finiteness and the mf property in c*-crossed products and the mf property in c -crossed products...

Finiteness and the MF Property in C∗-Crossed Products

Timothy Rainone

Texas A&M University, College StationDepartment of Mathematics

CBMSJune 2015, The University of Wyoming

First Principles

Throughout, Γ will always denote a countable discrete group, and A aseparable C ∗-algebra with unit 1A.

Fr will denote the free group on r generators, r ∈ {1, 2, . . . ,∞}.

A is said to have stable rank one, written sr(A) = 1, provided that theset of invertible elements GL(A) is norm-dense in A.

A is said to have real rank zero, written RR(A) = 0, provided thatAsa ∩ GL(A) is norm-dense in Asa.

A is exact if, for any short exact sequence 0→ B → C → D → 0 ofC ∗-algebras, 0→ A⊗ B → A⊗ C → A⊗ D → 0 is also exact.

By a C ∗-dynamical system, or an action Γ y A, we mean a triple(A, Γ, α) where α : Γ→ Aut(A) (s 7→ αs) is a group homomorphism.

T. Rainone (TAMU) 2015 2 / 23

First Principles

Given a C ∗-dynamical system (A, Γ, α), where A ⊂ B(H) is faithfullyrepresented, one may construct the reduced crossed productC ∗-algebra Aoλ Γ ⊂ B(H ⊗ `2(Γ)):

Consider the algebraic crossed product Aoalg Γ which is the complexlinear space of finitely supported functionsCc(Γ,A) = {

∑s∈F asus : F ⊂⊂ Γ, as ∈ A}, equipped with a twisted

multiplication and involution: for s, t ∈ Γ, a, b ∈ A

(aus)(but) = aαs(b)ust ,

(aus)∗ = αs−1(a∗)us−1 .

The ∗-algebra Aoalg,α Γ can then be faithfully represented asoperators on H ⊗ `2(Γ) via aus(ξ ⊗ δt) = α−1st (a)ξ ⊗ δst for ξ ∈ H

and s, t ∈ Γ. Completing with respect to the operator norm onB(H ⊗ `2(Γ)) gives Aoλ Γ.

(aus)∗ = αs−1(a∗)us−1 .

Finite and Infinite Algebras

Let A be a C ∗-algebra and Mn(A) ∼= Mn ⊗ A the C*-algebra of n × nmatrices over A.

Projections p, q ∈ A are Murray-von Neumann equivalent, writtenp ∼ q, if v∗v = p and vv∗ = q for some v ∈ A.

A unital algebra A is said to be infinite if there is a projection p ∈ Awith 1A ∼ p 6= 1A. Otherwise, A is said to be finite. If Mn(A) is finitefor every n ∈ N, A is called stably finite.

A projection p ∈ A is said to be properly infinite if there areprojections q, r ∈ A with q, r ≤ p, q ⊥ r and r ∼ p ∼ q. If 1A isproperly infinite then A is said to be properly infinite.

A simple algebra A is purely infinite if every non-zero hereditarysubalgebra of A contains an infinite projection. S. Zhang showed thatA is purely infinite iff RR(A) = 0 and every projection p in A isproperly infinite.

A unital algebra A is said to be infinite if there is a projection p ∈ Awith 1A ∼ p 6= 1A. Otherwise, A is said to be finite.

If Mn(A) is finitefor every n ∈ N, A is called stably finite.

A projection p ∈ A is said to be properly infinite if there areprojections q, r ∈ A with q, r ≤ p, q ⊥ r and r ∼ p ∼ q.

If 1A isproperly infinite then A is said to be properly infinite.

A simple algebra A is purely infinite if every non-zero hereditarysubalgebra of A contains an infinite projection.

S. Zhang showed thatA is purely infinite iff RR(A) = 0 and every projection p in A isproperly infinite.

MF Algebras

A subclass of stably finite algebras are the so called MF algebras,introduced by Blackadar and Kirchberg in 1997. A separableC ∗-algebra A is said to be MF if it can be expressed as a generalizedinductive system of finite-dimensional algebras, or equivalently,

A ↪→∞∏k=1

/ ∞⊕k=1

for some natural sequence (nk)k≥1 ⊂ N.

MF algebras are stably finite. The converse, however, is still open. Inparticular, is there a countable discrete group Γ for which C ∗λ(Γ) isnot MF? U. Haagerup and S. Thorbjørnsen showed in 2005 thatC ∗λ(Fr ) is MF.

Question: What property of Γ gives rise to an MF algebra C ∗λ(Γ)?LEF?

MF Algebras

A ↪→∞∏k=1

/ ∞⊕k=1

MF algebras are stably finite. The converse, however, is still open. Inparticular, is there a countable discrete group Γ for which C ∗λ(Γ) isnot MF?

U. Haagerup and S. Thorbjørnsen showed in 2005 thatC ∗λ(Fr ) is MF.

MF Algebras

A ↪→∞∏k=1

/ ∞⊕k=1

MF Algebras

A ↪→∞∏k=1

/ ∞⊕k=1

Question: What property of Γ gives rise to an MF algebra C ∗λ(Γ)?

MF Algebras

A ↪→∞∏k=1

/ ∞⊕k=1

K-Theory, Cuntz Semigroup

If a ∈ Mn(A)+ and b ∈ Mm(A)+, write a⊕ b for the matrixdiag(a, b) ∈ Mn+m(A)+.

Set M∞(A)+ =⊔

n≥1Mn(A)+; the set-theoretic direct limit of theMn(A)+ with connecting maps Mn(A)→ Mn+1(A) given bya 7→ a⊕ 0.

Write P(A) for the set of projections in A and setP∞(A) =

⊔n≥1 P(Mn(A)).

Elements a and b in M∞(A)+ are said to be Pedersen-equivalent,written a ∼ b, if there is a matrix v ∈ Mm,n(A) with v∗v = a andvv∗ = b. We say that a is Cuntz-subequivalent to (or Cuntz-smallerthan) b, written a - b, if there is a sequence (vk)k≥1 ⊂ Mm,n(A) with‖v∗k bvk − a‖ → 0 as k →∞. If a - b and b - a we say that a and bare Cuntz-equivalent and write a ≈ b.

One proves that ∼ and ≈ are equivalence relations on M∞(A)+ andthat a ∼ b implies a ≈ b.

Set M∞(A)+ =⊔

⊔n≥1 P(Mn(A)).

Set M∞(A)+ =⊔

⊔n≥1 P(Mn(A)).

Set M∞(A)+ =⊔

⊔n≥1 P(Mn(A)).

Set M∞(A)+ =⊔

⊔n≥1 P(Mn(A)).

Write V (A) = P∞(A)/ ∼, and [p] for the equivalence class ofp ∈ P∞(A).

Also set W (A) := M∞(A)+/ ≈ and write 〈a〉 for the class ofa ∈ M∞(A)+.

W (A) has the structure of a preordered abelian monoid with additiongiven by 〈a〉+ 〈b〉 = 〈a⊕ b〉 and preorder 〈a〉 ≤ 〈b〉 if a - b. Thismonoid W (A) will be referred to as the Cuntz semigroup of A.

With addition and ordering identical to that of W (A), V (A) is also apreordered abelian monoid. The ordering on V (A) agrees with thealgebraic ordering. Indeed, for p, q ∈ P∞(A), one shows that p - q iffp ∼ r ≤ q some r ∈ P∞(A) iff p ⊕ p′ ∼ q for some p′ ∈ P∞(A).

As a brief reminder, K0(A) = G(V (A)) the Grothendieck envelopinggroup of V (A) and [p]0 = γ([p]) where γ : V (A)→ K0(A) is thecanonical Grothendieck map.

When A is stably finite, K0(A) is an ordered abelian group withpositive cone K0(A)+ = γ(V (A)) and order unit [1]0.

We shall often require our algebras A to have cancellation, whichmeans that γ is injective: [p]0 = [q]0 implies p ∼ q. Algebras withstable rank one have cancellation.

An important notion throughout this talk is the idea of refinement. Asemigroup K has the Riesz refinement property if, whenever

n∑j=1

xj =m∑i=1

for members x1, . . . , xn, y1, . . . , ym ∈ K , there exist {zij}i ,j ⊂ Ksatisfying ∑

zij = xj and∑j

zij = yi

for each i and j . If A is a stably finite algebra with RR(A) = 0 then S.Zhang showed that K0(A)+ has the Riesz refinement property.

We shall often require our algebras A to have cancellation, whichmeans that γ is injective: [p]0 = [q]0 implies p ∼ q.

Algebras withstable rank one have cancellation.

n∑j=1

xj =m∑i=1

zij = xj and∑j

zij = yi

n∑j=1

xj =m∑i=1

zij = xj and∑j

zij = yi

n∑j=1

xj =m∑i=1

zij = xj and∑j

zij = yi

for each i and j .

If A is a stably finite algebra with RR(A) = 0 then S.Zhang showed that K0(A)+ has the Riesz refinement property.

n∑j=1

xj =m∑i=1

zij = xj and∑j

zij = yi

Induced K-theoretic Dynamics

If (G ,G+, u) is an ordered Abelian group with order unit u, wedenote by

Aut(G ,G+) = {β ∈ Aut(G ) : β(G+) = G+, β(u) = u}

the group of order automorphisms.

A state on (G ,G+, u) is a group homomorphism β : G → R withβ(G+) ⊂ R+ and β(u) = 1.

A C ∗-system (A, Γ, α), with A stably finite will induce an actionα̂ : Γ→ Aut(K0(A),K0(A)+), given by

α̂s([p]0 − [q]0) = [αs(p)]0 − [αs(q)]0.

In this case, a state β on (K0(A),K0(A)+, [1]) is Γ-invariant ifβ(α̂s(x)) = β(x) for every x ∈ K0(A) and s ∈ Γ.

α̂s([p]0 − [q]0) = [αs(p)]0 − [αs(q)]0.

MF Crossed Products

In many cases the induced K -theoretic dynamics sheds light on thestructure of the reduced crossed product.

If A is an AF algebra, the PVsequence reads:

r⊕j=1

K0(A)σ−→ K0(A)

ι̂−→ K0(Aoλ Fr ) −→ 0,

where σ(g1, . . . , gr ) =∑r

j=1(gj − α̂j(gj)). Set Hσ = im(σ) ≤ K0(A), sothat K0(A)/Hσ ∼= K0(Aoλ Fr ).

Theorem

Let A be a unital AF algebra and α : Fr → Aut(A) an action of the freegroup on r generators. Then the following are equivalent.

1 Hσ ∩ K0(A)+ = {0}.2 The reduced crossed product Aoλ Fr is MF.

3 The reduced crossed product Aoλ Fr is stably finite.

MF Crossed Products

In many cases the induced K -theoretic dynamics sheds light on thestructure of the reduced crossed product. If A is an AF algebra, the PVsequence reads:

r⊕j=1

K0(A)σ−→ K0(A)

ι̂−→ K0(Aoλ Fr ) −→ 0,

where σ(g1, . . . , gr ) =∑r

Theorem

MF Crossed Products

r⊕j=1

K0(A)σ−→ K0(A)

ι̂−→ K0(Aoλ Fr ) −→ 0,

where σ(g1, . . . , gr ) =∑r

j=1(gj − α̂j(gj)).

Set Hσ = im(σ) ≤ K0(A), sothat K0(A)/Hσ ∼= K0(Aoλ Fr ).

Theorem

MF Crossed Products

r⊕j=1

K0(A)σ−→ K0(A)

ι̂−→ K0(Aoλ Fr ) −→ 0,

where σ(g1, . . . , gr ) =∑r

Theorem

MF Crossed Products

r⊕j=1

K0(A)σ−→ K0(A)

ι̂−→ K0(Aoλ Fr ) −→ 0,

where σ(g1, . . . , gr ) =∑r

Theorem

Finiteness, Infiniteness, Paradoxicality, Dichotomy

There is a deep theme common to groups, dynamical systems andoperator algebras; that of finiteness, infiniteness, and proper infiniteness,the latter expressed in terms of paradoxical decompositions.

(Groups) The remarkable alternative theorem of Tarski establishes, fordiscrete groups, the dichotomy between amenability and paradoxicaldecomposability.

(Geometric Operator Algebras) If a discrete group Γ acts on itself byleft-translation, the Roe algebra C (βΓ) oλ Γ is properly infinite if andonly if Γ is Γ-paradoxical and this happens if and only if Γ isnon-amenable. (Rørdam-Sierakowski).

(von Neumann Algebras) All projections in a II1 factor are finite andthe ordering of Murray-von-Neumann subequivalence is determined bya unique faithful normal tracial state. Alternatively type III factorsadmit no traces since all non-zero projections therein are properlyinfinite.

(C∗-algebras) It was a longstanding open question whether thereexisted a unital, separable, nuclear and simple C∗-algebra which wasneither stably finite or purely infinite. M. Rørdam settled the issue byexhibiting a unital, simple, nuclear, and separable C∗-algebra Dcontaining a finite and infinite projection p, q. It follows thatA = qDq is unital, separable, nuclear, simple, and properly infinite,but not purely infinite.

However, the question is still open for unital, seprable, nuclear andsimple C∗-algebras of real rank zero.One reason why such a dichotomy is interesting is for classificationpurposes. The Elliott classification program has been relativelysuccessful in the stably finite case, but a more complete classifcationresult (Kirchberg, Phillips) exists in the purely infinite case modulo theUCT. Indeed, Kirchberg algebras (unital, separable, simple, nuclear,and purely infinite) are classified by their K -theory (or KK -theory).

(C∗-algebras) It was a longstanding open question whether thereexisted a unital, separable, nuclear and simple C∗-algebra which wasneither stably finite or purely infinite. M. Rørdam settled the issue byexhibiting a unital, simple, nuclear, and separable C∗-algebra Dcontaining a finite and infinite projection p, q. It follows thatA = qDq is unital, separable, nuclear, simple, and properly infinite,but not purely infinite.However, the question is still open for unital, seprable, nuclear andsimple C∗-algebras of real rank zero.

One reason why such a dichotomy is interesting is for classificationpurposes. The Elliott classification program has been relativelysuccessful in the stably finite case, but a more complete classifcationresult (Kirchberg, Phillips) exists in the purely infinite case modulo theUCT. Indeed, Kirchberg algebras (unital, separable, simple, nuclear,and purely infinite) are classified by their K -theory (or KK -theory).

(C∗-algebras) It was a longstanding open question whether thereexisted a unital, separable, nuclear and simple C∗-algebra which wasneither stably finite or purely infinite. M. Rørdam settled the issue byexhibiting a unital, simple, nuclear, and separable C∗-algebra Dcontaining a finite and infinite projection p, q. It follows thatA = qDq is unital, separable, nuclear, simple, and properly infinite,but not purely infinite.However, the question is still open for unital, seprable, nuclear andsimple C∗-algebras of real rank zero.One reason why such a dichotomy is interesting is for classificationpurposes. The Elliott classification program has been relativelysuccessful in the stably finite case, but a more complete classifcationresult (Kirchberg, Phillips) exists in the purely infinite case modulo theUCT. Indeed, Kirchberg algebras (unital, separable, simple, nuclear,and purely infinite) are classified by their K -theory (or KK -theory).

Constructing Infinite Crossed Products

A continuous action Γ y X of a discrete group on a locally compactspace is called a local boundary action if for every non-empty open setU ⊂ X there is an open set V ⊂ U and t ∈ Γ with t.V ( V .

Lacaand Spielberg showed that such actions yield infinite projections inthe reduced crossed product C0(X ) oλ Γ.

Adam Sierakowski remarked that the condition t.V ( V for somenon-empty open set V and group element t ∈ Γ is equivalent to theexistence of open sets U1,U2 ⊂ X and elements t1, t2 ∈ Γ such that

U1 ∪ U2 = X , t1.U1 ∩ t2.U2 = ∅ and t1.U1 ∪ t2.U2 6= X .

He generalized this by defining paradoxical actions. A transformationgroup (X , Γ) is n-paradoxical if there exist open subsetsU1, . . . ,Un ⊂ X and elements t1, . . . , tn ∈ Γ such that

n⋃j=1

Uj = X ,n⊔

tj .Uj ( X .

A continuous action Γ y X of a discrete group on a locally compactspace is called a local boundary action if for every non-empty open setU ⊂ X there is an open set V ⊂ U and t ∈ Γ with t.V ( V . Lacaand Spielberg showed that such actions yield infinite projections inthe reduced crossed product C0(X ) oλ Γ.

n⋃j=1

Uj = X ,n⊔

tj .Uj ( X .

n⋃j=1

Uj = X ,n⊔

tj .Uj ( X .

n⋃j=1

Uj = X ,n⊔

tj .Uj ( X .

Sierakowski showed that the algebra C (X ) oλ Γ is infinite provided that Xis compact and the action Γ y X is n-paradoxical for some n.

We wish tohave a similar result in the noncommutative setting.

Proposition

Let A be a unital C*-algebra and let α : Γ→ Aut(A) be an action which isW -paradoxical in the sense that there exist x1, . . . , xn ∈W (A) and groupelements t1, . . . , tn ∈ Γ with

n∑j=1

xj ≥ 〈1A〉 andn∑

α̂tj (xj) < 〈1A〉.

Then Aoλ Γ is infinite.

Perhaps what has been called paradoxical is misleading because, in asense, paradoxicality implies the idea of duplication of sets. Gleaning fromthe ideas explored by Kerr and Nowak, we define a notion of paradoxicaldecomposition with covering multiplicity.

Sierakowski showed that the algebra C (X ) oλ Γ is infinite provided that Xis compact and the action Γ y X is n-paradoxical for some n. We wish tohave a similar result in the noncommutative setting.

Proposition

n∑j=1

α̂tj (xj) < 〈1A〉.

Proposition

n∑j=1

α̂tj (xj) < 〈1A〉.

Proposition

n∑j=1

α̂tj (xj) < 〈1A〉.

Paradoxical Decompositions

Now we look at K-theoretic conditions that will characterize stably finiteversus purely infinite crossed products.

Definition

Let α : Γ→ Aut(A) be an action with its induced action α̂ on K0(A). Let0 6= x ∈ K0(A)+ and k > l > 0 be positive integers. We say x is(Γ, k , l)-paradoxical if there are x1, . . . , xn in K0(A)+ and t1, . . . , tn in Γsuch that

n∑j=1

xj ≥ kx andn∑

α̂tj (xj) ≤ lx .

If an element x ∈ K0(A)+ fails to be (Γ, k, l)-paradoxical for all integersk > l > 0 we call x completely non-paradoxical. The action α will becalled completely non-paradoxical if every member of K0(A)+ iscompletely non-paradoxical.

Definition

Let α : Γ→ Aut(A) be an action with its induced action α̂ on K0(A). Let0 6= x ∈ K0(A)+ and k > l > 0 be positive integers.

We say x is(Γ, k , l)-paradoxical if there are x1, . . . , xn in K0(A)+ and t1, . . . , tn in Γsuch that

n∑j=1

xj ≥ kx andn∑

Definition

n∑j=1

xj ≥ kx andn∑

Definition

n∑j=1

xj ≥ kx andn∑

Definition

n∑j=1

xj ≥ kx andn∑

If an element x ∈ K0(A)+ fails to be (Γ, k, l)-paradoxical for all integersk > l > 0 we call x completely non-paradoxical.

The action α will becalled completely non-paradoxical if every member of K0(A)+ iscompletely non-paradoxical.

Definition

n∑j=1

xj ≥ kx andn∑

Proposition

If (A, Γ, α) is a C ∗-system with stably finite reduced crossed productAoλ Γ. Then α is completely non-paradoxical.

For the converse we need some machinery. Analogous to the typesemigroup S(X , Γ) for Cantor systems studied by Rørdam and Sierakowski,we introduce a preordered Abelian monoid S(A, Γ, α) which correctlyreflects the above notion of paradoxicality, and then inevitably resort toTarski’s result tying the existence of non-trivial states on S(A, Γ, α) tonon-paradoxicality.

Definition

Let A be a C ∗-algebra, α : Γ→ Aut(A) an action. We define a relation onthe positive cone K0(A)+ as follows: for x , y ∈ K0(A)+, set x ∼α y if∃{uj}kj=1 ⊂ K0(A)+ and {tj}kj=1 ⊂ Γ, such that

k∑j=1

uj = x andk∑

α̂tj (uj) = y .

Proposition

Definition

k∑j=1

uj = x andk∑

α̂tj (uj) = y .

Proposition

Definition

k∑j=1

uj = x andk∑

α̂tj (uj) = y .

The preordered monoid S(A, Γ, α)

This relation is similar to the notion of equidecomposability in the settingof a group acting on a set, where the idea of refined partitions is key toestablishing transitivity. This translates to the Riesz refinement property inour context.

Proposition

Let A be a stably finite algebra with Riesz-refinement, and let α : Γ y Abe an action.

1 ∼α is an equivalence relation. Set S(A, Γ, α) = K0(A)+/ ∼α.

2 Addition [x ]α + [y ]α := [x + y ]α is well defined.

3 With the algebraic ordering, S(A, Γ, α) is an Abelian preorderedmonoid. For a Cantor system (X , Γ) with induced actionα : Γ y C (X ), S(C (X ), Γ, α) is isomorphic to the type semigroupS(X , Γ).

4 Given x ∈ K0(A)+, x is (Γ, k , l)-paradoxical iff k[x ]α ≤ l [x ]α inS(A, Γ, α).

The preordered monoid S(A, Γ, α)

This relation is similar to the notion of equidecomposability in the settingof a group acting on a set, where the idea of refined partitions is key toestablishing transitivity. This translates to the Riesz refinement property inour context.

Proposition

Let A be a stably finite algebra with Riesz-refinement, and let α : Γ y Abe an action.

1 ∼α is an equivalence relation. Set S(A, Γ, α) = K0(A)+/ ∼α.

2 Addition [x ]α + [y ]α := [x + y ]α is well defined.

3 With the algebraic ordering, S(A, Γ, α) is an Abelian preorderedmonoid. For a Cantor system (X , Γ) with induced actionα : Γ y C (X ), S(C (X ), Γ, α) is isomorphic to the type semigroupS(X , Γ).

4 Given x ∈ K0(A)+, x is (Γ, k , l)-paradoxical iff k[x ]α ≤ l [x ]α inS(A, Γ, α).

Some terminology

Let (W ,≤) be a preordered abelian monoid.

An element θ in W is said to properly infinite if 2θ ≤ θ.

W is said to be purely infinite if every member of W is properlyinfinite.

A state on W is a map ν : W → [0,∞] which is additive, respectsthe preordering ≤, and satisfies ν(0) = 0. If a state ν assumes a valueother than 0 or ∞, ν it said to be non-trivial.

The monoid W is said to be almost unperforated if, wheneverθ, η ∈W , and n,m ∈ N are such that nθ ≤ mη and n > m, thenθ ≤ η.

Suppose W is almost unperforated. If θ ∈W is (k, l)-paradoxical (kθ ≤ lθfor k > l > 0) then θ is properly infinite: