Mathematics

We give algebraic characterizations of the type I and CCR properties for locally compact second countable, ample Hausdorff groupoids in terms of subquotients of their Boolean inverse semigroups of compact open bisections. It yields in turn algebraic characterizations of both properties for inverse semigroups in terms of subquotients of their booleanizations.

In this note, we exhibit an example of a multiparameter CCR flow which is not cocycle conjugate to its opposite. This is in sharp contrast to the one parameter situation

We discuss the relative K-theory for a C ∗ -algebra, A , together with a C ∗ -subalgebra, A ′ ⊆A . The relative group is denoted K i ( A ′ ;A),i=0,1 , and is due to Karoubi. We present a situation of two pairs A ′ ⊆A and B ′ ⊆B are related so that there is a natural isomorphism between their respective relative K-theories. We also discuss applications to the case where A and B are C ∗ -algebras of a pair of locally compact, Hausdorff topological groupoids, with Haar systems.

In this paper we prove a K -homology index theorem for the Toeplitz operators obtained from the multishifts of the Bergman space on several classes of egg-like domains. This generalizes our theorem with Douglas and Yu on the unit ball.

When the reduced twisted C ∗ -algebra C ∗ r (G,c) of a non-principal groupoid G admits a Cartan subalgebra, Renault's work on Cartan subalgebras implies the existence of another groupoid description of C ∗ r (G,c) . In an earlier paper, joint with Reznikoff and Wright, we identified situations where such a Cartan subalgebra arises from a subgroupoid S of G . In this paper, we study the relationship between the original groupoids S,G and the Weyl groupoid and twist associated to the Cartan pair. We first identify the spectrum B of the Cartan subalgebra C ∗ r (S,c) . We then show that the quotient groupoid G/S acts on B , and that the corresponding action groupoid is exactly the Weyl groupoid of the Cartan pair. Lastly we show that, if the quotient map G→G/S admits a continuous section, then the Weyl twist is also given by an explicit continuous 2 -cocycle on G/S⋉B .

We investigate angles between Haagerup--Schultz projections of operators belonging to finite von Neumann algebras, in connection with a property analogous to Dunford's notion of spectrality of operators. In particular, we show that an operator can be written as the sum of a normal and an s.o.t.-quasinilpotent operator that commute if and only if the angles between its Haagerup--Schultz projections are uniformly bounded away from zero (and we call this the uniformly nonzero anlges property). Moreover, we show that spectrality is equivalent to this uniformly nonzero angles property plus decomposability. Finally, using this characterization, we construct an easy example of an operator which is decomposable but not spectral, and we show that Voiculescu's circular operator is not spectral (nor are any of the circular free Poisson operators).

We prove implications among the conditions in the title for an inclusion of a C*-algebra A in a C*-algebra B, and we also relate this to several other properties in case B is a crossed product for an action of a group, inverse semigroup or an étale groupoid on A. We show that an aperiodic C*-inclusion has a unique pseudo-expectation. If, in addition, the unique pseudo-expectation is faithful, then A supports B in the sense of the Cuntz preorder. The almost extension property implies aperiodicity, and the converse holds if B is separable. A crossed product inclusion has the almost extension property if and only if the dual groupoid of the action is topologically principal. Topologically free actions are always aperiodic. If A is separable or of Type I, then topological freeness, aperiodicity and having a unique pseudo-expectation are equivalent to the condition that A detects ideals in all intermediate C*-algebras. If, in addition, B is separable, then all these conditions are equivalent to the almost extension property.

We consider C*-algebras of finite higher-rank graphs along with their rotational action. We show how the entropy theory of product systems with finite frames applies to identify the phase transitions of the dynamics. We compute the positive inverse temperatures where symmetry breaks, and in particular we identify the subharmonic parts of the gauge-invariant equilibrium states. Our analysis applies to positively weighted rotational actions through a recalibration of the entropies.

Suppose A is a separable unital ASH C*-algebra, R is a sigma-finite II ∞ factor von Neumann algebra, and π,ρ:A→R are unital ∗ -homomorphisms such that, for every a∈A , the range projections of π(a) and ρ(a) are Murray von Neuman equivalent in R . We prove that π and ρ are approximately unitarily equivalent modulo K R , where K R is the norm closed ideal generated by the finite projections in R . We also prove a very general result concerning approximate equivalence in arbitrary finite von Neumann algebras.

We introduce a notion of approximate ideal structure for a C ∗ -algebra, and use it as a tool to study K -theory groups. The notion is motivated by the classical Mayer-Vietoris sequence, by the theory of nuclear dimension as introduced by Winter and Zacharias, and by the theory of dynamical complexity introduced by Guentner, Yu, and the author. A major inspiration for our methods comes from recent work of Oyono-Oyono and Yu in the setting of controlled K -theory of filtered C*-algebras; we do not, however, use that language in this paper. We give two main applications. The first is a vanishing result for K -theory that is relevant to the Baum-Connes conjecture. The second is a permanence result for the Künneth formula in C ∗ -algebra K -theory: roughly, this says that if A can be decomposed into a pair of subalgebras (C,D) such that C , D , and C∩D all satisfy the Künneth formula, then A itself satisfies the Künneth formula.