Mathematics

Building on previous papers by Anantharaman-Delaroche (AD) we introduce and study the notion of AD-amenability for partial actions and Fell bundles over discrete groups. We prove that the cross-sectional C*-algebra of a Fell bundle is nuclear if and only if the underlying unit fibre is nuclear and the Fell bundle is AD-amenable. If a partial action is globalisable, then it is AD-amenable if and only if its globalisation is AD-amenable. Moreover, we prove that AD-amenability is preserved by (weak) equivalence of Fell bundles and, using a very recent idea of Ozawa and Suzuki, we show that AD-amenabity is equivalent to an approximation property introduced by Exel.

We prove a functoriality result for the full C*-algebras of right-LCM monoids with respect to monoid inclusions that are closed under factorization and preserve orthogonality, and use this to show that if a right-LCM monoid is amenable in the sense of Nica, then so are its submonoids. As applications, we complete the classification of Artin monoids with respect to Nica amenability by showing that only the right-angled ones are amenable in the sense of Nica and we show that the Nica amenability of a graph product of right-LCM semigroups is inherited by the factors.

In this work we introduce and study a new notion of amenability for actions of locally compact groups on C ∗ -algebras. Our definition extends the definition of amenability for actions of discrete groups due to Claire Anantharaman-Delaroche. We show that our definition has several characterizations and permanence properties analogous to those known in the discrete case. For example, for actions on commutative C ∗ -algebras, we show that our notion of amenability is equivalent to measurewise amenability. We also give new characterizations of amenability even in the discrete case: in particular, we show that amenability is equivalent to the so-called quasi-central approximation property, a strong approximation property that was recently used by Suzuki in equivariant classification theory. We use our new notion of amenability to study when the maximal and reduced crossed products agree. One of our main results generalizes a theorem of Matsumura: we show that for an action of an exact locally compact group G on a locally compact space X the full and reduced crossed products C 0 (X) ⋊ max G and C 0 (X) ⋊ red G coincide if and only if the action of G on X is amenable. We also show that the analogue of this theorem does not hold for actions on noncommutative C ∗ -algebras. Finally, we study amenability as it relates to more detailed structure in the case of C ∗ -algebras that fibre over an appropriate G -space X , and the interaction of amenability with various regularity properties such as nuclearity, exactness, and the (L)LP, and the equivariant versions of injectivity and the WEP.

In this paper we study amenability, nuclearity and tensor products of C ∗ -Fell bundles by the method of induced representation theory.

Amenable actions of locally compact groups on von Neumann algebras are investigated by exploiting the natural module structure of the crossed product over the Fourier algebra of the acting group. The resulting characterisation of injectivity for crossed products generalises a result of Anantharaman-Delaroche on discrete groups. Amenable actions of locally compact groups on C ??-algebras are investigated in the same way, and amenability of the action is related to nuclearity of the corresponding crossed product. A survey is given to show that this notion of amenable action for C ??-algebras satisfies a number of expected properties. A notion of inner amenability for actions of locally compact groups is introduced, and a number of applications are given in the form of averaging arguments, relating approximation properties of crossed product von Neumann algebras to properties of the components of the underlying w ??-dynamical system. We use these results to answer a recent question of Buss-Echterhoff-Willett.

We establish several new characterizations of amenable W ∗ - and C ∗ -dynamical systems over arbitrary locally compact groups. In the W ∗ -setting we show that amenability is equivalent to (1) a Reiter property and (2) the existence of a certain net of completely positive Herz-Schur multipliers of (M,G,α) converging point weak* to the identity of G ⋉ ¯ M . In the C ∗ -setting, we prove that amenability of (A,G,α) is equivalent to an analogous Herz-Schur multiplier approximation of the identity of the reduced crossed product G⋉A , as well as a particular case of the positive weak approximation property of Bédos and Conti (generalized the locally compact setting). When Z( A ∗∗ )=Z(A ) ∗∗ , it follows that amenability is equivalent to the 1-positive approximation property of Exel and Ng. In particular, when A= C 0 (X) is commutative, amenability of ( C 0 (X),G,α) coincides with topological amenability the G -space (G,X) . Our results answer 2 open questions from the literature; one of Anantharaman--Delaroche, and one from recent work of Buss--Echterhoff--Willett.

Let E be a countable directed graph that is amplified in the sense that whenever there is an edge from v to w , there are infinitely many edges from v to w . We show that E can be recovered from C ∗ (E) together with its canonical gauge-action, and also from L K (E) together with its canonical grading.

We define a representation of the unitary group U(n) by metaplectic operators acting on L 2 ( R n ) and consider the operator algebra generated by the operators of the representation and pseudodifferential operators of Shubin class. Under suitable conditions, we prove the Fredholm property for elements in this algebra and obtain an index formula.

In the setting of modern mathematical logic and model theory, classification theory has been one of the landmark achievements of the field. Likewise, the classification of UHF-algebras and AF-algebras were substantial contributions to the field of operator algebra theory. These seemingly disparate topics of study in mathematics, model theory and operator algebras, have in recent years become closely related in many respects. I here attempt to bridge the gap between these two topics by discussing how operator algebraic classifications may be understood in terms of model-theoretic classification theory. This introductory article assumes basic familiarity with model theory and linear operator, but higher-level concepts are introduced when necessary. The focus of this introduction is conceptual and informal, and as such, many results are stated without proof, but relevant sources are cited for completeness. The reader should take this not as a detailed review, but rather as an overview of a general narrative thread connecting these two branches of modern mathematics.

We show that the set of projections in an operator system can be detected using only the abstract data of the operator system. Specifically, we show that if p is a positive contraction in an operator system V which satisfies certain order-theoretic conditions, then there exists a complete order embedding of V into B(H) mapping p to a projection operator. Moreover, every abstract projection in an operator system V is an honest projection in the C*-envelope of V . Using this characterization, we provide an abstract characterization for operator systems spanned by two commuting families of projection-valued measures and discuss applications in quantum information theory.