aa r X i v : . [ m a t h . OA ] N ov DIFFUSE TRACES AND HAAR UNITARIES
HANNES THIEL
Abstract.
We show that a tracial state on a unital C ∗ -algebra admits a Haarunitary if and only if it is diffuse, if and only if it does not dominate a tracialfunctional that factors through a finite-dimensional quotient. It follows that aunital C ∗ -algebra has no finite-dimensional representations if and only if eachof its tracial states admits a Haar unitary.More generally, we study when nontracial states admit Haar unitaries. Inparticular, we show that every state on a unital, simple, infinite-dimensional C ∗ -algebra admits a Haar unitary.We obtain applications to the structure of reduced free products. Notably,the tracial reduced free product of simple C ∗ -algebras is always a simple C ∗ -algebra of stable rank one. Introduction
Let A be a unital C ∗ -algebra with a tracial state τ : A → C . A unitary u ∈ A iscalled a Haar unitary (with respect to τ ) if τ ( u k ) = 0 for k ∈ Z \ { } . Equivalently,the sub- C ∗ -algebra C ∗ ( u ) of A generated by u is isomorphic to C ( T ) and therestriction of τ to C ∗ ( u ) corresponds to the normalized Lebesgue measure on T .Haar unitaries play important roles in various constructions and structure re-sults in operator algebras, starting with the fundamental fact that they are naturalgenerators of diffuse abelian *-subalgebras in separable, tracial von Neumann al-gebras. Haar unitaries also naturally arise in reduced group C ∗ -algebras, in groupvon Neumann algebras, and as generators of the irrational rotation C ∗ -algebras.Popa’s solution, [Pop87], to the commutant modulo compact operators prob-lem for general von Neumann algebras relies on the construction of suitable Haarunitaries in II factors. This construction was later refined, [Pop95], to prove the ex-istence of free independent sequences of Haar unitaries for irreducible subfactors ofII factors. Haar unitaries also play a prominent role in Voiculescu’s free probabilitytheory, [Dyk93], [DH01], in particular as the unitaries in the polar decompositionsof R-diagonal elements, [NS97], [NS06, Section 15].The main result of this paper characterizes when τ admits a Haar unitary: Theorem A (5.4) . The following are equivalent:(1) τ is diffuse (the normal extension τ : A ∗∗ → C vanishes on every minimalprojection; see Paragraph 3.1)(2) τ does not dominate a nonzero tracial functional that factors through afinite-dimensional quotient of A ;(3) there exists a unital (maximal) abelian sub- C ∗ -algebra C ( X ) ⊆ A such that τ induces a diffuse measure on X ; Date : 1 December 2020.2010
Mathematics Subject Classification.
Primary 46L05, 46L51; Secondary 46L09, 46L30.
Key words and phrases. C ∗ -algebras, traces, Haar unitaries, diffuse functionals, reduced freeproducts.The author was partially supported by the Deutsche Forschungsgemeinschaft (DFG, GermanResearch Foundation) under Germany’s Excellence Strategy EXC 2044-390685587 (MathematicsM¨unster: Dynamics-Geometry-Structure). (4) there exists a Haar unitary in A (with respect to τ ). Corollary B (5.5, 5.6) . A unital C ∗ -algebra has no finite-dimensional representa-tions if and only if each of its tracial states admits a Haar unitary.In particular, every tracial state on a unital, simple, infinite-dimensional C ∗ -algebra is diffuse and admits a Haar unitary.In Section 8, we apply these results to reduced group C ∗ -algebras. First, wesee that a discrete group G is infinite if and only if the canonical tracial state onthe reduced group C ∗ -algebra C ∗ red ( G ) admits a Haar unitary; see Proposition 8.1.Thus, if G is an infinite, locally finite group, then C ∗ red ( G ) contains a Haar unitary,while there exists no Haar unitary in C [ G ]; see Example 8.2. We also obtain acharacterization of nonamenability: Proposition C (8.3) . A discrete group G is nonamenable if and only if everytracial state on C ∗ red ( G ) admits a Haar unitary.In Section 9, we obtain applications to reduced free products. Given unital C ∗ -algebras A and B with faithful tracial states τ A and τ B , respectively, it is a well-studied problem to determine when the reduced free product of ( A, τ A ) and ( B, τ B )is simple or has stable rank one; see [DHR97, Dyk99]. By [Dyk99, Theorem 2],a sufficient condition is that there is an abelian subalgebra C ( X ) ⊆ A such that τ A induces a diffuse measure on X , and that B = C . Theorem A shows that thecondition on A is satisfied if and only if τ A is diffuse; see Theorem 9.1. As animportant special case, we get: Corollary D (9.2) . Let A and B be unital, simple C ∗ -algebras with tracial states τ A and τ B . Assume that A = C and B = C . Then the reduced free product of( A, τ A ) and ( B, τ B ) is simple, has stable rank one and a unique tracial state.It follows that the class of unital, simple, stable rank one C ∗ -algebras with uniquetracial state is closed under formation of reduced free products; see Corollary 9.3.Special cases of Theorem A have been shown before. Under the additional as-sumption that the C ∗ -algebra is abelian, it follows from [DHR97, Proposition 4.1(i)].For normal traces on von Neumann algebras, it is also well-known. In fact, givena normal trace τ on a diffuse von Neumann algebra M , every maximal abeliansubalgebra (masa) D ⊆ M is a diffuse sub-von Neumann algebra. It follows that τ | D is a diffuse trace and thus admits a Haar unitary (for instance by [DHR97,Proposition 4.1(i)]). In particular, every masa in M contains a Haar unitary.We point out that the analogous result does not hold for C ∗ -algebras: Givena diffuse trace on a unital C ∗ -algebra A , not every masa of A needs to contain aHaar unitary; see Example 5.7. This also indicates why the construction of a Haarunitary for a given diffuse trace is rather delicate. Methods.
The most difficult implication in Theorem A is to show that a diffusetracial state τ on a unital C ∗ -algebra A admits a Haar unitary. It suffices to con-struct a positive element in A with spectrum [0 ,
1] on which τ induces the Lebesguemeasure. To find such element, we first establish a correspondence between positiveelements in A and certain maps from ( −∞ ,
0] to the lattice O ( A ) of open projec-tions. More precisely, a positive element a induces a map f a : ( −∞ , → O ( A ),that sends t ≤ a + t ) + , and one can describe explic-itly which maps ( −∞ , → O ( A ) arise this way; see Proposition 2.6. Further, apositive element a has spectrum sp( a ) = [0 ,
1] and τ induces the Lebesgue measureon sp( a ) if and only if the associated map f a satisfies τ ( f a ( t )) = 1 + t for t ∈ [ − , f : ( −∞ , → O ( A ), it suffices to construct openprojections p t for dyadic rationals in [ − ,
0] such that τ ( p t ) = 1 + t and such that IFFUSE TRACES AND HAAR UNITARIES 3 p t ′ is compactly contained in p t in the sense of Definition 2.1, denoted p t ′ ≺ p t ,whenever t ′ < t . Such projections could easily be obtained by successive interpola-tion if we could show that for given p ≺ q in O ( A ) and t with τ ( p ) < t < τ ( q ), thereexists r ∈ O ( A ) such that p ≺ r ≺ q and τ ( r ) = t . In Lemma 4.5, we establishan approximate version of this interpolation result, which suffices to construct thedesired map ( −∞ , → O ( A ).The crucial assumption in Lemma 4.5 is that τ has ‘no gaps’ in the sense thatfor every open projection p , the set { τ ( p ′ ) : p ′ ∈ O ( A ) , p ′ ≤ p } is dense in[0 , τ ( p )]. The key observation is that this ‘no gaps’ property holds if (and onlyif) τ is nowhere scattered , which means that it gives no weight to scattered ideal-quotients; see Definition 3.5. This even holds for arbitrary positive functionals,and in Theorem 4.11 we provide several characterizations for a functional to benowhere scattered. The final ingredient is that a diffuse trace is nowhere scattered;see Proposition 3.9. Notation.
Given a C ∗ -algebra A , we use A + to denote the positive elements in A .By an ideal in a C ∗ -algebra we always mean a closed, two-sided ideal. Given a ∈ A ,we use sp( a ) to denote its spectrum, and we let supp( a ) denote the support pro-jection in A ∗∗ . Given a Hilbert space H , B ( H ) denotes the C ∗ -algebra of bounded,linear operators on H , and K ( H ) is the ideal of compact operators.2. Open projections
Let A be a C ∗ -algebra. A projection p ∈ A ∗∗ is said to be open if there exists anincreasing net in A + that converges to p in the weak*-topology; a projection p ∈ A ∗∗ is said to be closed if 1 − p is open; see [Ake69, Definition II.1]. Given an openprojection p , the sub- C ∗ -algebra pA ∗∗ p ∩ A of A is hereditary and p is the weak*-limit of any approximate unit of pA ∗∗ p ∩ A . Conversely, given a hereditary sub- C ∗ -algebra B ⊆ A , there exists a (unique) open projection p such that B = pA ∗∗ p ∩ A .For details we refer to [Ped79, p.77f].We let O ( A ) denote the collection of open projections in A ∗∗ , and we considerit as a subset of the complete lattice Proj( A ∗∗ ) of projections in A ∗∗ . By [Ake69,Proposition II.5], the infimum of an arbitrary family of closed projections is againclosed. Given any projection p ∈ A ∗∗ , this allows one to define its closure p asthe smallest closed projection that majorizes p . Dually, O ( A ) is closed under ar-bitrary suprema in Proj( A ∗∗ ). Given a family ( p j ) j of open projections, the openprojection W j p j corresponds to the hereditary sub- C ∗ -algebra of A generated by A ∩ S j p j A ∗∗ p j . We note that the infimum V j p j in Proj( A ∗∗ ) is in general strictlylarger than the open projection _ (cid:8) p ∈ O ( A ) : p ≤ p j for all j (cid:9) , which corresponds to the hereditary sub- C ∗ -algebra A ∩ T j p j A ∗∗ p j . Thus, O ( A ) isnaturally isomorphic to the lattice of hereditary sub- C ∗ -algebras studied in [AB15]. Definition 2.1.
We define an auxiliary relation ≺ on O ( A ) by setting p ≺ q for p, q ∈ O ( A ) if there exists a ∈ A + with p ≤ a ≤ q .One can show that p ≺ q if and only if p is compactly contained in q in thesense of [ORT11, Definition 3.6]. Using Theorem II.5 and Lemma III.1 in [Ake71],it follows that p ≺ q if and only if p ≤ q and there exists b ∈ A + such that p ≤ b .Thus, if A is unital, then p ≺ q if and only if p ≤ q . Lemma 2.2.
Let a be a positive, contractive element, and let e be a projection ina C ∗ -algebra. Then e ≤ a if and only if e = ea . Further, we have a ≤ e if and onlyif a = ae . HANNES THIEL
Proof. If e ≤ a ≤
1, then e ≤ eae ≤ e and so[ e (1 − a ) / ][ e (1 − a ) / ] ∗ = e (1 − a ) e = 0 . Hence, e (1 − a ) / = 0, which implies e (1 − a ) = 0, that is, a = ea . Conversely, if e = ea , then e = aea ≤ a , and so e = e / ≤ ( a ) / = a .Further, if a ≤ e , then 0 ≤ (1 − e ) a (1 − e ) ≤ (1 − e ) e (1 − e ) = 0 and so[ a / (1 − e )] ∗ [ a / (1 − e )] = 0. Hence a / (1 − e ) = 0, and we get a (1 − e ) = 0,that is, a = ae ; see [Bla06, Proposition II.3.3.2]. Conversely, if a = ae , then a = eae ≤ e = e . (cid:3) It follows from Lemma 2.2 that open projections p, q satisfy p ≺ q if and onlyif there exists a positive element a such that p = pa and a = aq . The next resultsummarizes basic properties of the relation ≺ . Lemma 2.3.
Let A be a C ∗ -algebra. Then the following statements hold:(1) Let p, q, r, s ∈ O ( A ) satisfy p ≤ q ≺ r ≤ s . Then p ≺ s .(2) Let p, q ∈ O ( A ) satisfy p ≺ q . Then there exists q ′ ∈ O ( A ) with p ≺ q ′ ≺ q .Proof. Statement (1) is obvious. To verify (2), let a ∈ A + satisfy p ≤ a ≤ q . Then p = pa and a = aq . Let f, g : R → [0 ,
1] be continuous functions such that f takesthe value 0 on [0 , ] and the value 1 on [1 , ∞ ), while g takes the value 0 on { } and the value 1 on [ , ∞ ). Note that f g = f . Set q ′ = supp( f ( a )) ∈ O ( A ). Then p ≤ g ( a ) ≤ q ′ ≤ f ( a ) ≤ q , which shows that p ≺ q ′ ≺ q . (cid:3) The following definition is inspired by the notion of paths in Q -semigroups (cer-tain directed complete, partially ordered, abelian semigroups equipped with anauxiliary relation) from [APT19, Paragraph 2.12]. Definition 2.4. A path in O ( A ) is an order-preserving map f : ( −∞ , → O ( A )satisfying the following conditions:(1) f ( t ) = sup { f ( t ′ ) : t ′ < t } for every t ∈ ( −∞ , f ( t ′ ) ≺ f ( t ) for all t ′ < t in ( −∞ , f is bounded if there exist t ∈ ( −∞ ,
0] such that f ( t ) = 0. Wethen define the length of f as l ( f ) := sup {| t | : f ( t ) = 0 } .Every positive element in A defines a bounded path in O ( A ): Lemma 2.5.
Let A be a C ∗ -algebra and a ∈ A + . Define f a : ( −∞ , → O ( A ) by f a ( t ) := supp(( a + t ) + ) for t ∈ ( −∞ , . Then f a is a bounded path in O ( A ) with length l ( f a ) = k a k . Below, we show that every bounded path in O ( A ) is induced by an elementin A + . It follows that positive elements in A correspond to bounded paths in O ( A ). However, the (pointwise) order on paths does not correspond to the usualorder on A + , but to the spectral order introduced by Olsen in [Ols71]. We notethat the next result is closely related to [Ake71, Theorem I.1]. Proposition 2.6.
Let A be a C ∗ -algebra and let f : ( −∞ , → O ( A ) be a boundedpath. Then there exists a unique positive element a ∈ A such that f = f a , that is, f ( t ) = supp(( a + t ) + ) for every t ∈ ( −∞ , .Proof. Uniqueness: Let a, b ∈ A + . By [Ols71, Theorem 3], a is dominated by b inthe spectral order if and only if a n ≤ b n for every n ≥
1. In particular, f a ≤ f b implies a ≤ b , and f a = f b implies a = b . Existence:
Set Z ≤ := Z ∩ ( −∞ , n ≥
1. For each k ∈ Z ≤ , using that f ( k − n ) ≺ f ( k n ), we obtain a n,k ∈ A + such that f ( k − n ) ≤ a n,k ≤ f ( k n ) . IFFUSE TRACES AND HAAR UNITARIES 5
For t ≤ k − n , we have f ( t ) ≤ a n,k . By Lemma 2.2, this implies that a n,k commuteswith f ( t ). Similarly, we obtain that a n.k commutes with f ( t ) for every t ≥ k n . Inparticular, a n,k commutes with f ( l n ) for every l ∈ Z ≤ . We set a n := X k = −∞ n a n,k , which is a finite sum since a n,k = 0 for k < n l ( f ), where l ( f ) is the lenght of f .We deduce that a n commutes with f ( l n ) for every l ∈ Z ≤ .For each m ≥ n , we have X k ∈ Z ≤ n f ( k − n ) ≤ a m ≤ X k ∈ Z ≤ n f ( k n ) . (1)Hence, k a n − a n +1 k ≤ n for each n , and it follows that ( a n ) n is a Cauchy sequence.We set a := lim n a n , which is a positive element in A .Let n ≥
1. It follows from (1) that X j = −∞ n f ( j − n ) ≤ a ≤ X j = −∞ n f ( j n ) . For each j , since a m commutes with f ( j n ) for m ≥ n , it follows that a commuteswith f ( j n ). Consequently, a commutes with P j = −∞ n f ( j − n ). Given commuting self-adjoint elements x, y in a C ∗ -algebra that satisfy x ≤ y , and given t ∈ R ,it follows that ( x + t ) + ≤ ( y + t ) + . (For noncommuting elements, this does notnecessarily hold.) Given k ∈ Z ≤ , we have X j = −∞ n f ( j n ) − k n + = k − X j = −∞ n f ( j n ) , and therefore k − X j = −∞ n f ( j n ) ≤ ( a − k n ) + ≤ k X j = −∞ n f ( j n ) . It follows that f ( k − n ) ≤ supp(( a − k n ) + ) ≤ f ( k n )for every n ≥ k ∈ Z , k ≤
0. Using condition (1) in the definition of apath, we obtain f ( k n ) = sup l ≥ f ( l k − n + l ) ≤ supp(( a − k n ) + ) , and so supp(( a − k n ) + ) = f ( k n ) for every n ≥ k ∈ Z , k ≤
0. It followsthat f agrees with f a at every nonpositive dyadic number. Since both f and f a arepaths, we deduce that f = f a . (cid:3) Diffuse and nowhere scattered functionals
In this section, we first recall basic properties of diffuse and atomic function-als. We then introduce the main technical concept of this paper: a positive func-tional is nowhere scattered if it gives no weight to scattered ideal-quotients; seeDefinition 3.5. Equivalently, the functional gives no weight to elementary ideal-quotients, or it vanishes on minimal projections in quotients of the algebra; seeProposition 3.7. (A projection p in a C ∗ -algebra A is said to be minimal it p = 0and pAp = C p .) In the next section, we prove that nowhere scattered functionalsadmit Haar unitaries. HANNES THIEL
We show that every diffuse functional is nowhere scattered; see Proposition 3.9.A C ∗ -algebra is type I if and only if every of its nowhere scattered states is diffuse;see Proposition 3.12. In forthcoming work, we will study the class of C ∗ -algebraswith the property that every positive functional is nowhere scattered. Let A be a C ∗ -algebra. By a positive functional on A we mean a positive,linear map ϕ : A → C . Every positive functional is bounded and therefore extendsuniquely to a normal, positive functional A ∗∗ → C which we also denote by ϕ .A positive functional ϕ is diffuse if ϕ ( e ) = 0 for every minimal projection e in A ∗∗ . Equivalently, ϕ ( z at ) = 0, where z at ∈ A ∗∗ denotes the supremum of allminimal projections in A ∗∗ . A positive functional ϕ is atomic if ϕ (1 − z at ) = 0.The notions of atomic and diffuse functionals on a C ∗ -algebra were introduced byPedersen in [Ped71] using the concept of Baire operators. It is straightforward toverify that the definitions in [Ped71] are equivalent to the ones above, and alsoequivalent to [Jen77, Definition 1.1].By [Ped71, Proposition 4], a positive functional ϕ is atomic if and only if ϕ = P ∞ k =1 α k ϕ k for a sequence ( ϕ k ) k of pure states and positive coefficients ( α k ) k with P k α k < ∞ ; see also [Jen77, Theorem 1.2].Every positive functional ϕ admits a unique decomposition as a sum of an atomic,positive functional ϕ a and a diffuse, positive functional ϕ d . With z at ∈ A ∗∗ asabove, we have ϕ a ( a ) = ϕ ( az at ) , and ϕ d ( a ) = ϕ ( a (1 − z at ))for a ∈ A . It follows that a positive functional is diffuse if and only if it does notdominated a nonzero multiple of a pure state.In the next result, we use π ϕ : A → B ( H ϕ ) to denote the GNS-representationassociated to a positive functional ϕ , and we let π ϕ ( A ) ′′ ⊆ B ( H ϕ ) denote thegenerated von Neumann algebra. Recall that a von Neumann algebra is diffuse ifit contains no minimal projections. It is called atomic (sometimes ‘purely atomic’)if its unit is the supremum of minimal projections; equivalently, it is a product oftype I factors. Lemma 3.2.
Let ϕ : A → C be a positive functional on a C ∗ -algebra A . Then ϕ is diffuse (atomic) if and only if π ϕ ( A ) ′′ is diffuse (atomic).Proof. Let s ϕ denote the support projection of ϕ in A ∗∗ , and let c ( s ϕ ) denote itscentral cover. We have π ϕ ( A ) ′′ ∼ = c ( s ϕ ) A ∗∗ ;see for example [Bla06, III.2.2.23f]. Let z at ∈ A ∗∗ denote the supremum of allminimal projections in A ∗∗ .Now ϕ is diffuse if and only if ϕ ( z at ) = 0, which is equivalent to s ϕ ≤ − z at .Since z at is central, this is also equivalent to c ( s ϕ ) ≤ − z at . Given a centralprojection z ∈ A ∗∗ , note that zA ∗∗ is diffuse if and only if z ≤ − z at . Thus, ϕ isdiffuse if and only if c ( s ϕ ) A ∗∗ is diffuse. The atomic case is proved analogously. (cid:3) Example 3.3.
Let X be a locally compact, Hausdorff space. Positive functionalson C ( X ) naturally correspond to bounded, positive Borel measures on X . Givena measure µ , the corresponding functional ϕ µ : C ( X ) → C is given by ϕ µ ( f ) = R X f ( x ) dµ ( x ). Then ϕ µ is diffuse if and only if µ is diffuse, that is, has not atoms. A C ∗ -algebra A is scattered if every positive functional on A is atomic; see[Jen77, Definition 2.1]. This is known to be equivalent to many other properties.For example, A is scattered if and only if every self-adjoint element in A has count-able spectrum, if and only if A has a composition series ( K α ) ≤ α ≤ β such that thesuccessive quotients K α +1 /K α are elementary (a C ∗ -algebra is elementary if it is IFFUSE TRACES AND HAAR UNITARIES 7 isomorphic to the algebra of compact operators on some Hilbert space); see [GK18,Theorem 1.4] and [Jen78, Theorem 2].Given a C ∗ -algebra A , an ideal-quotient of A is a (closed, two-sided) ideal of aquotient of A . Using the correspondence between ideals (quotients) of a C ∗ -algebraand open (closed) subsets of its primitive ideal space, it follows that ideal-quotientsof A correspond to locally closed subsets of the primitive ideal space of A . Definition 3.5.
A positive functional ϕ : A → C on a C ∗ -algebra A is nowherescattered if k ϕ | I k = k ϕ | J k for all ideals I ⊆ J ⊆ A such that J/I is scattered.
Remark 3.6.
Let ϕ : A → C be a positive functional on a C ∗ -algebra A . Givenideals I ⊆ J ⊆ A , let p, q ∈ O ( A ) be the corresponding central open projections.Then k ϕ | I k = ϕ ( p ) and k ϕ | J k = ϕ ( q ). Thus, we have k ϕ | I k = k ϕ | J k if and only if ϕ ( p ) = ϕ ( q ), that is, ϕ ( q − p ) = 0. Proposition 3.7.
Let A be a C ∗ -algebra, and let ϕ : A → C be a positive functional.Then the following are equivalent:(1) ϕ is nowhere scattered;(2) k ϕ | I k = k ϕ | J k for all ideals I ⊆ J ⊆ A such that J/I is elementary;(3) for every ideal I ⊆ A and every minimal projection e ∈ A/I , we have ϕ ( e ) = 0 (viewing e ∈ A/I ⊆ ( A/I ) ∗∗ ⊆ A ∗∗ ).Proof. To show that (3) implies (2), let I ⊆ J ⊆ A be ideals such that J/I iselementary. Let p, q ∈ O ( A ) denote the central, open projections correspondingto I and J . We can choose an approximate unit of finite-rank projections ( e λ ) λ in J/I . It follows from the assumption that ϕ ( e λ ) = 0 for each λ . We get ϕ ( q − p ) = ϕ (cid:0) sup λ e λ (cid:1) = 0 . To show that (2) implies (1), let I ⊆ J ⊆ A be ideals such that J/I is scattered.Let π : J → J/I denote the quotient map. By [Jen78, Theorem 2], there exists acomposition series ( K α ) ≤ α ≤ β for J/I such that K α +1 /K α is elementary for each α < β . In particular, K = { } and K β = J/I . For each α ≤ β , let p α ∈ O ( A ) bethe central open projection corresponding to the ideal π − ( K α ).Using transfinite induction, we show that ϕ ( p α − p ) = 0 for each α ≤ β . Thisis clear for α = 0. Assuming that it holds for some α , we use that K α +1 /K α iselementary and thus ϕ ( p α +1 − p α ) = 0 to deduce that ϕ ( p α +1 − p ) = 0. If α is alimit ordinal and we have ϕ ( p α ′ − p ) = 0 for every α ′ < α , then ϕ ( p α − p ) = ϕ (cid:18) sup α ′ <α ( p α ′ − p ) (cid:19) = sup α ′ <α ϕ ( p α ′ − p ) = 0 . Thus, ϕ ( p β − p ) = 0, and so k ϕ | I k = ϕ ( p ) = ϕ ( p β ) = k ϕ | J k . To show that (1) implies (3), let I ⊆ A be an ideal, and let e ∈ A/I be aminimal projection. Let π : A → A/I denote the quotient map. Let K ⊆ A/I denote the sub- C ∗ -algebra generated by all minimal projections in A/I . By [GK18,Theorem 1.2], K is scattered and an ideal of A/I . Thus, if p and q denote thecentral, open projections corresponding to the ideals I and π − ( K ), then ϕ ( q − p ) =0. Since e ≤ q − p , it follows that ϕ ( e ) = 0. (cid:3) Lemma 3.8.
Let A be a C ∗ -algebra, let B ⊆ A be a hereditary sub- C ∗ -algebra,and let ϕ : A → C be a nowhere scattered, positive functional. Then ϕ | B is nowherescattered. HANNES THIEL
Proof.
We use the characterization of unscattered functionals through minimal pro-jections in quotients from condition (3) in Proposition 3.7. Let I ⊆ B be an ideal,and let e ∈ B/I be a minimal projection. We need to show that ϕ | B ( e ) = 0. Let J ⊆ A be the ideal of A generated by I , and let π : A → A/J denote the quotientmap. We have I = B ∩ J , and so π ( B ) is a hereditary sub- C ∗ -algebra of A/J and π ( B ) ∼ = B/I . Thus, e is also a minimal projection of A/J . Since ϕ is nowherescattered, we get ϕ ( e ) = 0, and so ϕ | B ( e ) = 0. (cid:3) Proposition 3.9.
Every diffuse, positive functional is nowhere scattered.Proof.
Let A be a C ∗ -algebra, and let ϕ : A → C be a diffuse, positive functional.Let I ⊆ A be an ideal, and let e ∈ A/I be a minimal projection. Then e is also aminimal projection in ( A/I ) ∗∗ , and thus in A ∗∗ . Since ϕ is diffuse, we get ϕ ( e ) = 0.By Proposition 3.7, it follows that ϕ is nowhere scattered. (cid:3) The converse of Proposition 3.9 does not hold. In fact, we will show that a C ∗ -algebra is not type I if and only if it has a nowhere scattered state that is pure (andtherefore not diffuse); see Proposition 3.12.A nowhere scattered functional is diffuse if it is also tracial (Proposition 5.3) orif it is a normal functional on a von Neumann algbera (Lemma 7.1). Lemma 3.10.
Let ϕ be a pure state on a C ∗ -algebra A , and let π ϕ : A → B ( H ϕ ) be the induced GNS-representation. Then ϕ is nowhere scattered if and only if π ϕ ( A ) ∩ K ( H ϕ ) = { } .Proof. Set L := ker( π ϕ ). The restriction of a pure state to an ideal is either a purestate or zero. Thus, given an ideal I ⊆ A , we either have k ϕ | I k = 1 (which happensprecisely if I is not contained in L ) or ϕ | I = 0 (which happens if and only if I ⊆ L ).To show that forward implication, assume that π ϕ ( A ) ∩ K ( H ϕ ) = { } . Since π ϕ is irreducible, it follows that K ( H ϕ ) ⊆ π ϕ ( A ); see [Bla06, Corollary IV.1.2.5] Set J := π − ϕ ( K ( H ϕ )). Then J/L ∼ = K ( H ϕ ) is elementary and k ϕ | L k = 0 < k ϕ | J k ,showing that ϕ is not nowhere scattered.To show the converse implications, assume that ϕ is not nowhere scattered.Choose ideals I ⊆ J ⊆ A such that J/I is scattered and such that k ϕ | I k < k ϕ | J k .This forces k ϕ | J k = 1, and consequently J is not contained in L . It follows that therestriction of π ϕ to J is a nonzero, irreducible representation. Since J is scattered,and therefore of type I, it follows that π ϕ ( J ) contains a nonzero, compact operator.Hence, π ϕ ( A ) ∩ K ( H ϕ ) = { } . (cid:3) Lemma 3.11.
Let ϕ and ψ be positive functionals on a C ∗ -algebra satisfying ψ ≤ ϕ .Assume that ϕ is nowhere scattered. Then ψ is nowhere scattered.Proof. Let I ⊆ J ⊆ A be ideals such that J/I is scattered. Let p and q be thecentral, open projections corresponding to I and J , respectively. Then k ψ | J k − k ψ | I k = ψ ( q − p ) ≤ ϕ ( q − p ) = k ϕ | J k − k ϕ | I k = 0 , as desired. See also Remark 3.6. (cid:3) Proposition 3.12.
Let A be a C ∗ -algebra. Then A is of type I if and only if everynowhere scattered state on A is diffuse.Proof. To prove the forward implication, assume that A is of type I. By Proposition 3.9,it remains to verify that every nowhere scattered functional is diffuse. To prove thecontraposition, let ϕ be a positive functional on A that is not diffuse. Then thereexists a pure state ψ and t > ψ ≤ tϕ ; see Paragraph 3.1. Since A istype I, we have π ψ ( A ) ∩ K ( H ψ ) = { } and it follows from Lemma 3.10 that ψ isnot nowhere scattered. By Lemma 3.11, neither is tϕ , which implies that ϕ is notnowhere scattered. IFFUSE TRACES AND HAAR UNITARIES 9
To show that backward implication, assume that A is not of type I. Then thereexists a pure state ϕ such that π ϕ is not GCR, that is, π ϕ ( A ) ∩ K ( H ϕ ) = { } . ByLemma 3.10, ϕ is nowhere scattered, yet not diffuse. (cid:3) Haar unitaries characterize nowhere scattered functionals
The main result of this section is Theorem 4.11, which provides several charac-terizations for a functional to be nowhere scattered. Most interestingly, a positivefunctional A → C is nowhere scattered if and only if the minimal unitization of ev-ery hereditary sub- C ∗ -algebra of A contains a Haar unitary. It follows in particularthat every positive functional on a unital, simple, infinite-dimensional C ∗ -algebraadmits a Haar unitary; see Corollary 4.14.Every positive functional ϕ : A → C extends uniquely to a normal, positivefunctional A ∗∗ → C , which we also denote by ϕ . The induced map O ( A ) → [0 , ∞ ), p ϕ ( p ), is order-preserving and satisfies ϕ (sup j p j ) = sup j ϕ ( p j ) for everyincreasing net ( p j ) j in O ( A ). Given p ∈ O ( A ), it is easy to see that p is thesupremum of the set { p ′ ∈ O ( A ) : p ′ ≺ p } . However, this set is not necessarilyupward-directed. Nevertheless, we have: Proposition 4.1.
Let ϕ : A → C be a positive functional, let p, q ∈ O ( A ) satisfy p ≺ q , and let ε > . Then there exists q ′ ∈ O ( A ) such that p ≺ q ′ ≺ q, and ϕ ( q ) − ε ≤ ϕ ( q ′ ) . In particular, we have ϕ ( q ) = sup (cid:8) ϕ ( q ′ ) : q ′ ∈ O ( A ) , q ′ ≺ q (cid:9) . Proof.
Choose a ∈ A + such that p ≤ a ≤ q . Let C ∗ ( a, q ) denote the sub- C ∗ -algebra of A ∗∗ generated by a and q , and let sp( a ) denote the spectrum of a , whichis a closed subset of [0 , C ∗ ( a, q ) and C (sp( a )). The restriction of ϕ to C ∗ ( a, q ) induces a measure µ ϕ on sp( a ), whichwe view as a measure on [0 , s, t such that0 ≤ s < t ≤ , and µ ϕ (( s, t )) < ε. Then choose continuous functions f, g, h : [0 , → [0 ,
1] that take the value 0 on[0 , s ], that take the value 1 on [ t, f = f g and g = gh . Theelements f ( a ), g ( a ) and h ( a ) belong to A + and satisfy p ≤ f ( a ) , f ( a ) = f ( a ) g ( a ) , and g ( a ) = g ( a ) h ( a ) ≤ q. Set z := supp( q − h ( a )) ∈ A ∗∗ . To see that z is an open projection, let ( b j ) j be an increasing net in A + with q = sup j b j . Then (( q − h ( a )) b j ( q − h ( a )) j is anincreasing net in A + with supremum ( q − h ( a )) . Then z = supp (cid:0) q − h ( a ) (cid:1) = supp (cid:0) ( q − h ( a )) (cid:1) = sup j supp (cid:0) ( q − h ( a )) b j ( q − h ( a )) (cid:1) , which shows that z ∈ O ( A ). We have q − h ( a ) ≤ z and therefore µ ϕ ([ t, ≤ ϕ ( q − h ( a )) ≤ ϕ ( z ) . Set w := supp( f ( a )) ∈ O ( A ). We have f ( a ) ≤ w and therefore µ ϕ ([0 , s ]) ≤ ϕ ( f ( a )) ≤ ϕ ( w ) . Note that f ( a ) = f ( a ) g ( a ) implies that w = wg ( a ). Similarly, it follows from q − h ( a ) = ( q − g ( a ))( q − h ( a )) that z = ( q − g ( a )) z . Hence, wz = w ( q − g ( a )) z = 0 . We have µ ϕ ([0 , ϕ ( q ) and therefore ϕ ( w + z ) ≥ µ ϕ ([0 , s ] ∪ [ t, µ ϕ ([0 , − µ ϕ (( s, t )) > ϕ ( q ) − ε. Let ( c j ) j be an increasing net in A + with supremum z . Using that ϕ ( z ) =sup j ϕ ( c j ), choose j such that ϕ ( q ) − ε < ϕ ( w ) + ϕ ( c j ) . For each δ > c j − δ ) + be the element obtained by applying functionalcalculus for the function t max { , t − δ } to z j . Using that c j = sup δ> ( c j − δ ) + ,choose δ > ϕ ( q ) − ε < ϕ ( w ) + ϕ (cid:0) ( c j − δ ) + (cid:1) . Let e : [0 , → [0 ,
1] be a continuous function with e (0) = 0 and taking thevalue 1 on [ δ, (cid:0) ( z j − δ ) + (cid:1) ≤ e (cid:0) ( z j − δ ) + (cid:1) ≤ z, w ≤ g ( a ) , and zg ( a ) = 0 . Thus, q ′ := supp(( z j − δ ) + ) + w is an open projection satisfying q ′ ≤ e (cid:0) ( z j − δ ) + (cid:1) + g ( a ) ≤ q and thus q ′ ≺ q . Further, we have p ≤ f ( a ) ≤ w ≤ q ′ and thus p ≺ q ′ . Lastly, we have ϕ ( q ) − ε < ϕ ( w ) + ϕ (cid:0) ( c j − δ ) + (cid:1) ≤ ϕ ( w ) + ϕ (cid:0) supp(( c j − δ ) + ) (cid:1) = ϕ ( q ′ ) , as desired. (cid:3) Given a positive functional ϕ : A → C , it is well-known that the set L ϕ := (cid:8) a ∈ A : ϕ ( a ∗ a ) = 0 (cid:9) is a closed, left ideal; see [Ped79, Theorem 3.3.3]. Since a ∈ A satisfies ϕ ( a ∗ a ) = 0if and only if ϕ (supp( a ∗ a )) = 0, we have L ϕ := (cid:8) a ∈ A : ϕ (supp( a ∗ a )) = 0 (cid:9) .Analogously, L ∗ ϕ = { a ∈ A : ϕ (supp( aa ∗ )) = 0 } is a closed, right ideal, and L ϕ ∩ L ∗ ϕ = (cid:8) a ∈ A : ϕ (supp( a ∗ a )) = ϕ (supp( aa ∗ )) = 0 (cid:9) is a hereditary sub- C ∗ -algebra of A . The next result shows that L ϕ ∩ L ∗ ϕ is evena (closed, two-sided) ideal if 0 is isolated in ϕ ( O ( A )). It actually suffices that ϕ ( O ( A )) contains a sufficiently large gap near zero. Lemma 4.2.
Let A be a C ∗ -algebra, let ϕ : A → C be a nonzero, positive functional,and let δ ∈ (0 , k ϕ k ) such that ( δ, δ ] ∩ ϕ ( O ( A )) = ∅ . Then I := (cid:8) a ∈ A : ϕ (supp( aa ∗ )) , ϕ (supp( a ∗ a )) ≤ δ (cid:9) is an ideal in A .Proof. Set U := (cid:8) p ∈ O ( A ) : ϕ ( p ) ≤ δ (cid:9) , and z := _ U ∈ O ( A ) . Given p, q ∈ U , we have ϕ ( p ∨ q ) ≤ ϕ ( p + q ) ≤ δ. The assumption on the gap in ϕ ( O ( A )) implies that p ∨ q belongs to U . Thus, U is upward-directed and using that ϕ : A ∗∗ → C is a normal functional, we get ϕ ( z ) = ϕ ( _ U ) = sup p ∈ U ϕ ( p ) ≤ δ. Thus, z is the largest element in U . It follows that I is the hereditary sub- C ∗ -algebra of A corresponding to z .To show that I is a two-sided ideal, let U ( e A ) denote the subgroup of unitariesin the minimal unitization of A that are connected to the unit. Given a ∈ I and IFFUSE TRACES AND HAAR UNITARIES 11 u ∈ U ( e A ), choose a continuous path [0 , → U ( e A ), t u t , with u = 1 and u = u . For t ∈ [0 , u t aa ∗ u t ) = u t supp( aa ∗ ) u t . Thus, t supp( u t aa ∗ u t ) is a continuous path of open projections in A ∗∗ , and so t ϕ (supp( u t aa ∗ u t )) is a continuous map [0 , → R . The gap in ϕ ( O ( A )) impliesthat ϕ (supp( uaa ∗ u )) ≤ δ. Since also ϕ (supp( a ∗ uua )) = ϕ (supp( a ∗ a )) ≤ δ , we deduce that ua belongs to I .Using that every element in e A is a finite linear combination of elements in U ( e A ),it follows that I is a left ideal. Analogously, we obtain that I is a right ideal. (cid:3) Lemma 4.3.
Let A be a C ∗ -algebra, let ϕ : A → C be a nonzero, positive functional,and let δ ∈ (0 , k ϕ k ) such that ( δ, δ ] ∩ ϕ ( O ( A )) = ∅ . Then there exists an ideal I ⊆ A such that A/I is finite-dimensional and k ϕ | I k < k ϕ k .Proof. Set I := (cid:8) a ∈ A : ϕ (supp( aa ∗ )) , ϕ (supp( a ∗ a )) ≤ δ (cid:9) , which is an ideal in A by Lemma 4.2. Let z ∈ A ∗∗ denote the corresponding central,open projection. By assumption, we have δ < k ϕ k and therefore k ϕ | I k = ϕ ( z ) ≤ δ < k ϕ k . Set δ := ϕ ( z ). Then ( δ , δ ] ∩ ϕ ( O ( A )) = ∅ . Set B := A/I , and let π : A → B denote the quotient map. Using the natural identification of B ∗∗ with (1 − z ) A ∗∗ ,we define ψ : B → C as the composition B ⊆ B ∗∗ ∼ = (1 − z ) A ∗∗ ⊆ A ∗∗ ϕ −→ C . Claim: The functional ψ : B → C is faithful. To prove the claim, let b ∈ B + satisfy ψ ( b ) = 0. Lift b to find a ∈ A + with π ( a ) = b . Let p ∈ O ( A ) be the supportprojection of a . The natural isomorphism (1 − z ) A ∗∗ ∼ = B ∗∗ identifies (1 − z ) p withthe support projection of b , and we get ϕ (cid:0) (1 − z ) p (cid:1) = ψ (cid:0) supp( b ) (cid:1) = 0 . Since p ≤ z + (1 − z ) p , we obtain that ϕ ( p ) ≤ ϕ ( z ) + ϕ (cid:0) (1 − z ) p (cid:1) = δ ≤ δ. We get a ∈ I and so b = 0, which proves the claim. Claim: We have (0 , δ − δ ] ∩ ψ ( O ( B )) = ∅ . To prove the claim, assume that p ∈ O ( B ) satisfies ψ ( p ) ∈ (0 , δ − δ ]. We identify B ∗∗ with (1 − z ) A ∗∗ ⊆ A ∗∗ . Set¯ p := z + p . Then ¯ p is the open projection in A ∗∗ corresponding to the hereditarysub- C ∗ -algebra π − ( pB ∗∗ p ∩ B ) ⊆ A . We have ϕ (¯ p ) = ϕ ( z + p ) = δ + ψ ( p ) ∈ ( δ , δ ] . This contradicts that ( δ , δ ] ∩ ϕ ( O ( A )) = ∅ . The claim is proved.It follows from the above claims that ψ : B → C is faithful and that 0 is isolatedin ψ ( O ( B )). This implies that every positive element in B has finite spectrum.Using that a C ∗ -algebra is finite-dimensional if (and only if) every of its positiveelements has finite spectrum, we deduce that B is finite-dimensional. (cid:3) Given p ∈ O ( A ), we use [0 , p ] to denote the set { q ∈ O ( A ) : q ≤ p } . Lemma 4.4.
Let A be a C ∗ -algebra, and let ϕ : A → C be a nowhere scattered,positive functional. Then ϕ ([0 , p ]) is dense in [0 , ϕ ( p )] for every p ∈ O ( A ) . Proof.
Let p ∈ O ( A ). To reach a contradiction, assume that ϕ ([0 , p ]) is not densein [0 , ϕ ( p )]. Choose t ∈ [0 , ϕ ( p )) and δ > t, t + 2 δ ) ∩ ϕ ([0 , p ]) = ∅ . Wemay assume that t = sup (cid:8) ϕ ( p ′ ) : p ′ ∈ [0 , p ] , ϕ ( p ′ ) ≤ t (cid:9) , which allows us to choose p ′ ∈ O ( A ) satisfying p ′ ≤ p and t − δ < ϕ ( p ′ ). ApplyProposition 4.1 to obtain p ′′ ∈ O ( A ) satisfying p ′′ ≺ p ′ , and t − δ < ϕ ( p ′′ ) . Then p ′′ ≤ p ′ ≤ p , and we set q := p − p ′′ , which is an open projection. We have ϕ ( p ) ≥ t + 2 δ and ϕ ( p ′′ ) ≤ t , and therefore ϕ ( q ) = ϕ ( p ) − ϕ ( p ′′ ) ≥ ϕ ( p ) − ϕ ( p ′′ ) ≥ t + 2 δ − t = 2 δ. Claim: We have ( δ, δ ] ∩ ϕ ([0 , q ]) = ∅ . To prove the claim, assume that an openprojection q ′ ≤ q satisfies ϕ ( q ′ ) ∈ ( δ, δ ]. Since p ′′ and q are orthogonal and satisfy p ′′ + q ≤ p , we get that p ′′ + q ′ ∈ [0 , p ]. On the other hand, ϕ ( p ′′ + q ′ ) = ϕ ( p ′′ ) + ϕ ( q ′ ) ∈ ( t − δ, t ] + ( δ, δ ] ⊆ ( t, t + 2 δ ] , which contradicts that ( t, t + 2 δ ] ∩ ϕ ([0 , p ]) = ∅ . This proves the claim.Set B := qA ∗∗ q ∩ A , the hereditary sub- C ∗ -algebra of A corresponding to q .By construction, the restriction of ϕ to B satisfies the assumptions of Lemma 4.3.Hence, ϕ | B is not nowhere scattered, which contradicts Lemma 3.8. (cid:3) Lemma 4.5.
Let A be a C ∗ -algebra, let ϕ : A → C be a positive functional. Assumethat ϕ ([0 , z ]) is dense in [0 , ϕ ( z )] for every z ∈ O ( A ) . Let p, ˜ p, q ∈ O ( A ) and t, ˜ t ∈ [0 , satisfy p ≺ ˜ p ≺ q, and ϕ (˜ p ) ≤ t < ˜ t ≤ ϕ ( q ) . Then there exist r, ˜ r ∈ O ( A ) satisfying p ≺ r ≺ ˜ r ≺ q, and t ≤ ϕ ( r ) ≤ ϕ (˜ r ) ≤ ˜ t. Proof.
Choose ε > t < ˜ t − ε. Apply Proposition 4.1 to obtain e ∈ O ( A ) such that p ≺ e ≺ ˜ p, and ϕ (˜ p ) − ε ≤ ϕ ( e ) . Then e ≺ q , which allows us to set z := q − e ∈ O ( A ). We have q ≤ ˜ p + z andtherefore 0 < ˜ t − ϕ (˜ p ) ≤ ϕ ( q ) − ϕ (˜ p ) ≤ ϕ ( z ) . By assumption, we obtain z ′ ∈ O ( A ) such that z ′ ≤ z, and ϕ ( z ′ ) ∈ (˜ t − ϕ (˜ p ) − ε, ˜ t − ϕ (˜ p )] . Set f := ˜ p ∨ z ′ ∈ O ( A ). Then ϕ ( f ) = ϕ (˜ p ∨ z ′ ) ≤ ϕ (˜ p ) + ϕ ( z ′ ) ≤ ϕ (˜ p ) + ˜ t − ϕ (˜ p ) = ˜ t. On the other hand, we have e + z ′ ≤ f and therefore ϕ ( f ) ≥ ϕ ( e ) + ϕ ( z ′ ) ≥ ϕ (˜ p ) − ε + ˜ t − ϕ (˜ p ) − ε = ˜ t − ε > t. We also have p ≺ ˜ p ≤ f . Applying Proposition 4.1 twice, we find ˜ r and then r suchthat p ≺ r ≺ ˜ r ≺ f, and t < ϕ ( r ) . Then r and ˜ r have the desired properties. (cid:3) IFFUSE TRACES AND HAAR UNITARIES 13
Lemma 4.6.
Let A be a C ∗ -algebra, let ϕ : A → C be a positive functional. Assumethat ϕ ([0 , p ]) is dense in [0 , ϕ ( p )] for every p ∈ O ( A ) . Then there exist a path f : ( −∞ , → O ( A ) (in the sense of Definition 2.4) satisfying f ( −k ϕ k ) = 0 and ϕ ( f ( t )) = k ϕ k + t for t ∈ [ −k ϕ k , . By Proposition 2.6, f corresponds to a positive element a ∈ A with spectrum [0 , k ϕ k ] on which ϕ induces the Lebesgue measure.Proof. We may assume that ϕ is nonzero, and by rescaling we may also assumethat k ϕ k = 1. Using Lemma 4.5, we inductively find p ( n ) k , ˜ p ( n ) k ∈ O ( A ) for n ≥ k = 1 , . . . , n such that p ( n )1 ≺ ˜ p ( n )1 ≺ p ( n )2 ≺ ˜ p ( n )2 ≺ . . . ≺ p ( n )2 n ≺ ˜ p ( n )2 n ≺ n ≥
1, and such that p ( n ) k ≺ p ( n +1)2 k ≺ ˜ p ( n +1)2 k ≺ p ( n +1)2 k +1 ≺ ˜ p ( n +1)2 k +1 ≺ p ( n ) k +1 and k − n < ϕ ( p ( n ) k ) ≤ ϕ (˜ p ( n ) k ) ≤ k − n + n +1 for each n ≥ k = 1 , . . . , n .Given n ≥ k ∈ { , . . . , n } , we have p ( n ) k ≺ p ( n +1)2 k ≺ p ( n +2)2 k ≺ ... ≺ p ( n + m )2 m k ≺ . . . and we set g ( k n ) := sup m p ( n + m )2 m k ∈ O ( A ) . Note that this is well-defined, since if k n = k ′ n ′ for some other k ′ , n ′ ≥
1, then thesequence ( p ( n ′ + m )2 m k ′ ) m either contains ( p ( n + m )2 m k ) m as a subsequence (if n ′ ≤ n ) or viceverse (if n ≤ n ′ ). Setting g (0) = 0, we have defined g ( t ) for every dyadic rationalin [0 , Claim 1: Let n ≥ and k ∈ { , . . . , n } . Then ϕ ( g ( k n )) = k n . Indeed, for each m ≥
1, we have k n − n + m = m k − n + m ≤ ϕ (cid:0) p ( n + m )2 m k (cid:1) ≤ m k − n + m + n + m +1 ≤ k n , and therefore ϕ (cid:0) g ( k n ) (cid:1) = ϕ (cid:0) sup m p ( n + m )2 m k (cid:1) = sup m ϕ (cid:0) p ( n + m )2 m k (cid:1) = k n . Claim 2: Let t ′ , t ∈ [0 , be dyadic rationals satisfying t ′ < t . Then g ( t ′ ) ≺ g ( t ) . To prove the claim, choose n ≥ k ′ , k ∈ { , . . . , n } such that t ′ = k ′ n and t = k n . For each m ≥
1, we have p ( n + m )2 m k ′ ≺ p ( n + m − m − k ′ +1 ≺ p ( n + m − m − k ′ +1 ≺ . . . ≺ p ( n +1)2 k ′ +1 ≺ p ( n ) k ′ +1 and therefore g ( t ′ ) = g ( k ′ n ) = sup m p ( n + m )2 m k ′ ≤ p ( n ) k ′ +1 ≤ p ( n ) k ≺ p ( n +1)2 k ≤ sup m p ( n + m )2 m k = g ( k n ) = g ( t ) , which proves the claim.We now define f : ( −∞ , → O ( A ) by f ( t ) = 0 for t ≤ − f ( t ) := sup (cid:8) g ( k n ) : k n < t (cid:9) for t ∈ ( − , f is order-preserving, and that f ( t ) = sup { f ( t ′ ) : t ′ < t } for every t .To verify condition (2) of Definition 2.4, let t ′ < t ≤
0. If t ′ ≤ −
1, then f ( t ′ ) = 0 ≺ f ( t ). Otherwise, choose dyadic rationals s ′ , s ∈ (0 ,
1] such that1 + t ′ ≤ s ′ < s < t. Using Claim 2, we get f ( t ′ ) ≤ g ( s ′ ) ≺ g ( s ) ≤ f ( t ) . Finally, let t ∈ ( − , s k ) k of dyadic numbersin [0 , t ) with supremum 1 + t . Then f ( t ) = sup k g ( s k ). Using Claim 1, we get ϕ ( f ( t )) = ϕ (cid:0) sup k g ( s k ) (cid:1) = sup k ϕ ( g ( s k )) = sup k s k = 1 + t. By Proposition 2.6, there is a unique positive element a ∈ A such that f ( t ) =supp(( a + t ) + ) for each t ≤
0. Let σ ( a ) denote the spectrum of a , and let µ be themeasure on σ ( a ) induced by ϕ . For every t ≥
0, we have µ (( t, ∞ ) ∩ σ ( a )) = ϕ (supp(( a − t ) + )) = ϕ ( f ( − t )) = 1 − t, which implies that σ ( a ) = [0 ,
1] and that µ is the Lebesgue measure on σ ( a ). (cid:3) Combining Lemmas 4.4 and 4.6, we obtain:
Proposition 4.7.
Let A be a C ∗ -algebra, and let ϕ : A → C be a nowhere scattered,positive functional. Then there exists a positive element a ∈ A with spectrum [0 , k ϕ k ] on which ϕ induces Lebesgue measure. It will be convenient to generalize the notion of Haar unitaries to the setting ofpositive functionals that are not necessarily states or tracial.
Definition 4.8.
Let A be a unital C ∗ -algebra, and let ϕ : A → C be a positivefunctional. A Haar unitary in A with respect to ϕ is a unitary u ∈ A such that ϕ ( u k ) = 0 for every k ∈ Z \ { } . Proposition 4.9.
Let A be a unital C ∗ -algebra, and let ϕ : A → C be a positivefunctional. Then the following are equivalent:(1) There exists a Haar unitary in A with respect to ϕ .(2) There exists a positive element a ∈ A with spectrum [0 , k ϕ k ] on which ϕ induces the Lebesgue measure.(3) There exists a unital, abelian sub- C ∗ -algebra C ( X ) ⊆ A such that ϕ inducesa diffuse measure on X .(4) There exists a maximal abelian subalgebra (masa) D ⊆ A such that ϕ in-duces a diffuse measure on the spectrum of D .Proof. The statements hold for ϕ = 0. We may thus assume that ϕ is nonzero. Toshow that (2) implies (1), let a ∈ A be as in (2). Set u := exp(2 πi a k a k ), which is aunitary in A . Given k ∈ Z \ { } , we have ϕ ( u k ) = Z k a k exp(2 πi t k a k ) k dt = k a k Z exp(2 πit ) k dt = 0 . It is clear that (4) implies (3). To show that (3) implies (2), let C ( X ) ⊆ A beas in the statement. Set ψ := ϕ | C ( X ) : C ( X ) → C . By assumption, ψ is diffuse (seealso Example 3.3). By Proposition 3.9, ψ is nowhere scattered. Since, k ψ k = k ϕ k ,by applying Proposition 4.7, we obtain the desired positive element in C ( X ). (Thisalso follows from [DHR97, Proposition 4.1(i)].)To show that (1) implies (4), let u ∈ A be a Haar unitary. Then C ∗ ( u ), the sub- C ∗ -algebra of A generated by u , is naturally isomorphic to C ( T ), and ϕ inducesthe multiple of the Lebesgue measure on T with total mass k ϕ k . Choose anymasa D ⊆ A that contains u , and let X be a compact, Hausdorff space such that D ∼ = C ( X ). The inclusion C ( T ) ∼ = C ∗ ( u ) ⊆ D ∼ = C ( X ) corresponds to a surjective,continuous map X → T , and since ϕ induces a diffuse measure on T , it also doeson X . (See also Lemma 5.2.) (cid:3) IFFUSE TRACES AND HAAR UNITARIES 15
Lemma 4.10.
Let A be a unital C ∗ -algebra, let ϕ : A → C be a positive functionalthat admits a Haar unitary, and let I ⊆ A be an ideal such that A/I is scattered.Then k ϕ | I k = k ϕ k .Proof. Let π : A → A/I denote the quotient map. Choose a Haar unitary u in A with respect to ϕ , and let B ⊆ A be the sub- C ∗ -algebra generated by u . Then B ∼ = C ( T ), and the measure µ induced by ϕ on T is a multiple of the Lebesguemeasure.The ideal I ∩ B of B corresponds to a proper open subset U ⊆ T such that thequotient π ( B ) of B is naturally isomorphic to C ( T \ U ). Since A/I is scattered, sois π ( B ), and it follows that T \ U is countable. Hence, µ ( T \ U ) = 0, which impliesthat µ ( U ) = µ ( T ). We deduce that k ϕ | I k ≥ k ϕ | I ∩ B k = µ ( U ) = µ ( T ) = k ϕ k which implies the desired equality k ϕ | I k = k ϕ k . (cid:3) In the next result, given a hereditary subalgebra B ⊆ A with corresponding openprojection p ∈ A ∗∗ , we view the minimal unitization of B as e B = B + C p ⊆ A ∗∗ . Theorem 4.11.
Let A be a C ∗ -algebra, and let ϕ : A → C be a positive functional.Then the following are equivalent:(1) ϕ is nowhere scattered.(2) For every p ∈ O ( A ) , ϕ ([0 , p ]) is dense in [0 , ϕ ( p )] .(3) For every p ∈ O ( A ) and t ∈ [0 , ϕ ( p )) there exists p ′ ∈ O ( A ) with p ′ ≺ p and ϕ ( p ′ ) = t .(4) For every hereditary sub- C ∗ -algebra B ⊆ A there exists a Haar unitaryin e B .(5) For every ideal I ⊆ A there exists a Haar unitary in e I .Proof. By Lemma 4.4, (1) implies (2). By Lemma 4.6, (2) implies (3). It is clearthat (3) implies (2), and that (4) implies (5).To show that (5) implies (1), let I ⊆ J ⊆ A be ideals such that J/I is scattered.By assumption, there exists a Haar unitary in e J . The quotient e J/I is naturallyisomorphic to the forced unitization of
J/I and therefore scattered as well. ApplyingLemma 4.10 at the first step, we get k ϕ | I k = k ϕ | e J k = k ϕ | J k . To show that (2) implies (4), let B ⊆ A be a hereditary sub- C ∗ -algebra. ByLemma 3.8, ϕ | B is nowhere scattered. Applying Proposition 4.7, we obtain b ∈ B + with spectrum [0 , k ϕ | B k ] on which ϕ induces Lebesgue measure. Using that k ϕ | e B k = k ϕ | B k , it follows from Proposition 4.9 that e B contains a Haar unitary. (cid:3) Corollary 4.12.
Let A be a unital C ∗ -algebra that has no (nonzero) scattered ideal-quotients. Then every positive functional on A is nowhere scattered and thereforeadmits a Haar unitary. Example 4.13. A C ∗ -algebra A is purely infinite if it has no one-dimensionalrepresentations, and if an element a ∈ A + lies in the ideal generated by b ∈ A + ifand only if there exists a sequence ( r n ) n in A such that lim n →∞ k a − r n br ∗ n k = 0;see [KR00, Definition 4.1]. By [KR00, Theorem 4.19], pure infiniteness passes toideals and quotients. It follows that purely infinite C ∗ -algebras are not scattered,and that a purely infinite C ∗ -algebra has no scattered ideal-quotients.Hence, by Corollary 4.12, every state on a unital, purely infinite C ∗ -algebraadmits a Haar unitary.We point out the following important special case of Corollary 4.12. Corollary 4.14.
Let A be a unital, simple, nonelementary C ∗ -algebra. Then everypositive functionl ϕ : A → C admits a Haar unitary. Corollary 4.15.
Let A be a unital C ∗ -algebra. Assume that A contains a unitalsub- C ∗ -algebra B ⊆ A such that B has no scattered ideal-quotients. (For example, B is simple and nonelementary.) Then every positive functionl ϕ : A → C admitsa Haar unitary. Example 4.16.
Let A be a unital C ∗ -algebra of real rank zero that has no finite-dimensional representations. By [ER06, Corollary 2.4], A contains a unital, simple,nonelementary sub- C ∗ -algebra. Hence, by Corollary 4.15, every positive functionalon A admits a Haar unitary. Example 4.17.
Let ( δ n ) n ∈ Z be the standard orthonormal basis of ℓ ( Z ), and let U ∈ B ( ℓ ( Z )) be the bilateral shift satisfying U δ n = δ n +1 . Let ϕ : B ( ℓ ( Z )) → C bethe vector state induced by δ , that is, ϕ ( a ) = h aδ , δ i for a ∈ B ( ℓ ( Z )). Then ϕ ( U k ) = h U k δ , δ i = h δ k , δ i = ( , if k = 01 , if k = 0 . which shows that U is a Haar unitary with respect to ϕ . However, the ideal I := K ( ℓ ( Z )) of compact operators is elementary and satisfies k ϕ | I k = k ϕ k . Thus, ϕ isnot nowhere scattered.This shows that a positive functional admitting a Haar unitary is not necessarilynowhere scattered. In the next section, we show that this phenomenon does notoccur for tracial functionals.5. Traces admitting Haar unitaries
In this section, we prove the main result of this paper: A tracial state on a unital C ∗ -algebra admits a Haar unitary if and only if it is diffuse; see Theorem 5.4.By a trace on a C ∗ -algebra A we mean a positive functional τ : A → C thatis tracial: τ ( ab ) = τ ( ba ) for all a, b ∈ A . Every trace τ : A → C is bounded andtherefore extends uniquely to a normal trace A ∗∗ → C that we also denote by τ .Recall that τ is diffuse if τ ( e ) = 0 for every minimal projection e ∈ A ∗∗ . Lemma 5.1.
Let A be a C ∗ -algebra and let τ : A → C be a trace. Then the followingare equivalent:(1) τ is diffuse;(2) τ does not dominate a nonzero trace that factors through a finite-dimen-sional quotient of A ;(3) there is no surjective ∗ -homomorphism π : A → M n ( C ) (for some n ≥ )such that τ dominates a nonzero multiple of tr n ◦ π . where tr n denotes thetracial state on M n ( C ) .Proof. To show that (1) implies (2), assume that τ is diffuse, and let I ⊆ A be anideal such that A/I is finite-dimensional. Let z ∈ A ∗∗ be the central, open projec-tion corresponding to I . We have natural isomorphisms (1 − z ) A ∗∗ ∼ = ( A/I ) ∗∗ ∼ = A/I . Since τ vanishes on minimal projections, we get τ (1 − z ) = 0. It followsthat τ does not dominate a nonzero trace that factors through the quotient map A → A/I .It is clear that (2) implies (3). To show that (3) implies (1), let e ∈ A ∗∗ be aminimal projection. To reach a contradiction, assume that τ ( e ) >
0. Let c ( e ) ∈ A ∗∗ be the central cover of e . Then c ( e ) A ∗∗ is a type I factor. Since τ restricts to anonzero trace on c ( e ) A ∗∗ , we have c ( e ) A ∗∗ ∼ = M n ( C ) for some n . It follows that the IFFUSE TRACES AND HAAR UNITARIES 17 map π : A → M n ( C ), a c ( e ) a , is surjective. For each a ∈ A + , we have a ≥ c ( e ) a in A ∗∗ and therefore τ ( a ) ≥ τ ( c ( e ) a ) = τ ( c ( e )) · (tr n ◦ π )( a ) . Since τ ( c ( e )) ≥ τ ( e ) >
0, we have shown that τ dominates a nonzero multiple oftr n ◦ π , contradicting the assumption (3). (cid:3) Lemma 5.2.
Let A be a C ∗ -algebra, let τ : A → C be a trace, and let B ⊆ A be asub- C ∗ -algebra such that τ | B is diffuse and k τ | B k = k τ k . Then τ is diffuse.Proof. The inclusion map B → A induces a natural injective ∗ -homomorphism B ∗∗ → A ∗∗ . We let p ∈ A ∗∗ denote the projection that is the image of the unitof B ∗∗ . Note that p is the weak*-limit of any positive, increasing approximate unitof B in A ∗∗ . It follows that τ ( p ) = k τ | B k = k τ k = τ (1) , and thus τ (1 − p ) = 0.To show that τ is diffuse, let e ∈ A ∗∗ be a minimal projection, and let c ( e ) beits central cover. To reach a contradiction, assume that τ ( e ) >
0. Then c ( e ) A ∗∗ isa type I factor with a nonzero trace, and so c ( e ) A ∗∗ ∼ = M n ( C ) for some n . We let π : A → M n ( C ) denote the surjective ∗ -homomorphism a c ( e ) a .We distinguish two cases. If π ( B ) = { } , then c ( e ) p = 0, and it follows that e ≤ c ( e ) ≤ − p and therefore τ ( e ) = 0.If π ( B ) is nonzero, then π ( B ) is a nonzero finite-dimensional quotient of B .It follows that c ( e ) p belongs to B ∗∗ . Using that τ | B is diffuse we deduce that τ ( c ( e ) p ) = 0. It follows that τ ( c ( e )) = 0 and so τ ( e ) = 0. (cid:3) Proposition 5.3.
A trace on a C ∗ -algebra is diffuse if and only if it is nowherescattered.Proof. Let A be a C ∗ -algebra, and let τ : A → C be a trace. If τ is diffuse, thenit is nowhere scattered by Proposition 3.9. To show the converse, assume that τ isnowhere scattered. By Proposition 4.7, there exists a ∈ A + with spectrum [0 , k ϕ k ],on which ϕ induces Lebesgue measure. Let B ⊆ A be the sub- C ∗ -algebra generatedby a . Then B is commutative and ϕ | B is diffuse (see also Example 3.3). We have k ϕ | B k = k ϕ k , whence it follows from Lemma 5.2 that τ is diffuse. (cid:3) We summarize our findings for the case of a tracial state on a unial C ∗ -algebra.An analogous result holds for traces on nonunital C ∗ -algebras. Theorem 5.4.
Let A be a unital C ∗ -algebra, and let τ : A → C be a tracial state.Then the following are equivalent:(1) τ is diffuse;(2) the weak*-closure of A in the GNS-representation induced by τ is a diffusevon Neumann algebra;(3) τ is nowhere scattered;(4) τ does not dominate a nonzero trace that factors through a finite-dimen-sional quotient of A ;(5) there exists a ∈ A + with spectrum [0 , on which τ induces the Lebesguemeasure;(6) there exists a masa C ( X ) ⊆ A such that τ induces a diffuse measure on X ;(7) there exists a Haar unitary in A . Proof.
By Lemma 3.2, (1) and (2) are equivalent. By Proposition 5.3, (1) and (3)are equivalent. By Lemma 5.1, (1) and (4) are equivalent. By Theorem 4.11, (3)implies (7). By Proposition 4.9, (5), (6) and (7) are equivalent. Finally, it followsfrom Lemma 5.2 (see also Example 3.3) that (6) implies (1). (cid:3)
Corollary 5.5.
A unital C ∗ -algebra has no finite-dimensional representations ifand only if each of its tracial states admits a Haar unitary. Corollary 5.6.
Every trace on a unital, simple, nonelementary C ∗ -algebra is dif-fuse and admits a Haar unitary. We end this section with an example of a diffuse trace on a unital C ∗ -algebraand a masa that contains no Haar unitary. Example 5.7.
Let S : ℓ ( N ) → ℓ ( N ) be the one-sided shift, and let T := C ∗ ( S ) ⊆B ( ℓ ( N )) be the Toeplitz algebra. The compact operators K := K ( ℓ ( N )) are anideal in T with T / K ∼ = C ( T ), where T ⊆ C denotes the unit circle. We consider themasa ℓ ∞ ( N ) ⊆ B ( ℓ ( N )) and set B := ℓ ∞ ( N ) ∩ T . Then K ∩ B = c ( N ) ⊆ ℓ ∞ ( N ).Since the commutant of c ( N ) in B ( ℓ ( N )) is ℓ ∞ ( N ), we see that B is a masa in T .We let π : T → C ( T ) denote the quotient map. Let τ : C ( T ) → C be induced bythe normalized Lebesgue measure on T . Set ϕ := τ ◦ π , which is a diffuse tracialstate on T . Let ( e n ) n ∈ N be the standard basis in ℓ ( N ). We claim that ϕ ( a ) = lim n →∞ h ae n , e n i for each a ∈ T . Indeed, one can directly verify this for each S k ( S ∗ ) l for k, l ≥ T , the formulaholds for every a ∈ T . It follows that B = c ( N ) + C
1, and π ( B ) ⊆ C ( T ) containsonly the constant functions.Thus, every unitary u ∈ B satisfies | ϕ ( u ) | = 1. In particular, B contains noHaar unitary. To find a Haar unitary for ϕ , consider the function v : T → T satisfying v ( z ) = z for z with positive imaginary part, and satisfying v ( z ) = z − for z with negative imaginary part. Since v is of the form v = exp( ia ) for apositive element a ∈ C ( T ), we can lift v to a unitary u ∈ T with π ( u ) = v . Then ϕ ( u k ) = τ ( v k ) = R T v ( z ) k dz = 0 for k ∈ Z \ { } .6. States admitting Haar unitaries
Let ϕ : A → C be a positive functional on a unital C ∗ -algebra. In this section,we study when ϕ admits a Haar unitary. By Lemma 4.10, a necessary condition isthat ϕ gives no weight to scattered quotients of A . We conjecture that this is alsosufficient: Conjecture 6.1.
Let A be a unital C ∗ -algebra, and let ϕ : A → C be a positivefunctional. Then ϕ admits a Haar unitary if and only if there is no ideal I ⊆ A such that A/I is scattered and k ϕ | I k < k ϕ k .We confirm the conjecture in the case that ϕ is tracial (Theorem 5.4), and if A is a von Neumann algebra (Proposition 7.3).Recall that a topological space X is said to be T if for every x ∈ X the set { x } is closed. Lemma 6.2.
Let ϕ : A → C be a positive functional on a C ∗ -algebra A whoseprimitive ideal space is T . Then ϕ is nowhere scattered if and only if ϕ does notdominate a nonzero positive functional that factors through an elementary quotient.Proof. The forward implication is clear. To show the converse, assume that ϕ givesno weight to elementary quotients. Using that the primitive ideal space is T , it IFFUSE TRACES AND HAAR UNITARIES 19 follows that ϕ gives no weight to elementary ideal-quotients. By Proposition 3.7,this implies that ϕ is nowhere scattered. (cid:3) Proposition 6.3.
Let A be a unital C ∗ -algebra whose primitive ideal space is T ,and let ϕ : A → C be a positive functional. Then the following are equivalent:(1) ϕ is nowhere scattered;(2) ϕ admits a Haar unitary;(3) ϕ does not dominate a nonzero positive functional that factors through afinite-dimensional quotient.If A is also of type I , then these conditions are also equivalent to:(4) ϕ is diffuse.Proof. By Theorem 4.11, (1) implies (2). By Lemma 4.10, (2) implies (3). Sinceevery unital, elementary C ∗ -algebra is a matrix algebra and therefore finite-dimen-sional, it follows from Lemma 6.2 that (3) implies (1). If A is also of type I, thenthe equivalence of (1) and (4) follows from Proposition 3.12. (cid:3) Example 6.4.
Recall that a C ∗ -algebra A is liminal (also called CCR ) if for everyirreducible representation π : A → B ( H ) we have π ( A ) = K ( H ). Every liminal C ∗ -algebra is type I and its primitive ideal space is T . A unital C ∗ -algebra isliminal if and only if every of its irreducible representations is finite-dimensional.Proposition 6.3 verifies Conjecture 6.1 for liminal C ∗ -algebras.Recall that a C ∗ -algebra A is subhomogeneous if there exists n ∈ N such thatevery irreducible representation of A is at most n -dimensional. Every subhomoge-neous C ∗ -algebra is liminal. We obtain in particular that a positive functional on aunital, subhomogeneous C ∗ -algebra admits a Haar unitary if and only if it does notdominate a nonzero positive functional that factors through a finite-dimensionalquotient. 7. States on von Neumann algberas
In this section, we study when (normal) states on von Neumann algebras admitHaar unitaries. We first show that a normal state is diffuse if and only if it isnowhere scattered. We deduce that normal states on diffuse von Neumann alge-bras admit Haar unitaries. However, it turns out that diffuseness is not necessary.Indeed, the main result of this section, Theorem 7.4, shows that every state ona von Neumann algebra without finite-dimensional representations admits a Haarunitary. In particular, every state on B ( H ) admits a Haar unitary; see Remark 7.5. Lemma 7.1.
Let M be a von Neumann algebra, and let ϕ : M → C be a normal ,positive functional. Then the following are equivalent:(1) ϕ is diffuse (in the sense of Paragraph 3.1);(2) ϕ ( e ) = 0 for every minimal projection e ∈ M ;(3) ϕ is nowhere scattered.Proof. By Proposition 3.9, (1) implies (3). By Proposition 3.7, (3) implies (2). Toshow that (2) implies (1), assume that ϕ vanishes on every minimal projection in M , and let e be a minimal projection in M ∗∗ . Given a Banach space E , we use κ E : E → E ∗∗ to denote the natural inclusion. Let M ∗ denote the predual of M and set π := κ ∗ M ∗ : M ∗∗ ∼ = ( M ∗ ) ∗∗∗ → ( M ∗ ) ∗ ∼ = M. Then π is a ∗ -homomorphism satisfying π ◦ κ M = id M .Set ¯ e := π ( e ). Then ¯ e is a projection in M . To see that it is minimal, let x ∈ M .Since e is minimal in M ∗∗ , there exists λ ∈ C such that exe = λe . Then¯ ex ¯ e = π ( e ) π ( x ) π ( e ) = π ( exe ) = π ( λe ) = λ ¯ e. Thus, either ¯ e is zero, or ¯ e is a minimal projection in M . In either case, we have ϕ (¯ e ) = 0. Using that ϕ belongs to M ∗ , we get ϕ ( e ) = h κ M ∗ ( ϕ ) , e i M ∗∗∗ ,M ∗∗ = h ϕ, κ ∗ M ∗ ( e ) i M ∗ ,M = ϕ (¯ e ) = 0 . (cid:3) Recall that a von Neumann algbera is said to be diffuse if it contains no minimalprojections.
Proposition 7.2.
A von Neumann algebra M is diffuse if and only if every normalstate on M is diffuse.Proof. The forward implication follows from Lemma 7.1. To show the converse,assume that M is not diffuse. Choose a minimal projection e in M , and let c ( e ) beits central cover. Then c ( e ) M is a type I factor summand. Let H be a Hilbert spacesuch that c ( e ) M ∼ = B ( H ), and choose a unit vector ξ ∈ e ( H ). The correspondingvector state ϕ : B ( H ) → C , a
7→ h aξ, ξ i is normal and satisfies ϕ ( e ) = 1. Then M → C , a ϕ ( c ( e ) a ), is a normal state that is not diffuse. (cid:3) Let M be a diffuse von Neumann algebra, and let ϕ : M → C be a normal state.It follows from Lemma 7.1 and Proposition 7.2 that ϕ is nowhere scattered andtherefore admits a Haar unitary by Theorem 4.11. This is well-known and followsfor instance using that every maximal abelian sub- C ∗ -algebra (masa) D ⊆ M is adiffuse, abelian von Neumann algebra, that ϕ | D is a normal trace, and that everynormal trace on a diffuse, abelian von Neumann algebra admits a Haar unitary.Thus, every masa D ⊆ M contains a Haar unitary with respect to ϕ . (For C ∗ -algebras, this does not hold; see Example 5.7.) We note that this only holds for normal states. Indeed, given a masa D ⊆ M , we may choose a pure state on D and extend it to a state ψ on M . Then D contains no Haar unitary with respect to ψ . Nevertheless, in many cases we can find a different masa that contains a Haarunitary for ψ . Indeed, in Theorem 7.4 we will show that this is always possible if M has no finite-dimensional representations.Example 4.17 shows that a normal state on a von Neumann algebra may admita Haar unitary without being diffuse. Proposition 7.3.
Let M be a von Neumann algebra, and let ϕ : M → C be apositive functional. Then the following are equivalent:(1) ϕ admits a Haar unitary;(2) ϕ does not dominate a nonzero positive functional that factors through afinite-dimensional quotient of M .Proof. By Lemma 4.10, (1) implies (2). To show the converse, assume that ϕ givesno weight to finite-dimensional quotients of M . For each n ≥
1, let z n be the centralprojection in M such that z n M is the type I n summand of M . It follows from theassumption that the restriction of ϕ to z n M gives no weight to finite-dimensionalquotients. Since every irreducible representation of z n M is n -dimensional, we canapply Proposition 6.3 (see also Example 6.4) to obtain a Haar unitary u n ∈ z n M .Set z < ∞ := P ∞ n =1 z n and z ∞ := 1 − P ∞ n =1 z n . Set u := P ∞ n =1 u n ∈ z < ∞ M .Under the identification of z < ∞ M with Q ∞ n =1 z n M , the unitary u correspondts to( u n ) ∞ n =1 . It follows that u is a Haar unitary in z < ∞ M .If z ∞ = 0, then u is the desired Haar unitary. So assume that z ∞ = 0.Then z ∞ M admits no finite-dimensional representations. It follows that z ∞ M contains a unital, simple, nonelementary sub- C ∗ -algebra (for example, the hyperfi-nite II factor). Hence, the restriction of ϕ to z ∞ M admits a Haar unitary v ; seeCorollary 4.15. (Since von Neumann algebras have real rank zero, this also followsfrom Example 4.16.) Now u + v is the desired Haar unitary. (cid:3) IFFUSE TRACES AND HAAR UNITARIES 21
Theorem 7.4.
Let M be a von Neumann algebra. Then the following are equiva-lent:(1) M has no finite-dimensional representations;(2) every state on M admits a Haar unitary;(3) every tracial state on M admits a Haar unitary.Proof. It is clear that (2) implies (3), and that (3) implies (1). By Proposition 7.3,(1) implies (2). (cid:3)
Remark 7.5.
Let H be a separable, infinite-dimensional Hilbert space. By Theorem 7.4,every state on B ( H ) admits a Haar unitary and consequently restricts to a diffusestate on some masa. This should be contrasted with the result of Akemann andWeaver, [AW08], that the continuum hypothesis implies the existence of a purestate on B ( H ) that does not restrict to a pure state on any masa.8. Traces on reduced group C*-algebras
Let G be a discrete group, and let ℓ ( G ) be the associated Hilbert space withcanonical orthonormal basis ( δ g ) g ∈ G . The left-regular representation λ G is therepresentation of G on ℓ ( G ) that maps g ∈ G to the unitary u g ∈ B ( ℓ ( G ))satisfying u g δ h := δ gh for h ∈ G . The sub- C ∗ -algebra of B ( ℓ ( G )) generated by { u g : g ∈ G } is called the reduced group C ∗ -algebra of G , denoted by C ∗ red ( G ). Thevector δ ∈ ℓ ( G ) induces a canonical tracial state τ G : C ∗ red ( G ) → C given by τ G ( a ) := h aδ , δ i for a ∈ C ∗ red ( G ). We have τ G ( u ) = 1 and τ G ( u g ) = 0 for g ∈ G \ { } . It followsthat u g is a Haar unitary with respect to τ G if and only if g has infinite order in G .The von Neumann algebra generated by { u g : g ∈ G } is called the group vonNeumann algebra of G , denoted L ( G ). Proposition 8.1.
Let G be a discrete group. Then the following are equivalent:(1) G is infinite;(2) the trace τ G : C ∗ red ( G ) → C is diffuse;(3) the trace τ G : C ∗ red ( G ) → C admits a Haar unitary;(4) L ( G ) is diffuse.Proof. By [Dyk93, Proposition 5.1], (1) implies (4). Conversely, if G is finite,then L ( G ) is finite-dimensional and therefore not diffuse. By Theorem 5.4, (2)and (3) are equivalent. Note that L ( G ) is the weak*-closure of C ∗ red ( G ) underthe GNS-representation induced by τ G . Therefore, (2) and (4) are equivalent byLemma 3.2. (cid:3) Example 8.2.
Let G be an infinite, discrete group. Then τ G admits a Haar unitary.However, if G is a torsion group (such as G = T / Z ), then none of the canonicalunitaries u g ( g ∈ G ) is a Haar unitary since every element in G has finite order.If G is locally finite, then even more is true: There exists no Haar unitary for τ G in the group algebra C [ G ]. Indeed, given u ∈ C [ G ], since u has finite support andsince G is locally finite, there exists a finite subgroup F ⊆ G such that u belongsto C [ F ]. But C [ F ] is a finite-dimensional algebra and therefore does not containHaar unitaries. Thus, to find a Haar unitary for τ G , one really needs to go to thecompletion C ∗ red ( G ) of C [ G ]. Proposition 8.3.
Let G be a discrete group. Then the following are equivalent:(1) G is nonamenable;(2) C ∗ red ( G ) has no finite-dimensional representations;(3) every trace on C ∗ red ( G ) admits a Haar unitary. Proof.
Although the equivalence between (1) and (2) is well-known, we could notlocate it explicitly stated in the literature. In one direction, if G is amenable,then the trivial representation is weakly contained in λ G , which induces a one-dimensional representation of C ∗ red ( G ).Conversely, assume that C ∗ red ( G ) has a (unital) representations π : C ∗ red ( G ) → M n ( C ) for some n . Let tr n denote the tracial state on M n ( C ). Using the Arvesonextension theorem ([Bla06, Theorem II.6.9.12]), we obtain a completely positivecontraction ˜ π : B ( ℓ ( G )) → M n ( C ) that extends π . For each g ∈ G , we have˜ π ( u ∗ g u g ) = π (1) = 1 = π ( u g ) ∗ π ( u g ) = ˜ π ( u g ) ∗ ˜ π ( u g ) , which implies that u g belongs to the multiplicative domain of ˜ π , that is, ˜ π ( u ∗ g au g ) =˜ π ( u g ) ∗ ˜ π ( a )˜ π ( u g ) for every a ∈ B ( ℓ ( G )); see [Bla06, Proposition II.6.9.18]. Hence,the restriction of tr n ◦ ˜ π to ℓ ∞ ( G ) is an invariant mean on G , which shows that G is amenable. The maps are shown in the following commutative diagram: ℓ ∞ ( G ) (cid:31) (cid:127) / / B ( ℓ ( G )) ˜ π % % ❑❑❑❑❑ C ∗ red ( G ) ?(cid:31) O O π / / M n ( C ) tr n / / C . By Corollary 5.5, (2) and (3) are equivalent. (cid:3)
Let G be a discrete group. Consider the following properties:(1) G contains a subgroup isomorphic to F , the free group on two generators;(2) C ∗ red ( G ) contains a unital, simple, nonelementary sub- C ∗ -algebra;(3) every state on C ∗ red ( G ) admits a Haar unitary;(4) C ∗ red ( G ) has no finite-dimensional representations;(5) G is nonamenable.Then the following implications hold:(1) ⇒ (2) ⇒ (3) ⇒ (4) ⇔ (5) . Indeed, (1) implies that the simple, nonelementary C ∗ -algebra C ∗ red ( F ) unitallyembeds into C ∗ red ( G ); by Corollary 4.15, (2) implies (3); and by Proposition 8.3,(3) implies (4), which is equivalent to (5).Recall that G is C ∗ -simple if C ∗ red ( G ) is simple. Obviously, every C ∗ -simplegroup satisfies (2). By [OO14], there exist C ∗ -simple groups that have no noncyclic,free subgroups. Hence, the implication ’(1) ⇒ (2)’ can not reversed. What aboutthe other implications? Conjecture 6.1 predicts that (4) implies (3). Does (3)imply (2)? 9. Structure of reduced free products
Let A and B be unital C ∗ -algebras with faithful tracial states τ A and τ B , re-spectively. The reduced free product of ( A, τ A ) and ( B, τ B ) is the (unique) unital C ∗ -algebra C with faithful tracial state τ C and unital embeddings A ⊆ C and B ⊆ C such that τ C restrict to the given traces on A and B , such that A and B generated C as a C ∗ -algebra, and such that A and B are free with respect to τ C ,that is, τ C ( c c · · · c n ) = 0 whenever τ C ( c j ) = 0 for all j and either c , c , . . . ∈ A and c , c , . . . ∈ B , or vice versa.One can think of this construction as a generalization of the free product ofgroups: Given discrete groups G and H , the reduced free product of the reducedgroup C ∗ -algebras C ∗ red ( G ) and C ∗ red ( H ) with respect to their canonical tracialstates is naturally isomorphic to C ∗ red ( G ∗ H ). IFFUSE TRACES AND HAAR UNITARIES 23
It is a well-studied problem to determine when a reduced free product C is simpleor has stable rank one (that is, the invertible elements in C are dense). In [Avi82],Avitzour introduced the condition, later named after him, that there are unitaries u, v ∈ A and w ∈ B satisfying τ A ( u ) = τ A ( v ) = τ A ( uv ) = 0 , and τ B ( w ) = 0 . By [Avi82, Proposition 3.1], Avitzour’s condition implies that C is simple and hasa unique tracial state. By [DHR97, Theorem 3.8], Avitzour’s condition also impliesthat C has stable rank one.It is clear that Avitzour’s condition is satisfied if τ A and τ B admit Haar unitaries.Thus, it follows from Theorem 5.4 that the reduced free product of two C ∗ -algebraswith respect to diffuse (faithful) tracial states is a simple C ∗ -algebra of stable rankone and with unique tracial state. Using a result of Dykema, [Dyk99, Theorem 2],it even suffices that one trace is diffuse and the other algebra is nontrivial: Theorem 9.1.
Let A and B be unital C ∗ -algebras with faithful tracial states τ A and τ B , respectively. Assume that τ A is diffuse and that B = C . Then the reducedfree product of ( A, τ A ) and ( B, τ B ) is simple, has stable rank one and a uniquetracial state.Proof. By Theorem 5.4, A contains a unial, commutative sub- C ∗ -algebra C ( X )such that τ A induces a diffuse measure on X . This verifies the assumptions of[Dyk99, Theorem 2], which proves the result. (cid:3) Corollary 9.2.
Let A and B be unital, simple C ∗ -algebras with tracial states τ A and τ B , respectively. Assume that A = C and B = C . Then the reduced free productof ( A, τ A ) and ( B, τ B ) is simple, has stable rank one and a unique tracial state.Proof. If A is infinite-dimensional, then τ A is diffuse by Corollary 5.6 and the resultfollows from Theorem 9.1. The same argument applies if B is infinite-dimensional.If both A and B are finite-dimensional, then A ∼ = M m ( C ) and B ∼ = M n ( C ) forsome m, n ≥
2, and in this case one can directly verify that Avitzour’s condition issatisfied; see [DHR97, Proposition 4.1(iv)] (cid:3)
Corollary 9.3.
The class of unital, simple, stable rank one C ∗ -algebras with uniquetracial state is closed under reduced free products.Proof. Let A and B be two C ∗ -algebras in the considered class. If A ∼ = C , then thereduced free product is isomorphic to B , which belongs to the class, and similarlyif B ∼ = C . We may therefore assume that A = C and B = C . Now the result followsfrom Corollary 9.2. (cid:3) Let A and B be unital C ∗ -algebras with faithful states ϕ and ψ , respectively.Let ( C, γ ) be the reduced free product of (
A, ϕ ) and (
B, ψ ), which is defined anal-ogously to the tracial setting. One can show that γ is tracial if and only if ϕ and ψ are.The centralizer of A with respect to ϕ is defined as A ϕ := (cid:8) a ∈ A : ϕ ( ab ) = ϕ ( ba ) for all b ∈ A (cid:9) . Note that A ϕ is a unital sub- C ∗ -algebra of A , and the restriction of ϕ to A ϕ istracial. In this setting, Avitzour’s condition is that there exist unitaries u, v ∈ A ϕ and w ∈ B ψ satisfying ϕ ( u ) = ϕ ( v ) = ϕ ( uv ) = 0 , and ψ ( w ) = 0 . Avitzour’s condition still implies that the reduced free product is simple. By[Dyk99, Proposition 3.2], C is also simple if B = C and if there is a unital sub- C ∗ -algebra C ( X ) ⊆ A ϕ such that ϕ induces a diffuse measure on X . By Theorem 5.4, the condition on A ϕ is satisfied if and only if ϕ | A ϕ is a diffuse trace. In particular,we obtain that C is simple if ϕ is a diffuse trace on A and B = C . Proposition 9.5.
Let A and B be unital, simple, nonelementary C ∗ -algebras withstates ϕ and ψ , respectively, and let ( C, γ ) be their reduced free product. If ϕ and ψ are tracial, then C has stable rank one. If ϕ or ψ is not tracial, then C is properlyinfinite.Proof. The first statement follows from Corollary 9.2. To show the second state-ment, assume that ϕ or ψ is not tracial. Then γ is not tracial either. By Corollary 4.14, ϕ and ψ admit Haar unitaries. They do not necessarily lie in the centralizers of ϕ and ψ , but this is also not required to apply [DR98, Theorem 4], which gives that C is properly infinite. (cid:3) Problem 9.6.
Describe when the centralizer of a state contains a Haar unitary.
Remark 9.7.
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