Function algebras on a 2-dimensional quantum complex plane

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

Projective free algebras of continuous functions on compact abelian groups

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.

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.

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.

We begin by explicitly computing the quantum isometry groups for the classical spheres and show that these coincide with the commutative \(C^*\) algebra of continuous functions on their classical isometry groups. The rest of the chapter is devoted to the computation of the quantum group of orientation preserving isometries for two different families of spectral triples on the Podles’ spheres, one constructed by Dabrowski et al and the other by Chakraborty and Pal.

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

Непрерывной оболочкой называется операция, сопоставляющая произвольной инволютивной стереотипной алгебре $A$ ее ближайшую в некотором смысле стереотипную алгебру $\operatorname{\mathsf{Env}}_{\mathcal C}A$, свойства которой, улавливаемые гомоморфизмами во всевозможные $C^*$-алгебры, не отличаются от свойств $A$, но элементы которой различаются этими гомоморфизмами. Если $A$ - инволютивная стереотипная подалгебра в алгебре ${\mathcal C}(M)$ непрерывных функций на паракомпактном локально компактном топологическом пространстве $M$, то для того, чтобы ${\mathcal C}(M)$ была непрерывной оболочкой $A$ $\operatorname{\mathsf{Env}}_{\mathcal C}A={\mathcal C}(M)$ необходимо, но не достаточно, чтобы $A$ была плотна в ${\mathcal C}(M)$. В заметке анонсируется необходимое и достаточное условие для выполнения этого равенства: нужно, чтобы инволютивный спектр алгебры $A$ совпадал с $M$ с точностью до возможного ослабления топологии, при котором, однако, система компактов в $M$ остается прежней и топология на каждом компакте не меняется.

Universal continuous calculi are defined and it is shown that for every finite tuple of pairwise commuting Hermitian elements of a Su*‐algebra (an ordered *‐algebra that is symmetric, i.e., “strictly” positive elements are invertible and uniformly complete), such a universal continuous calculus exists. This generalizes the continuous calculus for ‐algebras to a class of generally unbounded ordered *‐algebras. On the way, some results about *‐algebras of continuous functions on locally compact spaces are obtained. The approach used throughout is rather elementary and especially avoids any representation theory.

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 provide a method for reconstructing the underlying locallycompact Hausdorff space of an algebra of continuous functions vanishing at infinity, using the norm attainment sets of continuous functions. As an application,it is demonstrated that an order isomorphism of norm attainment sets betweenspaces of continuous functions induces a homeomorphism between the underlying topological spaces, even without linearity. Moreover, it turns out that a linearmap between algebras of continuous functions is an order isomorphism of normattainment sets if and only if it is a scalar multiple of an isometric isomorphismprovided that the underlying topological spaces are not two-point sets. We alsopresent a counterexample to the above statement in the two-point setting.

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.