Functional calculus and ∗-regularity of a class of Banach algebras II

In this article, we define a natural Banach ∗-algebra for a C^∗-dynamical system (A,G,α) which is slightly bigger than L¹(G;A) (they are the same if A is finite-dimensional). We will show that this algebra is ∗-regular if G has polynomial growth. The main result in this article extends the two main results in [C.W. Leung, C.K. Ng, Functional calculus and ∗-regularity of a class of Banach algebras, Proc. Amer. Math. Soc., in press].     

On the topology of the Kasparov groups and its applications

In this paper we establish a direct connection between stable approximate unitary equivalence for *-homomorphisms and the topology of the KK-groups which avoids entirely C^*-algebra extension theory and does not require nuclearity assumptions. To this purpose we show that a topology on the Kasparov groups can be defined in terms of approximate unitary equivalence for Cuntz pairs and that this topology coincides with both Pimsner's topology and the Brown-Salinas topology. We study the generalized Rørdam group , and prove that if a separable exact residually finite dimensional C^*-algebra satisfies the universal coefficient theorem in KK-theory, then it embeds in the UHF algebra of type 2^∞. In particular such an embedding exists for the C^*-algebra of a second countable amenable locally compact maximally almost periodic group.     

Semigroup crossed products and the induced algebras of lattice-ordered groups

Let (G,G₊) be a quasi-lattice-ordered group with positive cone G₊. Laca and Raeburn have shown that the universal C^∗-algebra C^∗(G,G₊) introduced by Nica is a crossed product BG₊α_×G₊ by a semigroup of endomorphisms. The goal of this paper is to extend some results for totally ordered abelian groups to the case of discrete lattice-ordered abelian groups. In particular given a hereditary subsemigroup H₊ of G₊ we introduce a closed ideal IH₊ of the C^∗-algebra BG₊. We construct an approximate identity for this ideal and show that IH₊ is extendibly α-invariant. It follows that there is an isomorphism between C^∗-crossed products and B₊(G/H)β_×G₊. This leads to our main result that B₊(G/H)β_×G₊ is realized as an induced C^∗-algebra .     

Domains of infinitesimal generators of automorphism flows

We present a systematic characterization of the domain of a generator of a one parameter group on certain C∗-subalgebras of via finite-dimensional estimates. Our approach yields an example of a densely defined closed symmetric derivation on a C∗-subalgebras of whose domain is not closed with respect to the C¹-functional calculus. This completes and complements the earlier example of McIntosh (J. Funct. Anal. 30 (1977) 264). Our methods are partly based on the theory of adjoint C₀-semigroups.     

Simple compact quantum groups I

The notion of simple compact quantum group is introduced. As non-trivial (noncommutative and noncocommutative) examples, the following families of compact quantum groups are shown to be simple: (a) The universal quantum groups Bu(Q) for Q∈GL(n,C) satisfying , n?2; (b) The quantum automorphism groups Aaut(B,τ) of finite-dimensional C^∗-algebras B endowed with the canonical trace τ when dim(B)?4, including the quantum permutation groups Aaut(Xn) on n points (n?4); (c) The standard deformations Kq of simple compact Lie groups K and their twists , as well as Rieffel's deformation KJ.     

Realizations of AF-algebras as graph algebras, Exel-Laca algebras, and ultragraph algebras

We give various necessary and sufficient conditions for an AF-algebra to be isomorphic to a graph C^∗-algebra, an Exel-Laca algebra, and an ultragraph C^∗-algebra. We also explore consequences of these results. In particular, we show that all stable AF-algebras are both graph C^∗-algebras and Exel-Laca algebras, and that all simple AF-algebras are either graph C^∗-algebras or Exel-Laca algebras. In addition, we obtain a characterization of AF-algebras that are isomorphic to the C^∗-algebra of a row-finite graph with no sinks.     

The ideal property, the projection property, continuous fields and crossed products

Let A be the C^∗-algebra associated to an arbitrary continuous field of C^∗-algebras. We give a necessary and sufficient condition for A to have the ideal property and, if moreover A is separable, we give a necessary and sufficient condition for A to have the projection property. Some applications of these results are given. We also prove that “many” crossed products of commutative C^∗-algebras by discrete, amenable groups have the projection property, generalizing some of our previous results.     

Spectrum and analytical indices of the C-algebra of Wiener-Hopf operators

We study multivariate generalisations of the classical Wiener-Hopf algebra, which is the C^∗-algebra generated by the Wiener-Hopf operators, given by convolutions restricted to convex cones. By the work of Muhly and Renault, this C^∗-algebra is known to be isomorphic to the reduced C^∗-algebra of a certain restricted action groupoid, given by the action of Euclidean space on a certain compactification. Using groupoid methods, we construct composition series for the Wiener-Hopf C^∗-algebra by a detailed study of this compactification. We compute the spectrum, and express homomorphisms in K-theory induced by the symbol maps which arise by the subquotients of the composition series in analytical terms. Namely, these symbols maps turn out to be given by an analytical family index of a continuous family of Fredholm operators. In a subsequent paper, we also obtain a topological expression of these indices.     

Decomposition rank and absorbing extensions of type I algebras

We prove the following: Let A and B be separable C^*-algebras. Suppose that B is a type I C^*-algebra such that

(i)

B has only infinite dimensional irreducible *-representations, and

(ii)

B has finite decomposition rank.

If

0→B→C→A→0     

Generalized homothetic biorders

In this paper, we study the binary relations R on a nonempty N^∗-set A which are h-independent and h-positive (cf. the introduction below). They are called homothetic positive orders. Denote by B the set of intervals of R having the form [r,+∞[ with 0<r≤+∞ or ]q,∞[ with q∈Q_≥0. It is a Q_>0-set endowed with a binary relation > extending the usual one on R_>0 (identified with a subset of B via the map r?[r,+∞[). We first prove that there exists a unique map Φ_R:A×A→B such that (for all and all ) we have Φ(mx,ny)=mn⁻¹⋅Φ(x,y) and . Then we give a characterization of the homothetic positive orders R on A such that there exist two morphisms of N^∗-sets satisfying . They are called generalized homothetic biorders. Moreover, if we impose some natural conditions on the sets u₁(A) and u₂(A), the representation (u₁,u₂) is “uniquely” determined by R. For a generalized homothetic biorder R on A, the binary relation R₁ on A defined by is a generalized homothetic weak order; i.e. there exists a morphism of N^∗-sets u:A→B such that (for all ) we have . As we did in [B. Lemaire, M. Le Menestrel, Homothetic interval orders, Discrete Math. 306 (2006) 1669-1683] for homothetic interval orders, we also write “the” representation (u₁,u₂) of R in terms of u and a twisting factor.     

Separable states and positive maps

Using the natural duality between linear functionals on tensor products of C^∗-algebras with the trace class operators on a Hilbert space H and linear maps of the C^∗-algebra into B(H), we study the relationship between separability, entanglement and the Peres condition of states and positivity properties of the linear maps.     

Strongly quasibounded maximal monotone perturbations for the Berkovits-Mustonen topological degree theory

Let X be a real reflexive Banach space with dual X^∗. Let L:X⊃D(L)→X^∗ be densely defined, linear and maximal monotone. Let T:X⊃D(T)→X^∗2, with 0∈D(T) and 0∈T(0), be strongly quasibounded and maximal monotone, and C:X⊃D(C)→X^∗ bounded, demicontinuous and of type (S₊) w.r.t. D(L). A new topological degree theory has been developed for the sum L+T+C. This degree theory is an extension of the Berkovits-Mustonen theory (for T=0) and an improvement of the work of Addou and Mermri (for T:X→X^∗2 bounded). Unbounded maximal monotone operators with are strongly quasibounded and may be used with the new degree theory.     

Twisted first homology groups of the automorphism group of a free group

The automorphism group and outer automorphism group of a free group F_n of rank n act on the abelianized group H of F_n and the dual group H^* of H. The twisted first homology groups of and with coefficients in H and H^* are calculated.     

On the base dimension I and the property of universality

In Iliadis (2005) [13] for an ordinal α the notion of the so-called (bn-Ind?α)-dimensional normal base C for the closed subsets of a space X was introduced. This notion is defined similarly to the classical large inductive dimension Ind. In this case we shall write here I(X,C)?α and say that the base dimension I of the space X by the normal base C is less than or equal to α. The classical large inductive dimension Ind of a normal space X, the large inductive dimension Ind₀ of a Tychonoff space X defined independently by Charalambous and Filippov, as well as, the relative inductive dimension defined by Chigogidze for a subspace X of a Tychonoff space Y may be considered as the base dimension I of X by normal bases Z(X) (all closed subsets of X), Z(X) (all functionally closed subsets of X), and , respectively.In the present paper, we shall consider normal bases of spaces consisting of functionally closed subsets. In particular, we introduce new dimension invariant : for a space X, is the minimal element α of the class O∪{−1,∞}, where O is the class of all ordinals, for which there exists a normal base C on X consisting of functionally closed subsets such that I(X,C)?α. We prove that in the class of all completely regular spaces X of weight less than or equal to a given infinite cardinal τ such that there exist universal spaces. However, the following questions are open.(1) Are there universal elements in the class of all normal (respectively, of all compact) spaces X of weight ?τ with ?(2) Are there universal elements in the class of all Tychonoff (respectively, of all normal) spaces X of weight ?τ with Ind₀(X)?n∈ω? (Note that for a compact space X.)     

Unitary part of a contraction

For a contraction A on a Hilbert space H, we define the index j(A) (resp., k(A)) as the smallest nonnegative integer j (resp., k) such that ker(I−Aj∗Aj) (resp., ker(I−Ak*Ak)∩ker(I−AkAk∗)) equals the subspace of H on which the unitary part of A acts. We show that if , then j(A)?n (resp., k(A)?⌈n/2⌉), and the equality holds if and only if A is of class Sn (resp., one of the three conditions is true: (1) A is of class Sn, (2) n is even and A is completely nonunitary with ‖An−2‖=1 and ‖An−1‖<1, and (3) n is even and A=U⊕A^′, where U is unitary on a one-dimensional space and A^′ is of class Sn−1).     

The range of united K-theory

We prove that the united K-theory functor is a surjective functor from the category of real simple separable purely infinite C^∗-algebras to the category of countable acyclic CRT-modules. As a consequence, we show that every complex Kirchberg algebra satisfying the universal coefficient theorem is the complexification of a real C^∗-algebra.     

Improving the high order nonlinearity lower bound for Boolean functions with given algebraic immunity

Algebraic immunity is a recently introduced cryptographic parameter for Boolean functions used in stream ciphers. If pAI(f) and pAI(f⊕1) are the minimum degree of all annihilators of f and f⊕1 respectively, the algebraic immunity AI(f) is defined as the minimum of the two values. Several relations between the new parameter and old ones, like the degree, the r-th order nonlinearity and the weight of the Boolean function, have been proposed over the last few years.In this paper, we improve the existing lower bounds of the r-th order nonlinearity of a Boolean function f with given algebraic immunity. More precisely, we introduce the notion of complementary algebraic immunity defined as the maximum of pAI(f) and pAI(f⊕1). The value of can be computed as part of the calculation of AI(f), with no extra computational cost. We show that by taking advantage of all the available information from the computation of AI(f), that is both AI(f) and , the bound is tighter than all known lower bounds, where only the algebraic immunity AI(f) is used.     

A functional characterization of measures and the Banach-Ulam problem

For a measurable space (Ω,A), let ?_∞(A) be the closure of span{χA:A∈A} in ?_∞(Ω). In this paper we show that a sufficient and necessary condition for a real-valued finitely additive measure μ on (Ω,A) to be countably additive is that the corresponding functional ?μ defined by (for x∈?_∞(A)) is w^*-sequentially continuous. With help of the Yosida-Hewitt decomposition theorem of finitely additive measures, we show consequently that every continuous functional on ?_∞(A) can be uniquely decomposed into the ?₁-sum of a w^*-continuous functional, a purely w^*-sequentially continuous functional and a purely (strongly) continuous functional. Moreover, several applications of the results to measure extension are given.