Narrow Results

We isolate a large class of self-adjoint operators H whose essential spectrum is determined by their behavior at x ~ ∞ and we give a canonical representation of σ_ess(H) in terms of spectra of limits at infinity of translations of H.

We use constructive bounded Kasparov $K$-theory to investigate the numerical invariants stemming from the internal Kasparov products $K_i({𝒜})\times K{K}^i({𝒜},{ℬ})\to K_0({ℬ})\to ℝ$, $i=0,1$, where the last morphism is provided by a tracial state. For the class of properly defined finitely-summable Kasparov $({𝒜},{ℬ})$-cycles, the invariants are given by the pairing of $K$-theory of ${ℬ}$ with an element of the periodic cyclic cohomology of ${ℬ}$, which we call the generalized Connes–Chern character. When ${𝒜}$ is a twisted crossed product of ${ℬ}$ by ${\mathbb{Z}}^k$, ${𝒜}={ℬ}{⋊}_{\xi }^{\theta }{\mathbb{Z}}^k$, we derive a local formula for the character corresponding to the fundamental class of a properly defined Dirac cycle. Furthermore, when ${ℬ}=C(\mathrm{\Omega }){⋊}_{{\xi }^{\prime }}^{\varphi }{\mathbb{Z}}^j$, with $C(\mathrm{\Omega })$ the algebra of continuous functions over a disorder configuration space, we show that the numerical invariants are connected to the weak topological invariants of the complex classes of topological insulators, defined in the physics literature. The end products are generalized index theorems for these weak invariants, which enable us to predict the range of the invariants and to identify regimes of strong disorder in which the invariants remain stable. The latter will be reported in a subsequent publication.

We study C*-algebra endomorphims which are special in a weaker sense with respect to the notion introduced by Doplicher and Roberts. We assign to such endomorphisms a geometrical invariant, representing a cohomological obstruction for them to be special in the usual sense. Moreover, we construct the crossed product of a C*-algebra by the action of the dual of a (nonabelian, noncompact) group of vector bundle automorphisms. These crossed products supply a class of examples for such generalized special endomorphisms.

C*-endomorphisms arising from superselection structures with nontrivial center define a 'rank' and a 'first Chern class'. Crossed products by such endomorphisms involve the Cuntz–Pimsner algebra of a vector bundle having the above-mentioned rank, first Chern class and can be used to construct a duality for abstract (nonsymmetric) tensor categories versus group bundles acting on (nonsymmetric) Hilbert bimodules. Existence and unicity of the dual object (i.e. the 'gauge' group bundle) are not ensured: we give a description of this phenomenon in terms of a certain moduli space associated with the given endomorphism. The above-mentioned Hilbert bimodules are noncommutative analogs of gauge-equivariant vector bundles in the sense of Nistor–Troitsky.

For an inclusion N ⊆ M of II₁ factors, we study the group of normalizers and the von Neumann algebra it generates. We first show that imposes a certain "discrete" structure on the generated von Neumann algebra. By analyzing the bimodule structure of certain subalgebras of , this leads to a "Galois-type" theorem for normalizers, in which we find a description of the subalgebras of in terms of a unique countable subgroup of . Implications for inclusions B ⊆ M arising from the crossed product, group von Neumann algebra, and tensor product constructions will also be addressed. Our work leads to a construction of new examples of norming subalgebras in finite von Neumann algebras: If N ⊆ M is a regular inclusion of II₁ factors, then N norms M.

Let X be a Cantor set, and let A be a unital separable simple amenable -stable C*-algebra with rationally tracial rank no more than one, which satisfies the Universal Coefficient Theorem (UCT). We use C(X, A) to denote the algebra of all continuous functions from X to A. Let α be an automorphism on C(X, A). Suppose that C(X, A) is α-simple, [α|_1⊗A] = [id|_1⊗A] in KL(1 ⊗ A, C(X, A)), τ(α(1 ⊗ a)) = τ(1 ⊗ a) for all τ ∈ T(C(X, A)) and all a ∈ A, and for all u ∈ U(A) (where α^‡ and id^‡ are homomorphisms from U(C(X, A))/CU(C(X, A)) → U(C(X, A))/CU(C(X, A)) induced by α and id, respectively, and where CU(C(X, A)) is the closure of the subgroup generated by commutators of the unitary group U(C(X, A)) of C(X, A)), then the corresponding crossed product C(X, A) ⋊_α ℤ is a unital simple -stable C*-algebra with rationally tracial rank no more than one, which satisfies the UCT. Let X be a Cantor set and 𝕋 be the circle. Let γ : X × 𝕋ⁿ → X × 𝕋ⁿ be a minimal homeomorphism. It is proved that, as long as the cocycles are rotations, the tracial rank of the corresponding crossed product C*-algebra is always no more than one.

Let $X$ be an infinite compact metric space with finite covering dimension, let $A$ be a unital separable simple AH-algebra with no dimension growth, and denote by $C(X,A)$ the $C^{\ast }$-algebra of all continuous functions from $X$ to $A.$ Suppose that $\gamma :{\mathbb{Z}}^d\to \mathrm{\text{Aut}}(C(X,A))$ is a minimal group action and the induced ${\mathbb{Z}}^d$-action on $X$ is free. Under certain conditions, we show the crossed product $C^{\ast }$-algebra $C(X,A){⋊}_{\gamma }{\mathbb{Z}}^d$ has rational tracial rank zero and hence is classified by its Elliott invariant. Next, we show the following: Let $Y$ be a Cantor set, let $A$ be a stably finite unital separable simple $C^{\ast }$-algebra which is rationally TA${𝒮},$ where ${𝒮}$ is a class of separable unital $C^{\ast }$-algebras which is closed under tensoring with finite dimensional $C^{\ast }$-algebras and closed under taking unital hereditary sub-$C^{\ast }$-algebras, and let $\alpha \in \mathrm{\text{Aut}}(C(Y,A))$. Under certain conditions, we conclude that $C(Y,A){⋊}_{\alpha }\mathbb{Z}$ is rationally TA${𝒮}.$ Finally, we classify the crossed products of certain unital simple $C^{\ast }$-algebras by using the crossed products of $C(Y,A)$.

Let J(R) be the Jacobson radical of a ring R. Then R is called homogeneous semilocal if R/J(R) is simple artinian. The aim of this paper is to find necessary and sufficient conditions for the group rings and the crossed products to be homogeneous semilocal ring.

In 1993, Sim proved that all the faithful irreducible representations of a finite metacyclic group over any field of positive characteristic have the same degree. In this paper, we restrict our attention to non-modular representations and generalize this result for — (1) finite metabelian groups, over fields of positive characteristic coprime to the order of groups, and (2) finite groups having a cyclic quotient by an abelian normal subgroup, over number fields.

Let K be a normal subgroup of the finite group H. To a point of a K-interior H-algebra we associate a group extension, and we prove that this extension is isomorphic to an extension associated to the point given by the Brauer homomorphism. This may be regarded as a generalization and an alternative treatment of Dade's results [2, Section 12].