CLASSIFICATION AND $\mathcal{Z}$-STABILITY (Recent Developments on Classification Problems in Operator Algebras)

CLASSIFICATION

AND $Z$-STABILITY

NATHANIAL P. BROWN

ABSTRACT Inthis note wesurvey the currentstateof theclassification program withspecial_emPhasison

thereal rank zero case. Wetry to arguethata better understanding of $\mathcal{Z}$-stable algebras– those that

tensoriallyabsorbtheJiang-Su algebra-wouldleadtosignificantnewresults

1. INTRODUCTION

Inthis short survey Iwillgiveavery biasedandsubjective view of thecurrentstateof Elliott’sclassification

program. Iwill not prove any

new

results, but Ithink thespin may have somenovelty. Perhaps the main

pointIwish to make is that recent counterexamples ofAndrew Toms have convincinglydemonstratedthat, like it or not,

some

sort

_of

stabilizationis required in the

_{dassification}

program

Ever since his remarkable paper [15] giving a definitive counterexample to Elliott’s conjecture, Mikael

Rordam

has been suggesting thatwetry to classifyso-called $Z$-stablealgebras. That is, algebrassuch that

$A\otimes Z$ $\cong A$

where $Z$ is the Jiang-Su algebra$arrow$ the simple, unital, infinite dimensional, nuclear C’-algebra with Elliott

invariant isomorphic to that of

the

complex numbers (cf. [3]). In [2] it was shown that $A$ and $A\otimes Z$ have

isomorphic Elliott invariants if and only if$K_{0}(A)$ is weakly unperforated (i.e. $n$$\cdot$$x>0$ implies $x>0$ for

all $x$ $\in K_{0}(A))$. Hence, if$A$ and $A\otimes Z$ are classified by K-th ory and if$K_{0}(A)$ is weakly unperforated

then it necessarily followsthat$A$is$Z$-stable. Thus

Rordam’s

suggestion, thatwesimplyassumeZ-stability

and try to prove classification, is quite natural since $Z$-stability is a necessary condition for classification (in

the weakly unperforated case). However, many people (including myself) had psychological objections to assuming$Z$-stabilitysince

we

don’t know whenanalgebra satisfies this condition and so it feelsunnatural to

assumeit. Ontheotherhand,Andrew Toms has now

forced us

to face reality: One rreust

assume

Z-stability,

ingeneral, as there exists asimple, unital, AH algebra$A$with weakly unperforated$K_{0}(A)$ (evenstable rank one!) but which is not $Z$-stable (cf. [16],[17]). Wedon’t have to like it but the truth is the truthandthere

is no hope of classifying the non-\^i-stable AH algebras by their Elliott invariants (and $\mathcal{Z}$-stability is not

automaticeven for simpleAHalgebras with stablerank one and weakly unperforated$\mathrm{K}_{0}$-groups).

Given this unfortunate fact oflife, my perspective of the classification program has shifted and I want

to give my view of where

we

are and wherewe should go’. As I mentionedabove, this is a very subjective

survey andothers in the classification programmay disagree with the emphasis (or lack thereof) I put on

certain problems and results. My goal is not tostart arguments or offend but, rather, to highlight results

and directions which strike me as important. (Not surprisingly, the problems I think are most interesting also turn out tobethe most $” \mathrm{i}\mathrm{m}\mathrm{p}\mathrm{o}\mathrm{r}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}^{?}$’ ! ;-) Moreover,I willconcentrate mainlyonthereal rankzero case

so

there isno discussion of the existing ‘higherrank’ classification theorems.

Being lazy, I declare: all $\sigma$ algebrasznthis note are assumedto be unital, separable, simple and nuclear.

Also,Ineedto thankAndrew TomsandWilhelmWinterfor sharing preliminaryversions of theirworkwith

meand making helpful $\mathrm{s}\mathrm{u}\mathrm{g}\mathrm{g}\mathrm{e}\mathrm{s}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}/\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}$ tothe present

article.

2. KIRCHEERG-PHILLIPS CLASSIFICATION

To setthe stage,Iwantto quicklyrecallthetwomain steps in theKirchberg-Phillipsclassification theorem

(cf. [4], [5], [13]). Roughly speaking, the classification of purely infinite C’-algebras follows ffom two deep

results.

Theorem 21 ($\mathrm{O}_{\infty}$-StableImplies Classifiable). Assume$A\otimes \mathrm{O}_{\infty}\cong A_{\mathrm{J}}$ where

$O_{\infty}$ denotes the Cuntzalgebra

with infinitely many generators Then A is

_{classifiable.1}

This surveywaswrittenduringayear-long visit to TheUniversityofTokyo. Ithank them-especially Yasuyuki Kawahigashi

andNarutaka Ozawa-fortheir hospitality.

lRecallmy declarationonsimplicity, nuclearity,etc. Also,seethepapers referencedfor theprecisemeaningof‘classifiable’

NATHANIAL$\mathrm{P}$ BROWN

Theorem2.2 (PurelyInfiniteImplies$\mathrm{O}_{\infty}$-Stable). Assume A is purely

infinite.

Then stability$\iota s$automatic

-i.e. $A\otimes \mathrm{O}_{\infty}\cong A$.

Taken together these two theorems complete the purely infinite

case

ofElliott’s conjecture (modulo the UCT). I have chosen to separate them, however, to help motivate my current view of the stably finite case; we should concentrate onproving stably

_finite

analogues

_of

these two theorems.

I can’t pretend to know forsure what the ‘right’ analogues should be, but here are some very general versions. (I will give

more

tractablespecializations later.)

Question 2.3. Assume A is

_finite

and$Z$-stable Is A

classifiable?

As Iindicated, this problem has been posed by Mikael

Rordam

both privately and publicly (cf. [14]). It

isworth notingthat evenif

one

startswitha

non-Z-stable

algebra$A$it is easy to getawell behaved algebra

by replacing $A$ with $A\otimes Z$. (Jiang and Su showed that $Z$ $\otimes \mathcal{Z}$ $\cong Z$ $[3]$ and hence $Z$-stability is easy to

arrange.) I should also mention that

Rordam

has shown that finite plus $\mathcal{Z}$

-stable

implies stable rank one

and Blackadar’s fundamental comparison $\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{t}\mathrm{y}^{2}$. He can

even

characterize whensuch algebrashave real

rankzero [14] (Thesestatements depend

on

our

blanket assumption of nuclearityand simplicity ) Question 2.4 Find abstract conditions which imply Z-stabdity.

Since Tomshasshownthat$\mathcal{Z}$-stabilityisnotautomatic, there is

no

hope of achieving the ultimate analogue

of the ‘purely infiniteimplies $\mathrm{O}_{\infty}$-stable’ theorem above. Hencewe have to settle for weaker versionsand I

will stateafew possibilities lateron.

3. THE UNIQUE TRACE CASE

In light of Huaxin Lin’s celebrated classification theorem for tracially AF algebras (cf. [7]) our first analogues of Theorem 21 should be stated in the form ‘When does stable imply tracially$\mathrm{A}\mathrm{F}?$’Indeed, most

ofwhat follows will beasurvey of how muchprogresswehavemade onthis question. This is bothastrong indication that the classification of $Z$-stable algebras (with real rank zero) is possible and will hopefully

reinforce the idea thatwedesperatelyneed to find good analogues of Theorem 2.2.

Let’s restrict to the unique$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}$ caseand seewhat theclassificationprogramcurrentlylooks like. In my

opinionthe best result we have at the moment is due to Huaxin Lin [8] (though I have my owntake on

approximation properties of traces in [1]$)$.

Theorem 3.1. Up to $Z$-stabilization, the real rank zero, unique trace, inductive limit

of

type I case

of

Elliott’s program is complete. More precisely, $\mathrm{z}f$$A$ is an inductive limit

of

type I$\theta- algbras^{3}$ vnth unique

tracial state$\tau$ and

for

every$\epsilon>0$ there is aprojection$p\in A$ such that $\mathrm{r}(\mathrm{p})<\epsilon$. Then$A\otimes \mathcal{Z}$ is tracially

$AF$.

The proof ofthis resultdepends

on

three facts: (a) the typeI assumption implies that the unique $\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}$

satisfies a very strong approximation property (cf. [8], [1]) and tensoring with $Z$ forces (b) Blackadar’s

fundamental comparison property and (c) real rankzero [14]. Here

are

afew non-trivial corollaries.

Corollary 3.2. Let A be Villadsen’s example

of

anAH algebrawith stable rank greater than 1 [19], Then

$A\otimes Z$ is classifiable.

See Proposition 11 in [19] for aproofthat $A$ has projections of arbitrarily small $\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}$. It is admittedly

irritatingthatonemust tensorwith$Z$inordertoclassify,but Truthisnot

concerned

with human emotions.

In other words,several expertsbelievethatVilladsen’sexamples

are

not classifiable (bytheir Elliott invari-ants) and if thisturnsout tobecorrectthen the theoremabovewill bebestpossible. We don’t havetolike it, but that islife.

In the absence of projections it is hard for me to imaginethat we will prove very general

classification

results any time soon. For example, if$A$ is

an

inductivelimit of type Ialgebras with unique $\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}$ and few

(orno) projectionsthen I don’t

see

any reasonable strategyforclassifying$A\otimes \mathcal{Z}$atthe present time. On

the

other hand, if

one

allows aweaker stabilitythen the

classification

programis already complete. Say

that

$A$

is rationally stableif$A\cong A$

&

I, where$\mathcal{U}$ is the UHF algebra whose$\mathrm{K}_{0}$-group is isomorphic tothe

rational

numbers

Q.

$2_{\mathrm{T}\mathrm{h}\mathrm{i}\mathrm{s}}$

meansthat if$p$,$q$areprojectionsand$\tau(p\rangle$ $<\tau(q$}foralltracesthen$p$isequivalent toa subprojection of$q$.

Corollary 3.3. Up to rational stabilization, the

inductive

limit

_of

type $I_{J}$ unique trace case

of

Elliott’s

program zs complete. That is,

_if

A $\iota s$ an inductive

limit

of

type I algebras with unique trace then$A\otimes \mathcal{U}$ is

tracidly$AF$

.

Thanksto thedeep work of Q. Lin and Phillips [10]

we

thushave the following corollary.

Corollary 3.4. Let h:M $arrow M$ be

a

minimal diffeomorphism

of

_acompact

_manifold

_{M and}

_assume

_that$h$

has a unique invariant

measure.

Then $(C(M)>\mathrm{r}_{h}\mathbb{Z})$ cg$\mathcal{U}$ is tracially

AF.

I know you don’$\mathrm{t}$want totensorwith

a

UHFalgebra and I too would be thrilled if we only had to

tensor

with $Z$_. However,

as

Isaid above, it is hard for meto imagine_{howthe proofwould go}

so

_{Iamquite happy}

with the fact that, up torational stabilization, this

case

ofthe program isfinished. Of course, the results presented sofar beg the following two questions.

Question 3.5. Assume$A$ isan inductive limit

of

typeIalgebraswith unique trace and real rankzero. Is$A$

automatically$\mathcal{Z}$-stable ?

Note that thisquestion, ifanswered affirmatively, would imply that there

are

no Villadsentype examples

with real rank

zero

and hencethis isa non-trivial question. Indeed,

even

theAH case (rather than general type I) would be of significantinterest.

Question 3.6. Can one construct

an

inductive lzrnit

_of

type I algebras which has a unique trace and is

$\mathcal{Z}$-stable butwhich isnot

classifiable?

Doing this is probably quitehard

as

it would require, amongother things, inventinga newinvariant (all

theusualsuspectsarewell-behaved for$\mathcal{Z}$-stable algebras). If such counterexamples exist then it wouldshow

thatthe ’rational stabilization’ theorem we already have is actually the best possible result in general, I am

not suggesting that such counterexamples${}^{\mathrm{t}}\mathrm{s}\mathrm{h}\mathrm{o}\mathrm{u}1\mathrm{d}$’

exist (indeed, Ihave noclue) Iamjust pointing out that it is possibleourexisting ’classification upto stabilization’ theorems might be best possible for the general

class ofinductive limits of type I algebras with unique$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}$.

Finally, Iwill mention that

one

canformulate similar results under the weaker (but still quite restrictive) assumption that there are only countably many extreme traces Passing from one $\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}$ to a countable

number of extreme traces is just a technical argument (cf. [8]); the real challenge is passing to arbitrary tracial spaces,

4. GENERAL TRACIAL STATE SPACES

Now I want to discuss whatcanbe said in thecaseof arbitrary tracial state spaces. The resultsarequite nice as

are

the remaining problems. Inmy opinion, there aretworesults which deserve special recognition. The first has to do with AH algebras while the second treats finite decomposition rank in the sense of Kirchberg-Winter.

If we combine the results of [9], [11] and [14] then we get a very satisfactory result in the case ofAH

algebras with real rank zero. I must emphasize that there are no restrictions on dimension growth or the topology

_of

the base$spaces^{4}$ in the followingtheorem.

Theorem 4.1. Up to $Z$-stabilization, the real rankzero $AH$case

of

Elhott’sprogram $\iota s$complete. In other

words,

_if

$A$ is an$AHalgebra^{5}$ titith real rankzero then $A$(&Z is tracially$AF$.

Proof

WilhelmWinter pointed out that therewas

a

small gapin my original ’proof of this result. Namely,

I asserted that it followedimmediatelyfromthe three paperscited above and he correctly pointed out that it does not! Oops! ;-) (Thanks Wilhelm.) Contrary to my lazy blanket assumption, algebras appearing in this proofarenot necessarily simple.

Appealing to [11], we must show that $A\otimes Z$ is locally approximated by subalgebras which (a) have

Hausdorff spectrum and (b) aresubhomogeneous. Since $Z$ is an inductive limit of prime dimensiondrop

algebras and $A$ is locally approximated by homogeneous algebras, it suffices to establish two general facts:

First we must show that prime dimension drop algebras have

Hausdorff

spectrum and, second, that the

tensorproduct oftwoalgebraswith

Hausdorff

spectrum also has Hausdorffspectrum.

Recall that if$p$and $q$ are relatively prime integersthen the associated prime dimension drop algebra is

defined as

$\mathrm{Z}(\mathrm{P}, q)=$

{

$f\in C([0,1],$$\mathrm{M}\mathrm{P}(\mathrm{C})\otimes \mathrm{M}\mathrm{q}(\mathrm{C}))$: $f(0)\in M_{p}(\mathbb{C})$@1 and $f(1)\in 1\otimes M_{q}(\mathbb{C})$

}.

$4\mathrm{I}\mathrm{n}$

otherwords, the building blocks arecornersof algebras of the form $M_{n}(C(X))$where$X$is any compact metric

space-possiblyeveninfinitedimensional!

$5\mathrm{E}\mathrm{v}\mathrm{e}\mathrm{n}$

NATHANIAL$\mathrm{P}$ BROWN

A moments thought reveals that the center of$\mathrm{X}(\mathrm{p},$$q\rangle$is isomorphic to $C([0,1])$. A bit

more

contemplation

and

one

realizes that foreachprimitive ideal $J\triangleleft C([\mathrm{O}, 1])$, the ideal $J\mathrm{I}(p)$ $q)\triangleleft \mathrm{I}(p, q)$is alsoprimitive and,

moreover, this gives abijective map Prim(c ([0,$1])$) $arrow \mathrm{P}\mathrm{r}\mathrm{i}\mathrm{m}(\mathrm{X}(\mathrm{j}3, q))$ofprimitive idealspaces. (Surjectivity

follows from the fact that I$(p, q)/J\mathrm{I}(p, q)$ is a full matrix algebra.) That this map is

a

homeomorphism

follows easily from the definition of theclosure ofaset ofprimitive ideals. Hence the spectrum ofaprime

dimension drop algebra ishomeomorphicto $[0, 1]$.

Toshow that Hausdorffspectrum ispreservedundertaking tensor products it suffices torecall thegeneral fact that if$A$and $B$ are unitaltypeI C’-algebrasthen

$\overline{A\otimes B}\cong\hat{A}\mathrm{x}\hat{B}$.

This result

can

be found in [21]; for convenience wesketch a proofofthe

case we

are interestedin.

Assumeboth $A$and $B$ haveHausdorffspectrum. Since$A\otimes B$ is compact (our algebras

are

unital) and

$\hat{A})\langle\hat{B}$ is

Hausdorff itsufficesto show the existence ofa continuous bijection

$\overline{A\otimes B}arrow\hat{A}\mathrm{x}\hat{B}$_.

If$\pi$.$A\otimes Barrow B(H)$ isanirreduciblerepresentation then both$\pi(A\otimes 1)’$and$\pi(1\otimes B)’$ mustbefactors;since $A$and$B$ aretype$\mathrm{I}$, they must be full matrix algebrasor$B(K)$for

some

infinitedimensionalHilbertspace$K$.

In any

case

it follows that $\pi$ is unitarily equivalent tothetensor product of two irreducible representations

andthisevidently yieldsabijective mapA\otimes B\rightarrow \^A$\mathrm{x}\hat{B}$. The definition of convergenceofnetsof irreducible

representations readily implies continuity of this map.6 $\square$

I should mention that

one

need not tensor with $\mathcal{Z}$ if $A$ is known to satisfy Blackadar’s fundamental

comparison property. Hence the following questions

seem

quite natural.

Question 4.2. Does every AH algebra with real rank

zero

satisfy Blackadar’s

fundamental

comparison property l.?

Question 4.3. Is every AHalgebra with real rank zero arrtomatically$\mathcal{Z},stable^{\mathit{9}}$

Affirmative

answers

to these questions would, of course, be major contributions to the classification

program. For example, it would follow that real rank zero AH algebras always have weakly unperforated

$\mathrm{K}_{0}$-groups and stable rank one (without assumptionson dimension growth or topologyofbase spaces!). I

hopeyou will now agreethatwehave significant motivation to launch a full scale attack

on

finite analogues

of Theorem 2.2. I am not suggesting this is a simple problem, but I am saying that major cases ofthe

classification programwill be complete

as

soon as we do.

Inanother possible direction, it would be very nice to relax theAH assumptionabove toallowfor general subhomogeneousalgebras or, better yet, tyPeI.

Question 4,4.

_If

_A is an inductive limit

_of

type I algebras (or just subhomogeneous algebras) and has real

rank zero then does it

_follow

that $A\otimes Z$ is tracially$AF^{l}.$?

Finally, Iwant to stateatheoremwhich wasrecently proved by Wilhelm Winter [20]. For the definition of decomposition rankwe referto the original paper of Kirchberg-Winter [6]. For our purposes it suffices

to say that this is a generalization of classical covering dimension. However, the point to emphasize is that the definition does not

assume

any sort

_of

inductivelimit decomposition or tracial approximation by ’tractable’ _{subalgebras. It is}

_an

_{abstract hypothesis which is closely related toquasidiagonality.}

As

_{such, I}

find the following result particularly attractive. Moreover, it completes the classificationofinductive limits

of recursive subhomogeneous algebras with

no

dimension growth, at least _uP to $\mathcal{Z}$-stabilization (cf. [12,

Conjecture4.6]).

Theorem 4.5.

Assume

A has red rank

zero

and

_finite

decomposition rank. Then$A\otimes Z$ istracially AF.

Asbefore, this resultdares us to tackle thefollowing question.

Question 4.6. Is every algebra with realrank zero and

_finite

decomposition rankautomatically Z

stable

$\rho$

$\epsilon_{\mathrm{R}\mathrm{e}\mathrm{c}\mathrm{a}11}$

that$\pi\lambdaarrow\pi$ in

$\hat{C}$if andonlyif_$\lim$inf

5. SUMMARY

Please forgivethe repetition,but Iwill summarize.

Unfortunate Fact: Andrew Toms has conclusively demonstrated that it is necessary tosettle for

classifi-cation theorems ’upto stabilization’. This isthe best

we

can ever hopeforsince he has constructed an AH

algebrawithweakly unperforated$\mathrm{K}0$-group but which is not Z-stable.

Fortunate Fact: Combining the work of many hands, most notablyHuaxinLin, wehave nowcompleted a

numberof interesting

cases

of Elliott’s program‘upto$\mathcal{Z}$-stabilization’. The remaining problemsareprobably

not easy, but theyhavebeenisolated and

we

know whereto focus.

Future $\mathrm{F}\mathrm{a}\mathrm{c}\mathrm{t}(?)$: Somebodywill prove a wonderful theorem which shows that Zinstability is automatic for

large classesof (real rank zero) algebras thus completing a vast swath ofthe classification program.

See [18, Sections 2and 3] for general results which mayhelp withthis futurefact.

REFERENCES

1 N.P Brown, Invariantmeansand_finiterepresentation theory _of

c.

-algebras, preprint.

2. G. Gong, X. Jiang and H Su, Obstructions to$Z$-stabilityforunitalsimple$C$ -algebras Canad Math. Bull. 43 (2000),

418-426

3 X Jiang and H Su, On asimpleunital projectionless C’ -afgebra, Amer J Math. 121 (1999), 359-413.

4 E. Kirchberg, The _{classificationof}Purely_infinite

c.

-algebras using_KasParov’stheory, preprint

5. E. Kirchberg andNC. Phillips, Embedding _ofexactC’ -algebras in the Cuntzalgebra 02, J ReineAngew. Math. 525

(2OO0), 17-53

6 E Kirchberg andW Winter, Covering dimensionandquasidiagonality, Internat J Math. 15 (2004), 63-85

7. H. Lin, _{Classificationof}simple $C^{\alpha}$-algebras oftracialtopologicalrankzero, Duke Math. J125 (2004), 91119.

8–, Tracesand simple $C^{*}$-algebras with tracial topologicalrankzero, J. ReineAngew. Math 568 (2004),99–137.

9 –, Simple$AH$-algebrasofrealrank zero,Proc Amer Math. Soc 131 (2003),3813 3819

10. Q Lin andN.C Phillips, Thestructureof$C’$-algebras ofminimal diffeomorphisms, $\ln$preparatlon

11 P.W. Ng, Simple real rankzeroalgebras with locally_Hausdorffspectrum, preprint

12 NC Phillips, Real rankand property (SP)fordirect limitsofrecursivesubhomogeneowalgebras, preprint

13. –, Aclassificationtheorem fornuclearpurelyinfinite simple$C$ -algebras, Doc Math 5(2000),49-114

14 M. Rordam, The stable andreal rank_of$Z$-absorbing $\sigma$-algebras,InternatlonalJ Math 15(2004), 1065-1084.

15 $-\backslash A$ simple$C^{*}$-atgebm with afinite andaninfiniteprojection, ActaMath 191 (2003),109-142.

16 A S Tom s, Onthe independenceof$K$-theoryand stable rankforsimple $C^{\mathrm{r}}$-algebras, J Reine Angew. Math. (to appear)

17 –, On the Classificationproblem fornuclear

$C^{\mathrm{s}}$-algebras, preprint 18. AS. Tomsand W Winter,Stronglyselfabsorbrng $C^{*}$-algebras, preprint.

19 J. Villadsen, Onthestable rankofsimple C’-algebras, J Amer Math.Soc. I2 (1999),1091-1102

20 W.Winter, On theclassification ofsimple$\mathcal{Z}$-stable algebras with real rankzeroandfinitedecom position mnk, preprint

21. A Wolfsohn, The primitive spectrum ofa tensorproduct ofC’-algebras,Proc. Amer Math.Soc 19(1968),1094-1096.

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