The nuclear dimension of C*-algebras

By W. Winter, J. Zacharias. 2009

289 citations recorded by the source corpus.

We introduce the nuclear dimension of a C*-algebra; this is a noncommutative version of topological covering dimension based on a modification of the earlier concept of decomposition rank. Our notion behaves well with respect to inductive limits, tensor products, hereditary subalgebras (hence ideals), quotients, and even extensions. It can be computed for many examples; in particular, it is finite for all UCT Kirchberg algebras. In fact, all classes of nuclear C*-algebras which have so far been successfully classified consist of examples with finite nuclear dimension, and it turns out that finite nuclear dimension implies many properties relevant for the classification program. Surprisingly, the concept is also linked to coarse geometry, since for a discrete metric space of bounded geometry the nuclear dimension of the associated uniform Roe algebra is dominated by the asymptotic dimension of the underlying space.

Subfields: Algebra and Number Theory, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/37ac8ec7a22e201e5f407a831d48d6bcd989f665

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Citations in the corpus

The Rokhlin dimension of topological ℤm‐actions (2013, background): “zero up to δ, if for all a,b∈ Awith ab= 0 we have kψ(a)ψ(b)k ≤ δkakkbk. ψhas order zero if δ= 0. Order zero maps in particular appear in the deﬁnition of nuclear dimension and decomposition rank (see [11, 22]), which are important regularity properties for the classiﬁcation theory of nuclear C∗-alge”
UCT-Kirchberg algebras have nuclear dimension one (2014, unknown): “3(i) of [26] implies that each Bn⊕Cn has nuclear dimension 1.”
Rokhlin Dimension and C*-Dynamics (2012, methodology): “Covering dimension for topological spaces was generalized to the context of C∗algebras in [14] and [34], in which the related concepts of decomposition rank and nuclear dimension were introduced.”
Equivariant Kirchberg–Phillips-Type Absorption for Amenable Group Actions (2016, background): “Indeed, for the Jiang–Su algebra Z , which is the initial object in the category of strongly self-absorbing C∗-algebras [81], the most general analogue of the McDuff property is given by Z-stability and plays a major role in the Elliott program through the Toms–Winter regularity conjecture [15,82,83”
Nuclear dimension and -stability of pure C ∗ -algebras (2010, methodology): “The conjecture of Toms and the author was formulated in the not necessarily finite case in [43], replacing decomposition rank by nuclear dimension.”
Completely positive maps of order zero (2009, background): “They will serve a similar purpose in [14].”
On the classification of simple amenable C*-algebras with finite decomposition rank (2015, unknown): “) = (G0,(G0)+,u,G1,T,r). Deﬁnition 2.9. Let Aand Bbe C*-algebras. Recall ([19]) that a completely positive map ϕ: A→ Bis said to have order zero if ab= 0 ⇒ ϕ(a)ϕ(b) = 0, a,b∈ A. Deﬁnition 2.10 ([19], [33]). A C*-algebra Ahas nuclear dimension at most n, if there exists a net (Fλ,ψλ,ϕλ), λ∈ Λ, such t”
Nuclear dimension and $$\mathcal Z$$Z-stability (2014, background): “ only give a rough idea of what these notions mean; we will discuss them in greater detail in Section 1. Nuclear dimension is the C∗-algebraic analogue of Lebesgue covering dimension as introduced in [52]. This should be thought of as a topological property, phrased in terms of approximating Avia no”
Dimension, comparison, and almost finiteness (2017, background): “First we recall the definition of nuclear dimension [55].”
Strict comparison and $$ \mathcal{Z} $$-absorption of nuclear C∗-algebras (2011, unknown): “Winter conjectured that the properties of strict comparison, finite nuclear dimension, and Z-absorption are equivalent for unital separable simple infinite-dimensional nuclear C-algebras (see [21, 26] for example).”
Rokhlin dimension for actions of residually finite groups (2014, background): “ C∗-algebras of ﬁnite nuclear dimension. This extends the key results in [13] and [47]. But ﬁrst, we need a slightly more ﬂexible characterization of nuclear dimension: Proposition 3.10 (compare with [54]). Let Abe a C∗-algebra and r∈ Na natural number. Then dimnuc(A) ≤ r, if and only if the followi”