti-BPS walls is found to exist in noncompact spaces. Even in compactified space $S^1$, this nontrivial bound state persists above a critical radius of the compact dimension.

The covering dimension invariants

The Helgason Fourier transform for homogeneous vector bundles over compact Riemannian symmetric spaces—the local theory

In this paper we continue our study of the thermodynamics of large N gauge theories on compact spaces. We consider toroidal compactifications of pure SU(N) Yang-Mills theories and of maximally supersymmetric Yang-Mills theories dimensionally reduced to 0+1 or 1+1 dimensions, and generalizations of such theories where the adjoint fields are massive. We describe the phase structure of these theories as a function of the gauge coupling, the geometry of the compact space and the mass parameters. In particular, we study the behavior of order parameters associated with the holonomy of the gauge field around the cycles of the torus. Our methods combine analytic analysis, numerical Monte Carlo simulations, and (in the maximally supersymmetric case) information from the dual gravitational theories.

It is known by the Conley's theorem that the chain recurrent set CR(ϕ) of a deterministic flow ϕ on a compact metric space is the complement of the union of sets B(A) − A, where A varies over the collection of attractors and B(A) is the basin of attraction of A. It has recently been shown that a similar decomposition result holds for random dynamical systems (RDSs) on non-compact separable complete metric spaces, but under a so-called absorbing condition. In the present article, the authors introduce a notion of random chain recurrent sets for RDSs, and then prove the random Conley's theorem on non-compact separable complete metric spaces without the absorbing condition.

Every compact symmetric space $M$ admits a dual noncompact symmetric space $\check{M}$. When $M$ is a generalized Grassmannian, we can view $\check{M}$ as a open submanifold of it consisting of space-like subspaces \cite{HL}. Motivated from this, we study the embeddings from noncompact symmetric spaces to their compact duals, including space-like embedding for generalized Grassmannians, Borel embedding for Hermitian symmetric spaces and the generalized embedding for symmetric R-spaces. We will compare these embeddings and describe their images using cut loci.

Spectral decomposition for topologically Anosov homeomorphisms on noncompact and non-metrizable spaces

On questions of strict contractibility

A general combinatorial theorem for countably compact, noncompact spaces is given under the Proper Forcing Axiom. It follows that compact Hausdorff spaces of countable tightness are sequential under PFA, solving the Moore-Mrowka Problem. Other applications are also given.

We introduce various new type of recurrent sets for finitely generated semigroups on non-compact metric spaces that are conjugacy invariant, and obtain some basic properties of chain recurrent sets for semigroups via these new definitions. Moreover, we define the notion of weak shadowing property for finitely generated group actions on compact metric spaces, which is weaker than that of shadowing property, and prove the equivalence of the shadowing and weak shadowing properties for the finitely generated group actions on a generalized homogeneous space without isolated points.