Narrow Results

Let be the Jiang–Su algebra and let τ be its unique tracial state. We prove that for all , the following statements are equivalent:

(1) a is a finite sum of commutators.

(2) a is a sum of five commutators.

(3) τ(a) = 0.

For a directed graph G, we generalize the Catalan numbers by using the canonical generating partial isometries of the Cuntz–Krieger algebra for the transition matrix A^G of the directed edges of G. The generalized Catalan numbers enumerate the number of Dyck paths for the graph G. Its generating functions will be studied.

We describe primitive and prime ideals in the C*-algebra C*(E) of a graph E satisfying Condition (K), together with the topologies on each of these spaces. In particular, we find that primitive ideals correspond to the set of maximal tails disjoint union the set of finite-return vertices, and that prime ideals correspond to the set of clusters of maximal tails disjoint union the set of finite-return vertices.

We determine the K-theory of the C*-algebra C(SU_-1(2)), describe its spectrum, and compute its Haar state. As an application, we use Blanchard's technique in order to prove the existence of a continuous C*-bundle over [-1, 0) whose fiber at q is isomorphic to C(SU_q(2)).

We introduce and study a Rokhlin-type property for actions of finite groups on (not necessarily unital) C*-algebras. We show that the corresponding crossed product C*-algebras can be locally approximated by C*-algebras that are stably isomorphic to closed two-sided ideals of the given algebra. We then use this result to prove that several classes of C*-algebras are closed under crossed products by finite group actions with this Rokhlin property.

H. Dye showed that an isomorphism between the (discrete) unitary groups in two factors not of type I_n is implemented by a linear (or a conjugate linear) *-isomorphism of the factors. If φ is an isomorphism between the unitary groups of two unital C*-algebras, it induces a bijective map θ_φ between the sets of projections. For certain UHF-algebras, we construct an automorphism φ of their unitary group, such that θ_φ does not preserve the orthogonality of projections. For a large class of unital finite C*-algebras, we show that θ_φ is always an orthoisomorphism. If φ is a continuous automorphism of the unitary group of a UHF-algebra A, we show that φ is implemented by a linear or a conjugate linear *-automorphism of A.

Let $A=\underrightarrow{lim}{A}_n$ be an AF algebra, $G$ be a compact group. We consider inductive limit actions of the form $\alpha =\underrightarrow{lim}{\alpha }_n$, where ${\alpha }_n:G\curvearrowright A_n$ is an action on the finite-dimensional C*-algebra $A_n$ which fixes each matrix summand. We give a complete classification up to conjugacy of such actions using twisted equivariant K-theory.

Let U₀, U₁, …, U_n be a (finite or infinite) sequence of semigroups of isometries which act on the same separable Hilbert space H, n = 1, 2, …, ∞. {U_j} is said to be orthogonal if for all i ≠ j we have

We determine, in terms of and , the homotopy groups of certain groups of invertibles and of certain equivalence classes in the infinite Grassmann space on a Hilbert C*--module. These results provide various interpretations of .

We study a non-commutative generalization of Stone duality that connects a class of inverse semigroups, called Boolean inverse ∧-semigroups, with a class of topological groupoids, called Hausdorff Boolean groupoids. Much of the paper is given over to showing that Boolean inverse ∧-semigroups arise as completions of inverse ∧-semigroups we call pre-Boolean. An inverse ∧-semigroup is pre-Boolean if and only if every tight filter is an ultrafilter, where tight filters are defined by combining ideas of both Exel and Lenz. A simple necessary condition for a semigroup to be pre-Boolean is derived and a variety of examples of inverse semigroups are shown to satisfy it. Thus the polycyclic inverse monoids, and certain Rees matrix semigroups over the polycyclics, are pre-Boolean and it is proved that the groups of units of their completions are precisely the Thompson–Higman groups G_{n, r}. The inverse semigroups arising from suitable directed graphs are also pre-Boolean and the topological groupoids arising from these graph inverse semigroups under our non-commutative Stone duality are the groupoids that arise from the Cuntz–Krieger C*-algebras. An elementary application of our theory shows that the finite, fundamental Boolean inverse ∧-semigroups are just the finite direct products of finite symmetric inverse monoids. Finally, we explain how tight filters are related to prime filters setting the scene for future work.

The notion of a generalized scale emerged in recent joint work with Afsar–Brownlowe–Larsen on equilibrium states on $C^{\ast }$-algebras of right Least Common Multiple (LCM) monoids, where it features as the key datum for the dynamics under investigation. This work provides the structure theory for such monoidal homomorphisms. We establish the uniqueness of the generalized scale and characterize its existence in terms of a simplicial graph arising from a new notion of irreducibility inside right LCM monoids. In addition, the method yields an explicit construction of the generalized scale if existent. We discuss applications for graph products as well as algebraic dynamical systems and reveal a striking connection to Saito’s degree map.

We distinguish three classes of capacities on a C*-algebra: subadditive, additive and maxitive. A tightness notion for capacities, the vague and narrow topologies on the set of capacities are introduced. The vague space of additive capacities which are finite on compact projections is a noncommutative version of the usual vague space of Radon measures on a locally compact Hausdorff space X. We give criterions of vague and narrow relative compactness in various classes of capacities. This allows us to extend most classical compactness theorems for Radon measures. The set of bounded (resp. tight) maxitive capacities is in bijection with the set of positive q-upper semicontinuous (resp. strongly q-upper semicontinuous) operators. This allows us to define a vague (resp. narrow) large deviation principle for a net of capacities as a vague (resp. narrow) convergence of this net towards a maxitive capacity, generalizing the classical notion for Radon probability measures on X. Next, we apply compactness theorems in order to extend some results in large deviations theory.

The Horodeckis necessary and sufficient condition of separability of mixed states is generalized to arbitrary composite quantum systems.

We prove that all finite joint distributions of creation and annihilation operators in monotone and anti-monotone Fock spaces can be realised as Quantum Central Limit of certain operators in a $C^{*}$-algebra, at least when the test functions are Riemann integrable. Namely, the approximation is given by weighted sequences of creators and annihilators in discrete monotone $C^{*}$-algebras, the weights being related to the above cited test functions.

Two known constructions of rings R, each having the property that different sized matrix rings M_n(R) and M_m(R) are isomorphic but the free left R-modules _RRⁿ and _RR^m are not, are shown to be isomorphic. The first construction utilizes a direct limit, while the second involves an intersection process.

In this paper, we discuss wild embeddings like Alexanders horned ball and relate them to fractal spaces. We build a C*-algebra corresponding to a wild embedding. We argue that a wild embedding is the result of a quantization process applied to a tame embedding. Therefore, quantum states are directly the wild embeddings. Then we give an example of a wild embedding in the four-dimensional spacetime. We discuss the consequences for cosmology.

Equivalence classes of gapped Hamiltonians compatible with given symmetry constraints, such as those underlying topological insulators, can be defined in many ways. For the non-chiral classes modeled by vector bundles over Brillouin tori, physically relevant equivalences include isomorphism, homotopy, and K-theory, which are inequivalent but closely related. We discuss an important subtlety which arises in the chiral Class AIII systems, where the winding number invariant is shown to be relative rather than absolute as is usually assumed. These issues are then analyzed and reconciled in the language of K-theory.

The authors consider the irreducibility of the Cowen-Douglas operator T. It is proved that T is irreducible iff the unital C*-algebra generated by some non-zero blocks in the decomposition of T with respect to is .

Let M be a closed connected manifold and let D be an elliptic operator on M. Let G be a discrete countable group and let be a principal G-bundle. Connes and Moscovici showed that this data defines an analytic index ind_ℓ¹(G)(D) ∈ K₀(ℓ¹(G)). If B is a unital tracial C*-algebra, we give a formula for the trace of the image of ind_ℓ¹(G)(D) in K₀(B) under the map induced by a quasi-representation of G in B. As an application, we reprove and generalize a formula of Exel and Loring to surface groups.

We obtain a complete characterization of all orthogonality preserving operators from a JB*-algebra to a JB*-triple. If T : J → E is a bounded linear operator from a JB*-algebra (respectively, a C*-algebra) to a JB*-triple and h denotes the element T**(1), then T is orthogonality preserving, if and only if, T preserves zero-triple-products, if and only if, there exists a Jordan *-homomorphism such that S(x) and h operator commute and T(x) = h•_r(h) S(x), for every x ∈ J, where r(h) is the range tripotent of h, is the Peirce-2 subspace associated to r(h) and •_r(h) is the natural product making a JB*-algebra.

This characterization culminates the description of all orthogonality preserving operators between C*-algebras and JB*-algebras and generalizes all the previously known results in this line of study.