<Emphasis Type="Italic">C</Emphasis>*-Homomorphisms and duality of <Emphasis Type="Italic">C</Emphasis>*-discrete quantum groups

Let

Open image in new window

be a C*-discrete quantum group and let

Open image in new window

be the discrete quantum group associated with

Open image in new window

. Suppose that there exists a continuous action of

Open image in new window

on a unital C*-algebra

Open image in new window

so that

Open image in new window

becomes a

Open image in new window

-algebra. If there is a faithful irreducible vacuum representation π of

Open image in new window

on a Hilbert space H =

Open image in new window

with a vacuum vector Ω, which gives rise to a

Open image in new window

-invariant state, then there is a unique C*-representation (θ, H) of

Open image in new window

supplemented by the action. The fixed point subspace of

Open image in new window

under the action of

Open image in new window

is exactly the commutant of θ(

Open image in new window

).

     

Some Notes on <Emphasis Type="Italic">C</Emphasis>- and <Emphasis Type="Italic">C</Emphasis>*-quotients of Frames

We characterise C*-quotients and C-quotients of completely regular frames in terms of ?ech-Stone compactifications and Lindelöfications, respectively. The latter is a frame-theoretic result with no spatial counterpart. We also characterise C*-quotients and dense C-quotients of completely regular frames in terms of normal covers.     

On Homomorphisms between <Emphasis Type="Italic">C</Emphasis>*-Algebras and Linear Derivations on <Emphasis Type="Italic">C</Emphasis>*-Algebras

It is shown that every almost linear Pexider mappings f, g, h from a unital C*-algebra into a unital C*-algebra ℬ are homomorphisms when f(2 ⁿ _uy) = f(2 ⁿ u)f(y), g(2 ⁿ uy) = g(2 ⁿ u)g(y) and h(2 ⁿ uy) = h(2 ⁿ u)h(y) hold for all unitaries u ∈ , all y ∈ , and all n ∈ ℤ, and that every almost linear continuous Pexider mappings f, g, h from a unital C*-algebra of real rank zero into a unital C*-algebra ℬ are homomorphisms when f(2 ⁿ uy) = f(2 ⁿ u)f(y), g(2 ⁿ uy) = g(2 ⁿ u)g(y) and h(2 ⁿ uy) = h(2 ⁿ u)h(y) hold for all u ∈ {v ∈ : v = v* and v is invertible}, all y ∈ and all n ∈ ℤ. Furthermore, we prove the Cauchy-Rassias stability of *-homomorphisms between unital C*-algebras, and ℂ-linear *-derivations on unital C*-algebras. This work was supported by Korea Research Foundation Grant KRF-2003-042-C00008. The second author was supported by the Brain Korea 21 Project in 2005.     

On the <Emphasis Type="Italic">C</Emphasis>*-valued triangle equality and inequality in Hilbert <Emphasis Type="Italic">C</Emphasis>*-modules

We prove that for two elements x, y in a Hilbert C*-module V over a C*-algebra the C*-valued triangle equality |x + y| = |x| + |y| holds if and only if 〈x, y〉 = |x| |y|. In addition, if has a unit e, then for every x, y ∊ V and every ɛ > 0 there are contractions u, υ ∊ such that |x + y| ≦ u|x|u* + υ|y|υ* + ɛe.      

<Emphasis Type="Italic">l</Emphasis>-Groups <Emphasis Type="Italic">C</Emphasis>(<Emphasis Type="Italic">X</Emphasis>) in continuous logic

In the context of continuous logic, this paper axiomatizes both the class \(\mathcal {C}\) of lattice-ordered groups isomorphic to C(X) for X compact and the subclass \(\mathcal {C}^+\) of structures existentially closed in \(\mathcal {C}\); shows that the theory of \(\mathcal {C}^+\) is \(\aleph _0\)-categorical and admits elimination of quantifiers; establishes a Nullstellensatz for \(\mathcal {C}\) and \(\mathcal {C}^+\); shows that \(C(X)\in \mathcal {C}\) has a prime-model extension in \(\mathcal {C}^+\) just in case X is Boolean; and proves that in a sense relevant to continuous logic, positive formulas admit in \(\mathcal {C}^+\) elimination of quantifiers to positive formulas.     

Approximate <Emphasis Type="Italic">C</Emphasis>*-ternary ring homomorphisms

In this paper, we establish the generalized Hyers–Ulam–Rassias stability of C*-ternary ring homomorphisms associated to the Trif functional equation

<Emphasis Type="Italic">C</Emphasis>*-algebras generated by semigroups

In this paper we study C*-algebras generated by a commuting family of isometric operators. Such algebras naturally generalize the Toeplitz algebra. We investigate *-automorphisms and ideals of C*-algebras generated by semigroups.     

Tracial Limit of C^＊-algebras

A new limit of C^*-algebras, the tracial limit, is introduced in this paper. We show that a separable simple C^*-algebra A is a tracial limit of C^*-algebras in Z-(k) if and only if A has tracial topological rank no more than k. We present several known results using the notion of tracial limits.     

Smooth noncompact operators from <Emphasis Type="Italic">C</Emphasis>(<Emphasis Type="Italic">K</Emphasis>), <Emphasis Type="Italic">K</Emphasis> scattered

Let X be a Banach space, K be a scattered compact and T: B _C(K) → X be a Fréchet smooth operator whose derivative is uniformly continuous. We introduce the smooth biconjugate T**: B _C(K)** → X** and prove that if T is noncompact, then the derivative of T** at some point is a noncompact linear operator. Using this we conclude, among other things, that either is compact or that ℓ₁ is a complemented subspace of X*. We also give some relevant examples of smooth functions and operators, in particular, a C ^1,u -smooth noncompact operator from B _{c O} which does not fix any (affine) basic sequence. P. Hájek was supported by grants A100190502, Institutional Research Plan AV0Z10190503.     

<Emphasis Type="Italic">C</Emphasis>*-simplicity of <Emphasis Type="Italic">n</Emphasis>-periodic products

The C*-simplicity of n-periodic products is proved for a large class of groups. In particular, the n-periodic products of any finite or cyclic groups (including the free Burnside groups) are C*-simple. Continuum-many nonisomorphic 3-generated nonsimple C*-simple groups are constructed in each of which the identity xⁿ = 1 holds, where n ≥ 1003 is any odd number. The problem of the existence of C*-simple groups without free subgroups of rank 2 was posed by de la Harpe in 2007.     

<Emphasis Type="Italic">C</Emphasis>-distribution semigroups and <Emphasis Type="Italic">C</Emphasis>-ultradistribution semigroups in locally convex spaces

The main purpose of this paper is to study C-distribution semigroups and C-ultradistribution semigroups in the setting of sequentially complete locally convex spaces. We provide a few important theoretical novelties in this field and some interesting examples. Under consideration are stationary dense operators in a sequentially complete locally convex space.     

Discreteness criteria in <Emphasis Type="Italic">PU</Emphasis> (1, <Emphasis Type="Italic">n</Emphasis>; <Emphasis Type="Italic">C</Emphasis>)

In this paper, we study the discreteness for non-elementary subgroups in PU (1, n; C), and several discreteness criteria are obtained.     

<Emphasis Type="Italic">C</Emphasis>*-algebras which are Grothendieck spaces

Each monotoneσ-completeC*-algebra is a Grothendieck space.     

<Emphasis Type="Italic">C</Emphasis>*-algebras of generalized Hecke pairs

We generalize the concept of Hecke pairs and study representations of the corresponding C*-algebra. We introduce the notions of covariant pairs and matrix unit pairs of representations in this general setting and show that covariant pairs are exactly faithful matrix unit pairs.     

On the commutativity of <Emphasis Type="Italic">C</Emphasis>* -algebras

Let be a C* -algebra. Let f be a non-constant complex-valued continuous function defined on a closed interval I. We shall show that f densely spans As an application, is commutative if f(x)f(y)=f(y)f(x) for all self-adjoint elements x and y in with spectrums contained in I.Mathematics Subject Classification (1991):Primary 46L05     

On Quantum Dynamics on <Emphasis Type="Italic">C</Emphasis>*-Algebras

We study three related extremal problems in the space H of functions analytic in the unit disk such that their boundary values on a part γ₁ of the unit circle Γ belong to the space \(L_{{\psi _1}}^\infty ({\gamma _1})\)of functions essentially bounded on γ₁ with weight ψ₁ and their boundary values on the set γ₀ = Γ γ₁ belong to the space \(L_{{\psi _0}}^\infty ({\gamma _0})\)with weight ψ₀. More exactly, on the class Q of functions from H such that the \(L_{{\psi _0}}^\infty ({\gamma _0})\)-norm of their boundary values on γ₀ does not exceed 1, we solve the problem of optimal recovery of an analytic function on a subset of the unit disk from its boundary values on γ₁ specified approximately with respect to the norm of \(L_{{\psi _1}}^\infty ({\gamma _1})\). We also study the problem of the optimal choice of the set γ₁ for a given fixed value of its measure. The problem of the best approximation of the operator of analytic continuation from a part of the boundary by bounded linear operators is investigated.     

<Emphasis Type="Italic">C</Emphasis>*-algebras generated by cancellative semigroups

A C*-algebra generated by a commuting family of isometries is a natural generalization of the Toeplitz algebra. We study the *-automorphisms and invariant ideals of the C*-algebra geerated by a semigroup.     

Differences of idempotents in <Emphasis Type="Italic">C</Emphasis>*-algebras

Suppose that P and Q are idempotents on a Hilbert space H, while Q = Q* and I is the identity operator in H. If U = P ? Q is an isometry then U = U* is unitary and Q = I ? P. We establish a double inequality for the infimum and the supremum of P and Q in H and P ? Q. Applications of this inequality are obtained to the characterization of a trace and ideal F-pseudonorms on a W*-algebra. Let φ be a trace on the unital C*-algebra A and let tripotents P and Q belong to A. If P ? Q belongs to the domain of definition of φ then φ(P ? Q) is a real number. The commutativity of some operators is established.     

On topologically finite-dimensional simple <Emphasis Type="Italic">C</Emphasis>*-algebras

We show that, if a simple C*-algebra A is topologically finite-dimensional in a suitable sense, then not only K₀(A) has certain good properties, but A is even accessible to Elliott’s classification program. More precisely, we prove the following results:If A is simple, separable and unital with finite decomposition rank and real rank zero, then K₀(A) is weakly unperforated.If A has finite decomposition rank, real rank zero and the space of extremal tracial states is compact and zero-dimensional, then A has stable rank one and tracial rank zero. As a consequence, if B is another such algebra, and if A and B have isomorphic Elliott invariants and satisfy the Universal coefficients theorem, then they are isomorphic.In the case where A has finite decomposition rank and the space of extremal tracial states is compact and zero-dimensional, we also give a criterion (in terms of the ordered K₀-group) for A to have real rank zero. As a byproduct, we show that there are examples of simple, stably finite and quasidiagonal C*-algebras with infinite decomposition rank.Supported by: EU-Network Quantum Spaces - Noncommutative Geometry (Contract No. HPRN-CT-2002-00280) and Deutsche Forschungsgemeinschaft (SFB 478).     

On the computation of <Emphasis Type="Italic">C</Emphasis>* certificates

The cone of completely positive matrices C* is the convex hull of all symmetric rank-1-matrices xx ^T with nonnegative entries. While there exist simple certificates proving that a given matrix \({B\in C^*}\) is completely positive it is a rather difficult problem to find such a certificate. We examine a simple algorithm which—for a given input B—either determines a certificate proving that \({B\in C^*}\) or converges to a matrix \({\bar S}\) in C* which in some sense is “close” to B. Numerical experiments on matrices B of dimension up to 200 conclude the presentation.