Real Rank Research Articles

Rings and C*-algebras generated by commutators

We show that a unital ring is generated by its commutators as an ideal if and only if there exists a natural number N such that every element is a sum of N products of pairs of commutators. We show that one can take N≤2 for matrix rings, and that one may choose N≤3 for rings that contain a direct sum of matrix rings – this in particular applies to C*-algebras that are properly infinite or have real rank zero. For Jiang-Su-stable C*-algebras, we show that N≤6 can be arranged.For arbitrary rings, we show that every element in the commutator ideal admits a power that is a sum of products of commutators. Using that a C*-algebra cannot be a radical extension over a proper ideal, we deduce that a C*-algebra is generated by its commutators as a not necessarily closed ideal if and only if every element is a finite sum of products of pairs of commutators.

On unital absorbing extensions of C*-algebras of stable rank one and real rank zero

Suppose that B B is a separable stable C ∗ C^* -algebra with real rank zero, stable rank one and ( K 0 ( B ) , K 0 + ( B ) ) (\mathrm {K}_0(B), \mathrm {K}_0^+(B)) is weakly unperforated in the sense of Elliott [Internat. J. Math. 1 (1990), no. 4, pp. 361–380]. Let A A be a unital simple separable nuclear C ∗ \mathrm {C}^* -algebra. We show that B B has the corona factorization property and any unital extension of A A by B B is absorbing.

Total Cuntz semigroup, extension, and Elliott Conjecture with real rank zero

AbstractIn this paper, we exhibit two unital, separable, nuclear ‐algebras of stable rank one and real rank zero with the same ordered scaled total K‐theory, but they are not isomorphic with each other, which forms a counterexample to the Elliott Classification Conjecture for real rank zero setting. Thus, we introduce an additional normal condition and give a classification result in terms of the total K‐theory. For the general setting, with a new invariant, the total Cuntz semigroup [2], we classify a large class of ‐algebras obtained from extensions. The total Cuntz semigroup, which distinguishes the algebras of our counterexample, could possibly classify all the ‐algebras of stable rank one and real rank zero.

C*-algebras generated by representations of virtually nilpotent groups

We show that a C*-algebra generated by an irreducible representation of a finitely generated virtually nilpotent group satisfies the universal coefficient theorem and has real rank 0. This combines with previous joint work with Gillaspy and McKenney to show these C*-algebras are classified by their Elliott invariants. When we further assume the group is nilpotent we build explicit Cartan subalgebras that are closely related to the group and representation, although the Cartan subalgebras are generally not C*-diagonals.

Nowhere scattered multiplier algebras

We study sufficient conditions under which a nowhere scattered $\mathrm {C}^*$ -algebra $A$ has a nowhere scattered multiplier algebra $\mathcal {M}(A)$ , that is, we study when $\mathcal {M}(A)$ has no nonzero, elementary ideal-quotients. In particular, we prove that a $\sigma$ -unital $\mathrm {C}^*$ -algebra $A$ of (i) finite nuclear dimension, or (ii) real rank zero, or (iii) stable rank one with $k$ -comparison, is nowhere scattered if and only if $\mathcal {M}(A)$ is.

Sums of squares, Hankel index and almost real rank

Sums of squares, Hankel index and almost real rank

Real rank of extensions of $C^*$-algebras

Real rank of extensions of $C^*$-algebras

Braid group action and quasi-split affine 𝚤quantum groups I

This is the first of our papers on quasi-split affine quantum symmetric pairs ( U ~ ( g ^ ) , U ~ ı ) \big (\widetilde {\mathbf U}(\widehat {\mathfrak g}), \widetilde {{\mathbf U}}^\imath \big ) , focusing on the real rank one case, i.e., g = s l 3 \mathfrak g = \mathfrak {sl}_3 equipped with a diagram involution. We construct explicitly a relative braid group action of type A 2 ( 2 ) A_2^{(2)} on the affine ı \imath quantum group U ~ ı \widetilde {{\mathbf U}}^\imath . Real and imaginary root vectors for U ~ ı \widetilde {{\mathbf U}}^\imath are constructed, and a Drinfeld type presentation of U ~ ı \widetilde {{\mathbf U}}^\imath is then established. This provides a new basic ingredient for the Drinfeld type presentation of higher rank quasi-split affine ı \imath quantum groups in the sequels.

When amenable groups have real rank zero C∗-algebras

We investigate when discrete, amenable groups have [Formula: see text]-algebras of real rank zero. While it is known that this happens when the group is locally finite, the converse is an open problem. We show that if [Formula: see text] has real rank zero, then all normal subgroups of [Formula: see text] that are elementary amenable and have finite Hirsch length must be locally finite.

Sharp Hardy–Sobolev–Maz’ya, Adams and Hardy–Adams inequalities on quaternionic hyperbolic spaces and on the Cayley hyperbolic plane

The main purpose of this paper is to establish the higher order Poincaré– Sobolev and Hardy–Sobolev–Maz’ya inequalities on quaternionic hyperbolic spaces and on the Cayley hyperbolic plane using the Helgason–Fourier analysis on symmetric spaces. A crucial part of our work is to establish appropriate factorization theorems on these spaces, which can be of independent interest. To this end, we need to identify and introduce the “quaternionic Geller operators” and the “octonionic Geller operators”, which have been absent on these spaces. Combining the factorization theorems and the Geller type operators with the Helgason–Fourier analysis on symmetric spaces, some precise estimates for the heat and the Bessel–Green– Riesz kernels, and the Kunze–Stein phenomenon for connected real simple groups of real rank one with finite center, we succeed to establish the higher order Poincaré– Sobolev and Hardy–Sobolev–Maz’ya inequalities on quaternionic hyperbolic spaces and on the Cayley hyperbolic plane. The kernel estimates required to prove these inequalities are also sufficient to establish the Adams and Hardy–Adams inequalities on these spaces. This paper, together with our earlier works on real and complex hyperbolic spaces, completes our study of the factorization theorems, higher order Poincaré–Sobolev, Hardy–Sobolev–Maz’ya, Adams and Hardy–Adams inequalities on all rank one symmetric spaces of noncompact type.

On the dimension drop conjecture for diagonal flows on the space of lattices

Let X=G/Γ, where G is a Lie group and Γ is a lattice in G, let U be an open subset of X, and let {gt} be a one-parameter subgroup of G. Consider the set of points in X whose gt-orbit misses U; it has measure zero if the flow is ergodic. It has been conjectured that this set has Hausdorff dimension strictly smaller than the dimension of X. This conjecture is proved when X is compact or when G is a simple Lie group of real rank 1. In this paper we prove this conjecture for the case G=SLm+n(R), Γ=SLm+n(Z) and gt=diag(ent,…,ent,e−mt,…,e−mt), in fact providing an effective estimate for the codimension. The proof uses exponential mixing of the flow together with the method of integral inequalities for height functions on SLm+n(R)/SLm+n(Z). We also discuss an application to the problem of improving Dirichlet's theorem in simultaneous Diophantine approximation.

Symmetries of simple $A\mathbb{T}$-algebras

Let A be a unital simple A\mathbb{T} -algebra of real rank zero. Given an order two automorphism h: K_1(A)\to K_1(A) , we show that there is an order two automorphism \alpha : A\to A such that \alpha_{*0}=\mathrm {id} , \alpha_{*1}=h and the action of \mathbb{Z}_2 generated by \alpha has the tracial Rokhlin property. Consequently, C^*(A,\mathbb{Z}_2,\alpha) is a simple unital AH-algebra with no dimension growth, and with tracial rank zero. Thus our main result can be considered as the \mathbb{Z}_2 -action analogue of the Lin-Osaka theorem. As a consequence, a positive answer to a lifting problem of Blackadar is also given for certain split case.

Permanence of real rank zero from a centrally large subalgebra to a C*-algebra

Let A A be an infinite dimensional simple unital C*-algebra and B B be a centrally large subalgebra of A A . Archey and Phillips [J. Operator Theory 83 (2020), pp. 353–389] showed that A A has real rank zero when B B has real rank zero and stable rank one. In this paper, we have removed the restriction of B B has stable rank one. Using matrix decomposition by some suitable projections, we have shown that A A has real rank zero if B B has real rank zero.

Metrics on trace spaces

This article continues the investigation of the tracial geometry of classifiable C*-algebras that have real rank zero and stable rank one. Using the language of optimal transport, we describe several situations in which the distance between unitary orbits of ⁎-homomorphisms into such algebras can be computed in terms of tracial data. The domains we consider are certain (noncommutative) CW complexes, and the measurement is relative to a family of self-adjoint elements that are in a suitable sense tracially Lipschitz. As another application of the utility of this Lipschitz structure, we show how such elements can be repurposed to witness statistical features of endomorphisms in the classifiable category, in particular the tracial version of the (almost-sure) central limit theorem.

Stability of 𝚤canonical bases of irreducible finite type of real rank one

It has been known since their birth in Bao and Wang’s work that the ı \imath canonical bases of ı \imath quantum groups are not stable in general. In the author’s previous work, the stability of ı \imath canonical bases of certain quasi-split types turned out to be closely related to the theory of ı \imath crystals. In this paper, we prove the stability of ı \imath canonical bases of irreducible finite type of real rank 1 1 , for which the theory of ı \imath crystals has not been developed, by means of global and local crystal bases.

On the Range of Certain ASH Algebras of Real Rank Zero

On the Range of Certain ASH Algebras of Real Rank Zero

The Global Glimm Property

It is known that a C ∗ C^* -algebra with the Global Glimm Property is nowhere scattered (it has no elementary ideal-quotients), and the Global Glimm Problem asks if the converse holds. We provide a new approach to this long-standing problem by showing that a C ∗ C^* -algebra has the Global Glimm Property if and only if it is nowhere scattered and its Cuntz semigroup is ideal-filtered (the Cuntz classes generating a given ideal are downward directed) and has property (V) (a weak form of being sup-semilattice ordered). We show that ideal-filteredness and property (V) are automatic for C ∗ C^* -algebras that have stable rank one or real rank zero, thereby recovering the solutions to the Global Glimm Problem in these cases. We also use our approach to solve the Global Glimm Problem for new classes of C ∗ C^* -algebras.

Lower bounds on the rank and symmetric rank of real tensors

We lower bound the rank of a tensor by a linear combination of the ranks of three of its unfoldings, using Sylvester's rank inequality. In a similar way, we lower bound the symmetric rank by a linear combination of the symmetric ranks of three unfoldings. Lower bounds on the rank and symmetric rank of tensors are important for finding counterexamples to Comon's conjecture. A real counterexample to Comon's conjecture is a tensor whose real rank and real symmetric rank differ. Previously, only one real counterexample was known, constructed in a paper of Shitov. We divide the construction into three steps. The first step involves linear spaces of binary tensors. The second step considers a linear space of larger decomposable tensors. The third step is to verify a conjecture that lower bounds the symmetric rank, on a tensor of interest. We use the construction to build an order six real tensor whose real rank and real symmetric rank differ.

A total Cuntz semigroup for C*-algebras of stable rank one

In this paper, we show that for unital, separable C*-algebras of stable rank one and real rank zero, the unitary Cuntz semigroup functor and the functor K⁎ are natural equivalent. Then we introduce a refinement of the unitary Cuntz semigroup, say the total Cuntz semigroup, which is a new invariant for separable C⁎-algebras of stable rank one, is a well-defined continuous functor from the category of C⁎-algebras of stable rank one to the category Cu_. We prove that this new functor and the functor K_ are naturally equivalent for unital, separable, K-pure C⁎-algebras. Therefore, the total Cuntz semigroup is a complete invariant for a large class of C⁎-algebras of real rank zero.

Projective and conformal closed manifolds with a higher-rank lattice action

We prove global results about actions of cocompact lattices in higher-rank simple Lie groups on closed manifolds endowed with either a projective class of connections or a conformal class of pseudo-Riemannian metrics of signature (p, q), with \(\min (p,q) \ge 2\). Building on a recent article [17], provided that such a structure is locally equivalent to its model \({\textbf{X}}\), the main question treated here is the completeness of the associated \((G,{\textbf{X}})\)-structure. Because of the similarities between the model spaces of projective geometry and non-Lorentzian conformal geometry, a number of arguments apply in both contexts. We therefore present the proofs in parallel. The conclusion is that in both cases, when the real rank is maximal, the manifold is globally equivalent to either the model space \({\textbf{X}}\) or its double cover.