# Future Targets in the Classification Program for Amenable C*-Algebras (17w5127)

Arriving in Banff, Alberta Sunday, September 3 and departing Friday September 8, 2017

## Organizers

Zhuang Niu (University of Wyoming)

Aaron Tikuisis (University of Aberdeen)

George Elliott (University of Toronto)

## Objectives

The workshop will assess the major problems remaining in the classification program and its connections to other areas:

(i) The Toms--Winter conjecture, says that among Elliott algebras, the properties of finite nuclear dimension, Jiang--Su stability, and strict comparison of positive elements coincide. These three eclectic properties have varied pertinence in different settings, or constructions. Powerful structural consequences ensue for \(\text{C}^*\)-algebras known to have all three of these properties.

The diversity of the Toms--Winter properties makes this conjecture analogous to Connes's characterization of injectivity for von Neumann algebras, or the multifacetted characterization of group amenability. Many results provide partial confirmation (e.g., [1,ENST,MS,SWW,W]), often involving deep connections to von Neumann algebra theory at many levels. It is expected that further progress will require more interaction between \(\text{C}^*\)-algebra and von Neumann algebra research.

Besides the significant problem of settling the Toms--Winter conjecture, it is important to characterize when a \(\text C^*\)-algebra construction (e.g., from a minimal dynamical system) satisfies the conditions of the conjecture. There has been some progress: Giol and Kerr showed that certain subshift \(\text{C}^*\)-algebras don't satisfy any of the Toms--Winter properties, whereas various dimension-related hypotheses (e.g., involving mean dimension or Rokhlin dimension) imply certain Toms--Winter properties. A related problem is to understand the relationship between the nuclear dimension of a group \(\text{C}^*\)-algebra and the group structure. (Note that recent results from classification theory have solved the Rosenberg conjecture, that quasidiagonality of the reduced group \(\text{C}^*\)-algebra is equivalent to amenability of the group [TWW].)

(ii) The powerful classification result achieved in the last year applies to many naturally-occurring \(\text{C}^*\)-algebras, such as minimal crossed products of spaces of finite dimension (or with mean dimension zero). Among other things, it says that such \(\text{C}^*\)-algebras have nice models (limits of subhomogeneous \(\text{C}^*\)-algebras of low topological dimension). Can this structure be seen in the construction itself (as with irrational rotation algebras [EE])? Can one use the nice \(\text{C}^*\)-models to derive properties (such as Rokhlin-esque lemmas) about the underlying input object? A related question is: can one describe in dynamical terms when two systems give rise to isomorphic \(\text{C}^*\)-algebras?

Further questions deal with the range of \(\text{C}^*\)-algebras that can arise by dynamical constructions. It was recently shown that the important Jiang--Su algebra $\mathcal Z$ has a dynamical construction [DPS]. Which other \(\text{C}^*\)-algebras have dynamical models?

(iii) The Universal Coefficient Theorem (UCT) is the name given by Rosenberg and Schochet to a property of \(\text{C}^*\)-algebras which is crucial to \(\text{C}^*\)-algebra classification, as it connects K-theory to the indispensible technical tools provided by Kasparov's bivariant KK-theory. It has also been linked recently to quasidiagonality [TWW]. An question which has been open for over thirty years is: does the UCT hold for all separable amenable \(\text{C}^*\)-algebras?

This is an immensely difficult problem, although there are many opportunities for progress, adding to the \(\text{C}^*\)-algebras for which we know the UCT holds. Which \(\text{C}^*\)-crossed products satisfy the UCT? Do strongly self-absorbing \(\text{C}^*\)-algebras satisfy the UCT?

(iv) Building on work of Villadsen, R\o rdam and Toms surprisingly produced pairs of non-isomorphic Elliott algebras agreeing on traditional K-theoretic invariants [R1,T]. Their non-isomorphism can be seen from their Cuntz semigroups, a finer invariant. There are many known variations on these \(\text{C}^*\)-algebras, but they are understood only at a superficial level. A key question is whether the Cuntz semigroup is sufficiently sensitive to classify a meaningful class of these \(\text{C}^*\)-algebras (perhaps all Elliott algebras with trivial K$_1$-group?).

The distinguishing feature of these surprising examples is that they don't satisfy any conditions in the Toms--Winter conjecture (in short, they are ``poorly behaved''). There is an equally limited understanding of the full range of \(\text{C}^*\)-algebras that are poorly behaved in this sense, and their properties. While tensor products of poorly behaved \(\text{C}^*\)-algebras often become well behaved (satisfy the Toms--Winter conditions) (e.g., the \(\text{C}^*\)-algebras of minimal dynamical systems), the following is open: is there a poorly behaved Elliott algebra $A$ such that $A \cong A \otimes A$? No strongly self-absorbing \(\text{C}^*\)-algebras are poorly behaved, but is there a poorly behaved \(\text{C}^*\)-algebra with approximately inner flip? Is there a poorly behaved Elliott algebra with the same K-theory as $\mathbb C$? Is there a poorly behaved Elliott algebra with real rank zero? \vspace*{2mm}

The participant list (of 50 -- one must overbook) is chosen to include a good mix of researchers, from a number of talented Ph.D.\ students and postdocs, to established leaders in the field. Including organizers, the list includes seven ICM speakers, four Fellows of the AMS, and present or former editors of Crelle's Journal, Trans.\ Amer.\ Math.\ Soc., J.\ Funct.\ Anal., and J.\ Operator Theory (among other journals).

i) The Toms--Winter conjecture.

ii) What does classifiability of constructed \(\text{C}^*\)-algebras tell us about the underlying mathematical objects from which they were constructed?

iii) Which \(\text{C}^*\)-algebras satisfy the Universal Coefficient Theorem? (All amenable \(\text{C}^*\)-algebras?)

iv) Poorly behaved (high topological dimensional) Elliott algebras.

(i) The Toms--Winter conjecture, says that among Elliott algebras, the properties of finite nuclear dimension, Jiang--Su stability, and strict comparison of positive elements coincide. These three eclectic properties have varied pertinence in different settings, or constructions. Powerful structural consequences ensue for \(\text{C}^*\)-algebras known to have all three of these properties.

The diversity of the Toms--Winter properties makes this conjecture analogous to Connes's characterization of injectivity for von Neumann algebras, or the multifacetted characterization of group amenability. Many results provide partial confirmation (e.g., [1,ENST,MS,SWW,W]), often involving deep connections to von Neumann algebra theory at many levels. It is expected that further progress will require more interaction between \(\text{C}^*\)-algebra and von Neumann algebra research.

Besides the significant problem of settling the Toms--Winter conjecture, it is important to characterize when a \(\text C^*\)-algebra construction (e.g., from a minimal dynamical system) satisfies the conditions of the conjecture. There has been some progress: Giol and Kerr showed that certain subshift \(\text{C}^*\)-algebras don't satisfy any of the Toms--Winter properties, whereas various dimension-related hypotheses (e.g., involving mean dimension or Rokhlin dimension) imply certain Toms--Winter properties. A related problem is to understand the relationship between the nuclear dimension of a group \(\text{C}^*\)-algebra and the group structure. (Note that recent results from classification theory have solved the Rosenberg conjecture, that quasidiagonality of the reduced group \(\text{C}^*\)-algebra is equivalent to amenability of the group [TWW].)

(ii) The powerful classification result achieved in the last year applies to many naturally-occurring \(\text{C}^*\)-algebras, such as minimal crossed products of spaces of finite dimension (or with mean dimension zero). Among other things, it says that such \(\text{C}^*\)-algebras have nice models (limits of subhomogeneous \(\text{C}^*\)-algebras of low topological dimension). Can this structure be seen in the construction itself (as with irrational rotation algebras [EE])? Can one use the nice \(\text{C}^*\)-models to derive properties (such as Rokhlin-esque lemmas) about the underlying input object? A related question is: can one describe in dynamical terms when two systems give rise to isomorphic \(\text{C}^*\)-algebras?

Further questions deal with the range of \(\text{C}^*\)-algebras that can arise by dynamical constructions. It was recently shown that the important Jiang--Su algebra $\mathcal Z$ has a dynamical construction [DPS]. Which other \(\text{C}^*\)-algebras have dynamical models?

(iii) The Universal Coefficient Theorem (UCT) is the name given by Rosenberg and Schochet to a property of \(\text{C}^*\)-algebras which is crucial to \(\text{C}^*\)-algebra classification, as it connects K-theory to the indispensible technical tools provided by Kasparov's bivariant KK-theory. It has also been linked recently to quasidiagonality [TWW]. An question which has been open for over thirty years is: does the UCT hold for all separable amenable \(\text{C}^*\)-algebras?

This is an immensely difficult problem, although there are many opportunities for progress, adding to the \(\text{C}^*\)-algebras for which we know the UCT holds. Which \(\text{C}^*\)-crossed products satisfy the UCT? Do strongly self-absorbing \(\text{C}^*\)-algebras satisfy the UCT?

(iv) Building on work of Villadsen, R\o rdam and Toms surprisingly produced pairs of non-isomorphic Elliott algebras agreeing on traditional K-theoretic invariants [R1,T]. Their non-isomorphism can be seen from their Cuntz semigroups, a finer invariant. There are many known variations on these \(\text{C}^*\)-algebras, but they are understood only at a superficial level. A key question is whether the Cuntz semigroup is sufficiently sensitive to classify a meaningful class of these \(\text{C}^*\)-algebras (perhaps all Elliott algebras with trivial K$_1$-group?).

The distinguishing feature of these surprising examples is that they don't satisfy any conditions in the Toms--Winter conjecture (in short, they are ``poorly behaved''). There is an equally limited understanding of the full range of \(\text{C}^*\)-algebras that are poorly behaved in this sense, and their properties. While tensor products of poorly behaved \(\text{C}^*\)-algebras often become well behaved (satisfy the Toms--Winter conditions) (e.g., the \(\text{C}^*\)-algebras of minimal dynamical systems), the following is open: is there a poorly behaved Elliott algebra $A$ such that $A \cong A \otimes A$? No strongly self-absorbing \(\text{C}^*\)-algebras are poorly behaved, but is there a poorly behaved \(\text{C}^*\)-algebra with approximately inner flip? Is there a poorly behaved Elliott algebra with the same K-theory as $\mathbb C$? Is there a poorly behaved Elliott algebra with real rank zero? \vspace*{2mm}

The participant list (of 50 -- one must overbook) is chosen to include a good mix of researchers, from a number of talented Ph.D.\ students and postdocs, to established leaders in the field. Including organizers, the list includes seven ICM speakers, four Fellows of the AMS, and present or former editors of Crelle's Journal, Trans.\ Amer.\ Math.\ Soc., J.\ Funct.\ Anal., and J.\ Operator Theory (among other journals).

#### Bibliography

- [BBSTWW] J. Bosa, N. Brown, Y. Sato, A. Tikuisis, S. White, and W. Winter. Covering dimension of \(\text{C}^*\)-algebras and 2-coloured classification. arXiv:1506.03974.
- [DPS] R. Deeley, I. Putnam, and K. Strung. Constructing minimal homeomorphisms on point-like spaces and a dynamical presentation of the Jiang--Su algebra. arXiv:1503.03800.
- [ENST] G.A. Elliott, Z. Niu, L. Santiago, and A. Tikuisis. Decomposition rank of approximately subhomogeneous \(\text{C}^*\)-algebras. arXiv:1505.06100.
- [E] G.A. Elliott. The classification problem for amenable \(\text{C}^*\)-algebras. In
*Proceedings of the International Congress of Mathematicians, Vol.\ 1, 2 (Zürich, 1994)}, pages 922--932. Birkhäuser, Basel, 1995.* - [EE] G.A. Elliott and D.E. Evans. The structure of the irrational rotation {$\text{C}^*$}-algebra.
*Ann. of Math. (2)*, 138(3):477--501, 1993. - [EN] G.A. Elliott, and Z. Niu. On the classification of simple amenable \(\text{C}^*\)-algebras with finite decomposition rank. arXiv:1507.07876.
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*Invent. Math.*, 183(2):385--450, 2011. - [LN] H. Lin and Z. Niu. Lifting {$KK$}-elements, asymptotic unitary equivalence and classification of simple \(\text{C}^*\)-algebras.
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*Invent. Math.*To appear. arXiv:1403.0747. - [TWW] A. Tikuisis, S. White, and W. Winter. Quasidiagonality of nuclear \(\text{C}^*\)-algebras. arXiv:1509.08318.
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*Ann. of Math. (2)*, 167(3):1029--1044, 2008. - [V] J. Villadsen. On the stable rank of simple \(\text{C}^*\)-algebras.
*J. Amer. Math. Soc.*, 12(4):1091--1102, 1999. - [W] W. Winter. Nuclear dimension and {$\mathcal{Z}$}-stability of pure \(\text{C}^*\)-algebras.
*Invent. Math.*, 187(2):259--342, 2012.