Mathematics

Let p and q be multiplicatively independent integers. We show that the complex group ring of Z[ 1 pq ]⋊ Z 2 admits a unique C ∗ -norm. The proof uses a characterization, due to Furstenberg, of closed ×p− and ×q− invariant subsets of T .

We introduce a new invariant for C*-algebras of stable rank one that merges the Cuntz semigroup information together with the K_1-group information. This semigroup, termed the Cu_1-semigroup, is constructed as equivalence classes of pairs consisting of a positive element in the stabilization of the given C*-algebra together with a unitary element of the unitization of the hereditary subalgebra generated by the given positive element. We show that the Cu_1-semigroup is a well-defined continuous functor from the category of C*-algebras of stable rank one to a suitable codomain category that we write Cu^\sim. Furthermore, we compute the Cu_1-semigroup of some specific classes of C*-algebras. Finally, in the course of our investigation, we show that we can recover functorially Cu, K_1 and K_*:=K_0\oplus K_1 from Cu_1.

We introduce an equivariant version of the weak expectation property (WEP) at the level of operator modules over completely contractive Banach algebras A . We prove a number of general results---for example, a characterization of the A -WEP in terms of an appropriate A -injective envelope, and also a characterization of those A for which A -WEP implies WEP. In the case of A= L 1 (G) , we recover the G -WEP for G - C ∗ -algebras in recent work of Buss--Echterhoff--Willett. When A=A(G) , we obtain a dual notion for operator modules over the Fourier algebra. These dual notions are related in the setting of dynamical systems, where we show that a W ∗ -dynamical system (M,G,α) with M injective is amenable if and only if M is L 1 (G) -injective if and only if the crossed product G ⋉ ¯ M is A(G) -injective. Analogously, we show that a C ∗ -dynamical system (A,G,α) with A nuclear and G exact is amenable if and only if A has the L 1 (G) -WEP if and only if the reduced crossed product G⋉A has the A(G) -WEP.

We give a procedure to compute the rational homotopy groups of the group of quasi-unitaries of an AF-algebra. As an application, we show that an AF-algebra is K-stable if and only if it is rationally K-stable.

We study simply connected Lie groups G for which the hull-kernel topology of the primitive ideal space Prim(G) of the group C ∗ -algebra C ∗ (G) is T 1 , that is, the finite subsets of Prim(G) are closed. Thus, we prove that C ∗ (G) is AF-embeddable. To this end, we show that if G is solvable and its action on the centre of [G,G] has at least one imaginary weight, then Prim(G) has no nonempty quasi-compact open subsets. We prove in addition that connected locally compact groups with T 1 ideal spaces are strongly quasi-diagonal.

The problem of additivity of the Minimum Output Entropy is of fundamental importance in Quantum Information Theory (QIT). It was solved by Hastings in the one-shot case, by exhibiting a pair of random quantum channels. However, the initial motivation was arguably to understand regularized quantities and there was so far no way to solve additivity questions in the regularized case. The purpose of this paper is to give a solution to this problem. Specifically, we exhibit a pair of quantum channels which unearths additivity violation of the regularized minimum output entropy. Unlike previously known results in the one-shot case, our construction is non-random, infinite dimensional and in the commuting-operator setup. The commuting-operator setup is equivalent to the tensor-product setup in the finite dimensional case for this problem, but their difference in infinite dimensional setting has attracted substantial attention and legitimacy recently in QIT with the celebrated resolutions of Tsirelson's and Connes embedding problem. Likewise, it is not clear that our approach works in the finite dimensional setup. Our strategy of proof relies on developing a variant of the Haagerup inequality optimized for a product of free groups.

Let M⊂B(H) be a von Neumann algebra acting on the Hilbert space H . We prove that M is finite if and only if, for every x∈M and for all vectors ξ,η∈H , the coefficient function u↦⟨ux u ∗ ξ|η⟩ is weakly almost periodic on the topological group U M of unitaries in M (equipped with the weak or strong operator topology). The main device is the unique invariant mean on the C ∗ -algebra WAP( U M ) of weakly almost periodic functions on U M . Next, we prove that every coefficient function u↦⟨ux u ∗ ξ|η⟩ is almost periodic if and only if M is a direct sum of a diffuse, abelian von Neumann algebra and finite-dimensional factors. Incidentally, we prove that if M is a diffuse von Neumann algebra, then its unitary group is minimally almost periodic.

In this paper, we introduce two new approximation properties for étale groupoids, almost elementariness and (ubiquitous) fiberwise amenability, inspired by Matui's and Kerr's notions of almost finiteness. In fact, we show that, in their respective scopes of applicability, both notions of almost finiteness are equivalent to the conjunction of our two properties. These new properties stem from viewing étale groupoids as coarse geometric objects in the spirit of geometric group theory. Fiberwise amenability is a coarse geometric property of étale groupoids that is closely related to the existence of invariant measures on unit spaces and corresponds to the amenability of the acting group in a transformation groupoid. Almost elementariness may be viewed as a better dynamical analogue of the regularity properties of C*-algebras than almost finiteness, since, unlike the latter, the former may also be applied to the purely infinite case. To support this analogy, we show almost elementary minimal groupoids give rise to tracially Z-stable reduced groupoid C*-algebras. In particular, the reduced C*-algebras of second countable amenable almost finite groupoids in Matui's sense are Z-stable.

We show that Cantor minimal Z⋊ Z 2 -systems and essentially free amenable odometers are almost finite. We also compute the homology groups of Cantor minimal Z⋊ Z 2 -systems and show that the associated transformation groupoids satisfy the HK conjecture if and only if the action is free.

Inspired by Kerr's work on topological dynamics, we define tracial Z -stability for sub- C ∗ -algebras. We prove that for a countable discrete amenable group G acting freely and minimally on a compact metrizable space X , tracial Z -stability for the sub- C ∗ -algebra (C(X)⊆C(X)⋊G) implies that the action has dynamical comparison. Consequently, tracial Z -stability is equivalent to almost finiteness of the action, provided that the action has the small boundary property.