Quantum Isometry Groups of Classical and Quantum Spheres

We construct spectral triples for the C^*-algebra of continuous functions on the quantum SU(2) group and the quantum sphere. There has been various approaches towards building a calculus on quantum spaces, but there seems to be very few instances of computations outlined in chapter~6 of Connes' book. We give detailed computations of the associated Connes-de Rham complex and the space of L_2-forms.

We construct new families of spectral triples over quantum spheres, with a particular attention focused on the standard Podleś quantum sphere and twisted Dirac operators.

We study maps between uniformly closed algebras of complex-valued continuous functions which vanish at infinity on locally compact Hausdorff spaces. Without assuming linearity nor multiplicativity on the maps we show that they are isometrical isomorphisms as Banach space operators if they satisfy that the peripheral range of the product of the images of any two elements coincides with the peripheral range of the product of those elements. Furthermore, if the underlying algebras contain approximate identities, then they are isometrically isomorphic as Banach algebras, which is a generalization of a recent result of Luttman and Tonev for the case of uniform algebras. On the other hand it is not the case without assuming the existence of approximate identities; An example is given.

Quantum isometry groups of spectral triples associated with approximately finite-dimensional C ∗ C^* -algebras are shown to arise as inductive limits of quantum symmetry groups of corresponding truncated Bratteli diagrams. This is used to determine explicitly the quantum isometry group of the natural spectral triple on the algebra of continuous functions on the middle-third Cantor set. It is also shown that the quantum symmetry groups of finite graphs or metric spaces coincide with the quantum isometry groups of the corresponding classical objects equipped with natural Laplacians.

We give a description of the continuity ideals and the kernels of homomorphisms from the algebras of continuous functions on locally compact spaces into Banach algebras. We also construct families of prime ideals satisfying a certain intriguing property in the algebras of continuous functions.

Inductive limits of semiprojective C⁎-algebras

An analogue of the Riemannian Geometry for an ultrametric Cantor set (C,d) is described using the tools of Noncommutative Geometry. Associated with (C,d) is a weighted rooted tree, its Michon tree [29]. This tree allows to define a family of spectral triples (\mathcal{C}_{\rm Lip}(C),\mathcal{H},D) using the ℓ^2 -space of its vertices, giving the Cantor set the structure of a noncommutative Riemannian manifold. Here \mathcal{C}_{\rm Lip}(C) denotes the space of Lipschitz continuous functions on (C,d) . The family of spectral triples is indexed by the space of choice functions , which is shown to be the analogue of the sphere bundle of a Riemannian manifold. The Connes metric coming from the family of these spectral triples allows to recover the metric on C . The corresponding ζ -function is shown to have abscissa of convergence, s_0 , equal to the upper box dimension of (C,d) . Taking the residue at this singularity leads to the definition of a canonical probability measure on C , which in certain cases coincides with the Hausdorff measure at dimension s_0 . This measure in turn induces a measure on the space of choices. Given a choice, the commutator of D with a Lipschitz continuous function can be interpreted as a directional derivative. By integrating over all choices, this leads to the definition of an analogue of the Laplace–Beltrami operator. This operator has compact resolvent and generates a Markov semigroup which plays the role of a Brownian motion on C . This construction is applied to the simplest case, the triadic Cantor set, where: (i) the spectrum and the eigenfunctions of the Laplace–Beltrami operator are computed, (ii) the Weyl asymptotic formula is shown to hold with the dimension s_0 , (iii) the corresponding Markov process is shown to have an anomalous diffusion with \mathbb{E}(d(X_t,X_{t + δt})^2) ≃ δt \ln(1/δt) as δt ↓ 0 .

Weighted circle actions on the quantum Heeqaard 3-sphere are considered. The fixed point algebras, termed quantum weighted Heegaard spheres, and their representations are classified and described on algebraic and topological levels. On the algebraic side, coordinate algebras of quantum weighted Heegaard spheres are interpreted as generalised Weyl algebras, quantum principal circle bundles and Fredholm modules over them are constructed, and the associated line bundles are shown to be non-trivial by an explicit calculation of their Chern numbers. On the topological side, the C*-algebras of continuous functions on quantum weighted Heegaard spheres are described and their K-groups are calculated.

Projective free algebras of continuous functions on compact abelian groups

The well-behaved representations of the coordinate algebra of a 2-dimensional quantum complex plane are classified and a C*-algebra is defined which can be viewed as the algebra of continuous functions on the 2-dimensional quantum complex plane vanishing at infinity.

By a result of Nagy, the C^*-algebra of continuous functions on the q -deformation G_q of a simply connected semisimple compact Lie group G is KK-equivalent to C(G) . We show that under this equivalence the K-homology class of the Dirac operator on G_q , which we constructed in an earlier paper, corresponds to that of the classical Dirac operator. Along the way we prove that for an appropriate choice of isomorphisms between completions of U_q\mathfrak g and U\mathfrak g a family of Drinfeld twists relating the deformed and classical coproducts can be chosen to be continuous in q .

We compare topologization of certain algebras as topological algebras as well as metrizable semitopological algebras. The main aim of the paper is to prove that the algebra of continuous functions on a non Lindelof paracompact space is not topologizable as a metrizable semitopological algebra. This gives us further examples of locally convex topologizable algebras that are not topologizable as metrizable semitopological algebras. On the other hand we prove that the free algebra generated by א variables has at last max{2א0 ,א} topologies of metrizable semitopological locally convex algebras. We provide also, for all p ∈ (0,1] , an example of locally bounded topological algebra which is not topologizable as a metrizable semitopological locally p -convex algebra. This is an investigation in the direction of some results of J. Esterle, V. Muller, W. Żelazko and the author in previous papers. Mathematics subject classification (2010): 46H05.

We describe a class of MV-algebras which is a natural generalization of the class of “algebras of continuous functions”. More specifically, we're interested in the algebra of frame maps Hom\({_{\cal F}}\) (Ω(A), K) in the category T of frames, where A is a topological MV-algebra, Ω(A) the lattice of open sets of A, and K an arbitrary frame.

The ideal structure of C ( X ) C(X) , the algebra of continuous functions from a completely regular Hausdorff space X X to the scalars is analyzed by examining for fixed A ⊂ β X A \subset \beta X (the Stone-Čech compactification of X X ) the structure of the quotient I A / F A {I^A}/{F^A} , where I A [ F A ] {I^A}[{F^A}] is the ideal of maps f ∈ C ( X ) f \in C(X) for which \[ A ⊂ cl β X Z ( f ) [ A ⊂ int β X cl β X Z ( f ) ] . A \subset {\text {cl} _{\beta X}}Z(f)\quad [A \subset {\operatorname {int} _{\beta X}}{\text {cl} _{\beta X}}Z(f)]. \] Unless it vanishes, I A / F A {I^A}/{F^A} has no minimal or maximal ideals, and its Krull dimension is infinite. If J J is an ideal of C ( X ) C(X) strictly between F A {F^A} and I A {I^A} , there are ideals J _ \underline {J} and J ¯ \bar J of C ( X ) C(X) for which F A ⊂ J _ ⊂ J ⊂ J ¯ ⊂ I A {F^A} \subset \underline {J} \subset J \subset \bar J \subset {I^A} with all inclusions proper. For K ⊂ C ( X ) K \subset C(X) , let Z ( K ) = ∩ { cl β X Z ( f ) : f ∈ K } Z(K) = \cap \{ {\text {cl} _{\beta X}}Z(f):f \in K\} . If J ⫋ I J \subsetneqq I are ideals of C ( X ) C(X) with Z ( J ) = Z ( I ) Z(J) = Z(I) and if I I is semiprime, there is an ideal K K strictly between J J and I I . If I I and J J are Z Z -ideals, K K can be chosen to be of the form P ∩ I , P P \cap I, P a prime ideal of C ( X ) C(X) . The maximal ideals of a semiprime ideal I I of C ( X ) C(X) are of the form I q ∩ I , q ∈ β X − Z ( I ) {I^q} \cap I,q \in \beta X - Z(I) . If A ⊂ β X A \subset \beta X is closed, I A {I^A} is a finitely generated ideal iff A A is open.

Point derivations and analytic structure in the spectrum of a Banach algebra