COVERING DIMENSION AND QUASIDIAGONALITY

By Eberhard Kirchberg, W. Winter. 2002

192 citations recorded by the source corpus.

We introduce the decomposition rank, a notion of covering dimension for nuclear C*-algebras. The decomposition rank generalizes ordinary covering dimension and has nice permanence properties; in particular, it behaves well with respect to direct sums, quotients, inductive limits, unitization and quasidiagonal extensions. Moreover, it passes to hereditary subalgebras and is invariant under stabilization. It turns out that the decomposition rank can be finite only for strongly quasidiagonal C*-algebras and that it is closely related to the classification program.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/853ff40c6734c02a06ed74c9102a4b61b27c7d37

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In Algebra and Number Theory

1. W. Winter
2. G'abor Szab'o
3. Andrew S. Toms
4. David Kerr
5. G. Elliott
6. Yasuhiko Sato
7. Z. Niu
8. Stuart White

In Geometry and Topology

1. David Kerr
2. G'abor Szab'o
3. W. Winter
4. Andrew S. Toms
5. Yasuhiko Sato
6. Z. Niu
7. G. Elliott
8. Stuart White

In Mathematical Physics

1. W. Winter
2. G'abor Szab'o
3. David Kerr
4. Andrew S. Toms
5. Yasuhiko Sato
6. Stuart White
7. Aaron Tikuisis
8. Z. Niu

Citations in the corpus

• Nuclear dimension and $$\mathcal Z$$Z-stability (2014, background): “We should discuss this alongside the decomposition rank, the precursor to nuclear dimension from [20].”
• Localizing the Elliott conjecture at strongly self-absorbing C*-algebras (2007, background): “1.4 Examples: The only known examples of strongly self-absorbing C∗-algebras are UHF algebras ([19]) of infinite type (i.e., every prime number that occurs in the respective supernatural number occurs with infinite multiplicity – in this case, we also say the supernatural number is of infinite…”
• On the classification of simple amenable C*-algebras with finite decomposition rank (2015, background): “Furthermore, by [13], A ⊗ Q is stably finite and nuclear and so by [1] and [11], T(A) 6= Ø.”
• The classification of simple separable KK-contractible C*-algebras with finite nuclear dimension (2016, background): “φ(l) j ψ (l) j (1Ae)a−φ (l) j ψ (l) j (a)k = 0, l= 0,1,...,m, a∈ A. Note that φ(l) j : F (l) j → Ais of order zero, and the relation for an order zero map is weakly stable (see (P) and (P1) of 2.5 of [33]. On the other hand, if iis large enough, then σi◦φ (l) j satisfies the relation for order zero t”
• Regularity properties in the classification program for separable amenable C*-algebras (2007, background): “We refer the reader to [34] for details.”
• The Jiang–Su algebra revisited (2008, background): “Kirchberg and the second named author in [11].”
• Classifying crossed product C*-algebras (2013, background): “of certain touchstone C∗-algebras and order completeness of homological invariants, cf. [5] and [35]. When Andrew Toms and I began to study the tight connections between finite decomposition rank (cf. [14]), Z-stability (cf. [11, 24]) and strict comparison (cf. [23]), this was for very specific (and, ”
• Strongly self-absorbing C*-algebras (2005, methodology): “TER C∗-algebras as well as for D-stable C∗-algebras. In view of the classification results of [42], [43] and [44] it seems also natural to rephrase the question in terms of the decomposition rank (cf. [22] and [41]): Question: Does every separable, strongly self-absorbing limit D of recursive subhomo”
• Perturbations of nuclear C*-algebras (2009, background): “Indeed in [36] it is shown that a separable C∗-algebra with finite decomposition rank is necessarily quasidiagonal and so stably finite, while in [56] it is shown that the Cuntz algebras On (and all classifiable Kirchberg algebras) have finite nuclear dimension.”
• Decomposition rank and Z-stability (2008, background): “ γ. 10 WILHELM WINTER 1.10 We shall have use for the following consequence of Stinespring’s theorem, which is a standard tool to analyze completely positive approximations of nuclear C∗-algebras. See [22], Lemma 3.5, for a proof. Lemma: Let Aand Fbe C∗-algebras, b∈ Aa positive element of norm at mos”
• Minimal Dynamics and K-Theoretic Rigidity: Elliott’s Conjecture (2009, background): “Decomposition rank is lower semicontinuous with respect to inductive limits, whence AY has decomposition rank at most dim(X), too (see [10]).”
• Rokhlin dimension for compact group actions (2014, methodology): “5 in [22] to refine U to a d-decomposable covering V .”
• Decomposition rank of UHF-absorbing c* -algebras (2013, background): “The theorem above, together with the results in [13, 27, 33, 20], gives a partial affirmative answer to this conjecture in the following sense.”
• Dimension, comparison, and almost finiteness (2017, background): “X), as one can verify by a straightforward partition of unity argument using the formulation of covering dimension in term of the chromatic numbers of open covers (see the proof of Proposition 3.4 in [23]). Noting that fusqk = usαs(f)qk ∈Ak for every f ∈Υ and s∈Fsince ˆh i vanishes on Bi,0 for each ”