AF-embeddability for Lie groups with T 1 primitive ideal spaces
aa r X i v : . [ m a t h . OA ] M a y AF-EMBEDDABILITY FOR LIE GROUPSWITH T PRIMITIVE IDEAL SPACES
INGRID BELTIT¸ ˘A AND DANIEL BELTIT¸ ˘A
Abstract.
We study simply connected Lie groups G for which the hull-kerneltopology of the primitive ideal space Prim( G ) of the group C ∗ -algebra C ∗ ( G ) is T , that is, the finite subsets of Prim( G ) are closed. Thus, we prove that C ∗ ( G )is AF-embeddable. To this end, we show that if G is solvable and its action onthe centre of [ G, G ] has at least one imaginary weight, then Prim( G ) has nononempty quasi-compact open subsets. We prove in addition that connectedlocally compact groups with T ideal spaces are strongly quasi-diagonal. Introduction
Considerable attention has been recently paid to finite-dimensional approxima-tion properties of C ∗ -algebras, such as quasi-diagonality or embeddability into a C ∗ -algebra that is an inductive limit of finite-dimensional C ∗ -algebras —for short,AF-embeddability.In the present paper we study these properties for C ∗ -algebras that occur inthe noncommutative harmonic analysis, that is, the C ∗ -algebras of connected Liegroups. In sharp contrast to the C ∗ -algebras of countable discrete groups, we havealready found in [2] that there exist solvable (hence amenable) connected Lie groupswhose corresponding C ∗ -algebras are not quasi-diagonal, and in particular they arenot AF-embeddable. It turns out however that affirmative results can be obtainedunder natural topological assumptions on the primitive ideal space, as describedbelow.The primitive ideal space Prim( G ) of a locally compact group G is the primitiveideal space of the group C ∗ -algebra C ∗ ( G ), endowed with its hull-kernel topology.One of the main results of this paper is: If G is a simply connected Lie groupfor which Prim( G ) is T , then C ∗ ( G ) is AF-embeddable (Theorem 5.3). We recallthat the class of connected Lie groups G for which Prim( G ) is T was characterizedin [17] and [22] in Lie algebraic terms. Using that characterization, it turns outthat it simultaneously contains many differing classes of groups solvable Lie groupsthat are not necessarily of type I, such as the celebrated Mautner groups, Dixmiergroups, the diamond groups, the rigid motion groups, nilpotent Lie groups andcompact groups.If the topological condition T is not required, then C ∗ ( G ) is still AF-embeddablewhenever G is a connected and simply connected solvable Lie group and its actionon the centre of the commutator group [ G, G ] (which is nilpotent) has a purely
Key words and phrases. solvable Lie group; group C ∗ -algebra; quasi-orbit.2020 Mathematics Subject Classification.
Primary 22D25; Secondary 22E27. imaginary weight (Corollary 4.5). We recall that, without the weight conditions,connected, simply connected and solvable Lie groups may not be AF-embeddable(see [2, Th. 2.15]).Our approach to the above AF-embeddability result is based on a preliminaryinvestigation of the dual topology of the solvable Lie groups, a topic that, despitesome remarkable works, remains notoriously difficult even for some type I solvableLie groups. Our key technical result shows that if G is a second countable locallycompact group which has a closed normal subgroup L such that A := G/L isabelian and the closure of every orbit of G in b L is the spectrum of the restriction ofa unitary representation of G , then there is a homeomorphism of quasi-orbit spaces(Prim( G ) / b A ) ∼ ≃ (Prim( L ) /G ) ∼ . (See Theorem 3.6.) Using the results of [21], we show that the above conditions on G are satisfied for arbitrary connected and simply connected solvable Lie groups, with L = [ G, G ]. What the above homeomorphism facilitates for us is to check that, whenthe action of G action on the centre of L has at least one purely imaginary weight,Prim( G ) has no nonempty quasi-compact open subsets (Theorem 4.1), which fur-ther allows an application of a result of [12]. As an application of Theorem 5.3, wefinally prove that if A is the C ∗ -algebra of a simply connected solvable Lie group,then the following implication holds true: if the image of A in every irreduciblerepresentation is AF-embeddable, then A itself is AF-embeddable (Corollary 6.7).Apart for the case when A is a type I C ∗ -algebra, very few other instances of thatimplication seem to be known in the literature, although the analogous implicationfor quasi-diagonality has long been known and is fairly elementary.Our paper is organized as follows: In Section 2 we establish some preliminaryfacts belonging to three topics: topologies of the spaces of closed subsets of a topo-logical space, weak containment of representations, and the technique of weightsapplied to studying the topology of the space of quasi-orbits of a linear dynamicalsystem. In Section 3 we establish the basic homeomorphism theorem on quasi-orbitspaces mentioned above. In Section 4 we then obtain a key result on absence ofnon-empty open quasi-compact subsets of Prim( G ) if G is a simply connected solv-able Lie group such that the action of G action on the centre of [ G, G ] has at leastone purely imaginary weight. As a consequence, we obtain absence of non-emptyopen quasi-compact subsets of Prim( G ) if G is a simply connected solvable Liegroup of type R . Earlier results of this type can be found in [9, § C ∗ ( G ) if G is a simply connectedLie group for which Prim( G ) is T . Finally, in Section 6 we show that if G is asimply connected solvable Lie group, then the T condition on Prim( G ) is equiva-lent to the condition that the image of C ∗ ( G ) in every irreducible representation isAF-embeddable, and this hence implies that C ∗ ( G ) is itself AF-embeddable by theresults of Section 5. In addition, we prove that every connected, locally compactgroup with T primitive ideal space is strongly quasi-diagonal (Corollary 6.6). ROUPS WITH T PRIMITIVE IDEAL SPACE 3 Preliminaries
We prove here some preliminary results for later use. For the notions and nota-tion in C ∗ -algebras and topology, we refer the reader to [10], [6], [29] and [5].2.1. The space of closed subsets of a topological space.
We first recall thetopology defined on the set of all closed subsets of a topological space, seen as a setwith a partial ordering given by inclusion. For details we refer the reader to [13].
Definition 2.1.
Let X be a topological space. For F ⊆ X closed, we define ↓ F := { F ′ | F ′ ⊆ F, F ′ closed in X } . We denote by Cl ( X ) the space of all closed subsets of X endowed with its uppertopology, that is, the topology for which a sub-base of closed sets consists of X andsets of the form ↓ F , with F ⊆ X closed. See [13, Def. O-5.4].One of the main objects in our paper is the space of quasi-orbits of an action ofa topological group on a topological space. In these spaces the points are not nec-essarily closed, so we need to briefly recall the notion of T -ization of a topologicalspace. See [29, Ch. 6] for more details. Definition 2.2.
Let X be topological space, and consider the equivalence relation x ∼ y ⇐⇒ { x } = { y } . Then the T -ization of X is the quotient space X ∼ = X/ ∼ , with the corresponding quotient topology. The quotient map q : X → X/ ∼ iscalled the T -ization map .Part of the following lemma is known in some special cases (see [29, Lemma6.8]), but we give the proofs here for the sake of completeness. Lemma 2.3.
Let X be a topological space, and G a topological group with a con-tinuous action G × X → X , ( g, x ) g · x . Denote by r : X → X/G the canonicalquotient map, by q : X/G → ( X/G ) ∼ the T -ization map, and by Q : X → ( X/G ) ∼ , Q := q ◦ r the quasi-orbit map. Then we have: (i) The map r is continuous, open and surjective, and r − ( { r ( x ) } ) = G · x forevery x ∈ X . (ii) For every x, y ∈ X , { r ( x ) } = { r ( y ) } if and only if G · x = G · y . (iii) q and Q are continuous, open and surjective. (iv) The map ι : ( X/G ) ∼ → Cl ( X ) , given by ι ( Q ( x )) = G · x is well-defined, and a homeomorphism onto its image.Proof. (i) The maps r is continuous and surjective since it is a quotient map. Thefact that r is open is well-known: For every D open in X , r − ( r ( D )) = G · D ⊆ X is open, hence r ( D ) is open in the quotient topology.We have that G · x = r − ( { r ( x ) } ) ⊆ r − ( { r ( x ) } ), and this last set is closedsince r is continuous, hence G · x ⊆ r − ( { r ( x ) } ). INGRID BELTIT¸ ˘A AND DANIEL BELTIT¸ ˘A
It remains to prove the converse inclusion. We note that, since r is an openmap, the set G · x is G -invariant, that is, G · x = r − ( r ( G · x )). Therefore itscomplement X \ G · x is also G -invariant, thus r − ( r ( X \ G · x )) = X \ G · x . Since( X \ Gx ) ∩ G · x = ∅ , it follows then that r ( x ) r ( X \ G · x ) . (2.1)On the other hand, using that r is an open mapping, the subset r ( X \ G · x ) ⊆ X/G is open. Therefore (2.1) implies { r ( x ) } ∩ r ( X \ G · x ) = ∅ , hence r − ( { r ( x ) } ) ∩ r − ( r ( X \ G · x )) = ∅ . Using again that X \ G · x is G -invariant, it follows that r − ( { r ( x ) } ) ∩ ( X \ G · x ) = ∅ .Thus we get the inclusion r − ( { r ( x ) } ) ⊆ G · x .(ii) The assertion follows immediately from (i) for the implication ” ⇐ ” usingthat the map r is surjective.(iii) The map q is continuous and surjective since it is a quotient map. To provethat q is open, it suffices to show that q − ( q ( r ( D ))) is open for every D open in X .We have that q − ( q ( r ( D ))) = { r ( x ) | ∃ y ∈ D , q ( r ( x )) = q ( r ( y )) } = { r ( x ) | ∃ y ∈ D, { r ( x ) } = { r ( y ) }} = { r ( x ) | ∃ y ∈ D, G · x = G · y } , where in the last line we have used (ii) . If there is y ∈ D such that G · x = G · y ,then D ∩ G · x = ∅ , hence D ∩ G · x = ∅ . On the other hand, if D ∩ G · x = ∅ , thenfor y ∈ D ∩ G · x we have that G · x = G · y , hence G · x = G · y . Summing up, wehave obtained that q − ( q ( r ( D ))) = { r ( x ) | D ∩ G · x = ∅} = r ( G · D ) , therefore q − ( q ( r ( D ))) is open.The fact that Q is a continuous, open and surjective map follows immediately.(iv) The map ι is well-defined and injective since Q is surjective, while Q ( x ) = Q ( y ) if and only if { r ( x ) } = { r ( y ) } and if and only if G · x = G · y , by (ii).A sub-basis of open sets for the topology on Cl ( X ) is given by the sets of theform U ( D ) = { F ∈ Cl ( X ) | F ∩ D = ∅} , where D are open sets in X . For D ⊆ X open, ι − ( U ( D )) = { Q ( x ) | ι ( Q ( x )) ∈ U ( D ) } = { Q ( x ) | G · x ∩ D = ∅} = { Q ( x ) | G · x ∩ D = ∅} = Q ( G · D ) , hence ι − ( U ( D )) is open. It follows that ι is continuous. On the other hand, usingthe arguments above, ι ( Q ( D )) = { G · x | x ∈ D } = { G · x | G · x ∩ D = ∅} = ι (( X/G ) ∼ ) ∩ U ( D ) . ROUPS WITH T PRIMITIVE IDEAL SPACE 5
It follows that ι : ( X/G ) ∼ → ι (( X/G ) ∼ ) is open as well, hence ι is a homeomorphismonto its image (see [4, p. TGI.30]). (cid:3) Definition 2.4.
Whenever the conditions Lemma 2.3 are satisfied we denote(
X/G ) ≈ := ι (( X/G ) ∼ ) ⊆ Cl ( X )with the induced topology. Hence ( X/G ) ≈ is a homeomorphic copy of ( X/G ) ∼ .2.2. Upper topology and inner hull-kernel topology.
Let G be a secondcountable locally compact group G , with its C ∗ -algebra C ∗ ( G ). We denote byPrim( G ) the space of primitive ideals in C ∗ ( G ) with the hull-kernel topology. Wedenote by b G the space of equivalence classes [ π ] of irreducible unitary represen-tations π of G endowed with the hull-kernel topology, that is, the inverse imagetopology for the canonical map κ : b G → Prim( G ), κ ([ π ]) = Ker π . We also notethat, by the definition of the topology of b G , the surjective mapping κ is both openand closed. Here and throughout this paper, for every continuous unitary represen-tation π of G , we denote also by π its corresponding nondegenerate ∗ -representationof C ∗ ( G ). In particular, we identify b G = \ C ∗ ( G ).We denote by T ( G ) the set of all equivalence classes of unitary representationsof G in separable complex Hilbert spaces. For any S , S ⊆ T ( G ), we write S (cid:22) S ( S is weakly contained in S ) if T π ∈S Ker π ⊇ T π ∈S Ker π , and S ≈ S ( S is weaklyequivalent to S ) if S (cid:22) S and S (cid:22) S .On T ( G ) we consider the inner hull-kernel topology, which restricted to b G is thehull-kernel topology. (See [11].) We recall that for every ∗ -representation T of the C ∗ -algebra C ∗ ( G ) there exists a unique closed subset spec ( T ) ⊆ b G —the spectrumof the unitary equivalence class of ∗ -representations [ T ]—which is weakly equivalentto [ T ], and spec ( T ) = { [ τ ] ∈ b G | τ (cid:22) T } . (See [10, Def. 3.4.6].)We note that for simplicity, for a unitary representation T of G , we sometimeswrite T instead of [ T ]. Remark 2.5.
The inner hull-kernel topology of T ( G ) is the coarsest topology forwhich the mapping spec : T ( G ) → Cl ( b G )is continuous. Lemma 2.6.
Let G be a second countable locally compact group. Define the map b κ : Cl ( b G ) → Cl (Prim( G )) , b κ ( F ) := { κ ([ π ]) | [ π ] ∈ F} for all F ∈
Cl ( b G ) . Then b κ is continuous.Proof. The map b κ takes values in Cl (Prim( G )) since κ : b G → Prim( G ) is a closedmap. To prove that b κ is continuous, it is enough to show that for every S ∈
Cl (Prim( G )), the set b κ − ( ↓ S ) is closed in Cl ( b G ). By a simple computation we see INGRID BELTIT¸ ˘A AND DANIEL BELTIT¸ ˘A that b κ − ( ↓ S ) = ↓ κ − ( S ), and the assertion above follows since κ − ( S ) is a closedsubset of b G . (cid:3) Definition 2.7.
We keep the notation above. For every ∗ -representation T of the C ∗ -algebra C ∗ ( G ) we define the support of the unitary equivalence class [ T ] ∈ T ( G )by supp ( T ) := b κ (spec ( T )) ∈ Cl (Prim( G ))) . Remark 2.8.
The map supp : T ( G ) → Cl (Prim( G )) is continuous. This is aconsequence of the definition, Remark 2.5 and Lemma 2.6. Lemma 2.9.
Let L be a locally compact group, and let κ : b L → Prim( L ) , π Ker π be its canonical map. Then for T ∈ T ( L ) and A ∈ Cl ( b L ) , spec ( T ) = A if and onlyif supp ( T ) = κ ( A ) .Proof. This follows from the fact that, since κ is an closed and open surjective map, κ − ( κ ( A )) = A , and supp ( T ) = κ (spec ( T )) for every A ∈ Cl ( b L ). (cid:3) Lemma 2.10.
Let G , L be locally compact groups such that G has a continuousaction G × C ∗ ( L ) → C ∗ ( L ) by automorhisms of C ∗ ( L ) , and let κ : b L → Prim( L ) , π Ker π be the canonical map corresponding to L . Then the map κ ≈ : ( b L/G ) ≈ → (Prim( L ) /G ) ≈ , G · π G · κ ( π ) is a homeomorphism.Proof. We recall the canonical identification \ C ∗ ( L ) = b L , therefore G acts contin-uously on b L and Prim( L ). Then for every A ⊆ b L , A = κ − ( κ ( A )), since κ isopen, closed and surjective. It follows that for A , A ⊆ b L , A = A if and only if κ ( A ) = κ ( A ). Thus for π , π ∈ b L we have that G · π = G · π ⇔ κ ( G · π ) = κ ( G · π ) ⇔ G · κ ( π ) = G · κ ( π ) . Therefore the map κ ≈ is well-defined and injective. Since κ is surjective, κ ≈ issurjective, as well. Hence κ ≈ is bijective.The map of κ ≈ is open and continuous since the diagram b L κ / / (cid:15) (cid:15) Prim( L ) (cid:15) (cid:15) ( b L/G ) ≈ κ ≈ / / (Prim( L ) /G ) ≈ is commutative, where the down-arrows are the maps b L → ( b L/G ) ≈ , π G · π , andPrim( L ) → (Prim( L ) /G ) ≈ , P 7→ G · P , which are open and continuous, while κ iscontinuous, closed and open. Hence κ ≈ is a homeomorphism. (cid:3) Actions of Lie groups on abelian Lie groups, and the absence ofquasi-compact open sets in the space of quasi-orbits.
Let G be a topologicalgroup with its center Z . For every unitary irreducible representation π : G → B ( H π )there exists a character χ π : Z → T with π ( g ) = χ π ( g )id H π for every g ∈ Z . The ROUPS WITH T PRIMITIVE IDEAL SPACE 7 character χ π actually depends on the unitary equivalence class of π rather than on π itself, hence we obtain a well-defined mapping R G : b G → b Z, [ π ] χ π . (2.2) Lemma 2.11.
Let G be a separable locally compact group with its center Z . Thenthe following assertions hold: (i) The mapping R G : b G → b Z is surjective and continuous. (ii) There exists a surjective continuous mapping
Res GZ : Prim( G ) → b Z satisfying Res GZ (Ker π ) = R G ([ π ]) for every [ π ] ∈ b G . (iii) If G is amenable, then R G : b G → b Z and Res GZ : Prim( G ) → b Z are open maps.Proof. (i) By [10, Prop. 18.1.5] we obtain that, for every subset S ⊆ b G , R G mapsthe closure of S into the closure of R G ( S ), hence R G is continuous.To prove that R G is surjective, let χ ∈ b Z be arbitrary and let π χ : G → B ( H )be the unitary representation induced from χ . Since G is separable, it follows by[11, Th. 4.5] that π | Z is weakly equivalent to the orbit of χ under the naturalaction of G on Z . However, since Z is the center of G , that G -orbit of χ is thesingleton { χ } ⊆ b Z , and therefore R G ([ π χ ]) = χ .(ii) The mapping Res GZ exists and is continuous by (i), using [29, Lemma C.6] or[29, Cor. 6.15].(iii) As noted above, the G -orbit of any χ ∈ b Z is { χ } ⊆ b Z hence, since b Z isHausdorff, it follows that G acts minimally on b Z . Then, since G is amenable, forany χ ∈ b G and π ∈ b G one has π (cid:22) Ind GZ ( χ ) if and only if χ (cid:22) π | Z by [14, Th. 3.3].Taking into account the definition of R G : b G → b Z , we then obtain( R G ) − ( χ ) = { π ∈ b G | χ (cid:22) π | Z } = { π ∈ b G | π (cid:22) Ind GZ ( χ ) } = spec (Ind GZ ( χ )) . Therefore (Res GZ ) − ( χ ) = supp (Ind GZ ( χ )) =: S ( χ )for arbitrary χ ∈ b Z . The mapping S : b Z → Cl (Prim( G )) defined this way iscontinuous as a a composition of continuous maps. For every C ∈ Cl (Prim( G )) wehave (Res GZ ) ♯ ( C ) := { χ ∈ b Z | ( R G ) − ( χ ) ⊆ C } = S − ( ↓ C ) , hence (Res GZ ) ♯ ( C ) ⊆ b Z is a closed subset since ↓ C ⊆ Cl (Prim( G )) is a closed subsetand the map S is continuous. Since Res GZ : Prim( G ) → b Z is surjective, it thenfollows that Res GZ is an open mapping. Finally, R G = (Res GZ ) ◦ κ is open since κ : b G → Prim( G ) is an open map. (cid:3) Remark 2.12.
Lemma 2.11 provides an alternative approach to [9, Prop. 4.1] thatdoes not need continuous fields of C ∗ -algebras. The connection with the earlierapproach is that, if the locally compact group G is amenable, then one has thenatural open continuous surjective mapping Res GZ : Prim( G ) → b Z by Lemma 2.11,hence C ∗ ( G ) has the structure of a continuous C ∗ -bundle on b Z by [29, Th. C.26].Before proceeding we need a definition. INGRID BELTIT¸ ˘A AND DANIEL BELTIT¸ ˘A
Definition 2.13.
Let K be a connected Lie group with its Lie algebra k , V afinite-dimensional real vector space and ρ : K → End( V ) a representation of K on V . By a slight abuse of notation, for every T ∈ End( V ) we denote by againby T its corresponding extension to a C -linear operator on the complexification V C := C ⊗ R V , that is, the operator id C ⊗ T is denoted again by T . Then an R -linear functional λ : k → C is a called a weight of the representation ρ if thereexists w ∈ V C \ { } for which d ρ ( X ) w = λ ( X ) w for every X ∈ k . We say that λ isa purely imaginary weight if λ ( k ) ⊆ i R .For instance, if K is a solvable Lie group and the spectrum of d ρ ( X ) is containedin i R for every X ∈ k , then there exists a purely imaginary weight of ρ as a directconsequence of Lie’s theorem on representations of solvable Lie algebras. If K isactually a nilpotent Lie group, then every weight of ρ is purely imaginary if and onlyif the spectrum of d ρ ( X ) is contained in i R for every X ∈ k , by the weight-spacedecomposition of representations of nilpotent Lie algebras. Remark 2.14.
Let Z be an abelian Lie group, connected and simply connected.We may then assume that Z is a finite dimensional real vector space. Let K alocally compact group, and α : K → End( Z ) be a continuous representation; itinduces a continuous action K × b Z → b Z . On the other hand b Z can be identifiedwith Z ∗ , by ξ → χ ξ , with χ ξ ( z ) = e i h ξ,z i , where h· , ·i : Z ∗ × Z → R is the dualitybracket. With this identification, α induces a representation α ∗ : K → End( Z ∗ ),which is fact the contragredient of α . Lemma 2.15.
Let Z be an abelian Lie group, connected and simply connected, K aconnected and simply connected Lie group with its Lie algebra k , and a continuousrepresentation α : K → End( Z ) . Assume that the action d α : k → End( Z ) hasat least a purely imaginary weight. Then ( Z ∗ /K ) ∼ contains no non-empty openquasi-compact subsets.Proof. With the notations in Remark 2.14, Z is a vector space, b Z ≃ Z ∗ , andthe hypothesis implies that the action d α ∗ : k → End( Z ∗ ) has at least a purelyimaginary weight.Let λ : k → i R be a purely imaginary weight of d α ∗ . By the weight space decom-position of Z ∗ with respect to the Lie algebra representation d α ∗ : k → End( Z ∗ ),as discussed in [3, § Z ∗ ⊆ Z ∗ satisfying the following conditions: • d α ∗ ( k ) Z ∗ ⊆ Z ∗ and dim R Z ∗ /Z ∗ = 2 when λ
0, or dim R Z ∗ /Z ∗ = 1when λ ≡ • When λ
0, the space Z ∗ := Z ∗ /Z ∗ has the structure of a C -vector spacefor which ( ∀ y ∈ k )( ∀ x ∈ z ) d α ∗ ( y ) x ∈ λ ( y ) x + Z ∗ . (2.3)There is then the commutative diagram Z ∗ f / / q (cid:15) (cid:15) Z ∗ q (cid:15) (cid:15) ( Z ∗ /K ) ∼ e f / / Z ∗ / exp λ ( k ) ROUPS WITH T PRIMITIVE IDEAL SPACE 9 where f : Z ∗ → Z ∗ , x x + Z ∗ ,q : Z ∗ → ( Z ∗ /K ) ∼ , q ( x ) := Kxq : Z ∗ → Z ∗ / exp λ ( k ) , q ( x + Z ∗ ) := (exp λ ( k )) x + Z ∗ , and e f : ( Z ∗ /K ) ∼ → Z ∗ / exp λ ( k ) , e f ( Kx ) := (exp λ ( k )) x. Here λ ( k ) = i R when λ
0, hence exp λ ( k ) = T ; otherwise λ ( k ) = 0 andexp λ ( k ) = { } . Then the mappings q and q are surjective, continuous, and openby [29, Lemma 6.12], while f is clearly surjective, continuous, and open. It thenfollows by the above commutative diagram that e f is surjective, continuous, andopen. Moreover, when λ Z ∗ is a 1-dimensional complex vector space, andthere is a homeomorphism Ψ : Z ∗ / exp λ ( k ) → [0 , ∞ ); otherwise λ ( k ) = 0 and weobtain a homeomorphism Ψ : Z ∗ / exp λ ( k ) → R . Since neither [0 , ∞ ) nor R haveopen quasi-compact subsets, this completes the proof. (cid:3) Quasi-orbit spaces
In the present section, unless otherwise mentioned, we keep the following nota-tion and assumption.
Setting 3.1.
Let G be a fixed second countable locally compact group and assumethat the following conditions hold:(1) There is a closed normal subgroup L of G such that G/L is abelian.(2) For every
P ∈
Prim( L ) there is T ∈ b G such that supp ( T | L ) = G · P . Remark 3.2.
Condition (2) in Setting 3.1 is equivalent to the following condition.(2’) For every π ∈ b L there is T ∈ b G such thatspec ( T | L ) = G · π. This is follows from Lemma 2.9.
Lemma 3.3.
Let G be as above. (i) For every T ∈ b G , there is π ∈ b L such that supp ( T | L ) = G · Ker π. (ii) For π , π ∈ b L , G · π = G · π if and only if G · Ker π = G · Ker π , and ifand only if the representations Ind GL ( π ) and Ind GL ( π ) are weakly equivalent.Proof. The assertion (i) is immediate from [14, Thm. 2.1].(ii) By Lemma 2.10, G · π = G · π if and only if G · Ker π = G · Ker π .It it thus enough to show that G · π = G · π if and only if the representationsInd GL ( π ) and Ind GL ( π ) are weakly equivalent. If π ∈ G · π , then π G · π , by[10, Thm. 3.4.10]. Thus, Ind GL ( π ) Ind GL ( π ) by [11, Thm. 4.2], since Ind GL ( g · π ) =Ind GL ( π ) for every g ∈ G . (See [7, Lemma 2.1.3].) The converse implication followsdirectly from [11, Thm. 4.5]. (cid:3) For the next results, we use the notation and definitions in Subsection 2.2.
Lemma 3.4.
The map S : (Prim( L ) /G ) ≈ → Cl (Prim( G )) , G · Ker π supp (Ind GL ( π )) is continuous.Proof. We first remark that S is well-defined by Lemma 3.3(ii). Denote by Q the map b L → (Prim( L ) /G ) ≈ , Q ( π ) = G · Ker π , that is continuous, surjective andopen by Lemma 2.3(iii), (iv) and Lemma 2.10. We have the commutative diagram b L Q (cid:15) (cid:15) Ind GL / / T ( G ) supp (cid:15) (cid:15) spec / / Cl ( b G ) b κ x x qqqqqqqqqqq (Prim( L ) /G ) ≈ S / / Cl (Prim( G )) , where all the mappings are continuous. If D is an open subset in Cl (Prim( G )), then( S ◦ Q ) − ( D ) is open in b L , hence Q (( S ◦ Q ) − ( D )) is open in (Prim( L ) /G ) ≈ . Onthe other hand, since Q is surjective we have that Q (( S ◦ Q ) − ( D )) = S − ( D ).We have thus obtained that S − ( D ) is open in (Prim( L ) /G ) ≈ whenever D is openin Cl (Prim( G )), hence S is continuous. (cid:3) Lemma 3.5.
The map R : Prim( G ) → (Prim( L ) /G ) ≈ , Ker T supp ( T | L ) is well defined, continuous and surjective.Proof. (i) From Lemma 3.3(i) it follows that for every T ∈ b G , there is π ∈ b L such that supp ( T | L ) = G · Ker π . Hence the map R : b G → (Prim( L ) /G ) ≈ , T supp ( T | L ) is well-defined. It is also continuous, since the restriction map b G →T ( L ), b G → T | L is continuous, and supp : T ( L ) → Cl (Prim( L )) is continuous. Onthe other hand, if T , T ∈ b G are such that Ker T = Ker T , then R ( T ) = R ( T ),therefore the map R is well-defined and we have the commutative diagram b G R (cid:15) (cid:15) κ / / Prim( G ) R w w ♦♦♦♦♦♦♦♦♦♦♦♦ (Prim( L ) /G ) ≈ . Since the canonical map κ : b G → Prim( G ), κ ([ T ]) = Ker T , is continuous and open,it follows that R is continuous as well.Surjectivity follows by the assumption (2) in Setting 3.1. (cid:3) We now introduce some notation needed in Theorem 3.6 below. Set A := G/L and let p : G → A be the canonical quotient homomorphism. The character group b A has a continuous action by automorphisms of C ∗ ( G ), b A × C ∗ ( G ) → C ∗ ( G ) , ( χ, ϕ ) χ · ϕ ROUPS WITH T PRIMITIVE IDEAL SPACE 11 where ( χ · ϕ )( g ) := χ ( p ( g )) ϕ ( g ) if χ ∈ b A , ϕ ∈ L ( G ), and g ∈ G . This gives riseto continuous actions of b A on Prim( G ) and on \ C ∗ ( G ), respectively. Taking intoaccount the canonical identification \ C ∗ ( G ) ≃ b G , the corresponding action of b A onclasses of irreducible representation of G is given by b A × b G → b G, ( χ, T ) χ · T = ( χ ◦ p ) ⊗ T where, for any unitary irreducible representation T : G → B ( H ) and any χ ∈ b A one defines χ · T : G → B ( H ), ( χ · T )( g ) := χ ( p ( g )) T ( g ). As usual, we denoteby (Prim( G ) / b A ) ∼ and ( b G/ b A ) ∼ the quasi-orbit spaces corresponding to the aboveactions of b A on Prim( G ) and on b G , respectively, and by (Prim( G ) / b A ) ≈ and ( b G/ b A ) ≈ their homeomorphic copies in Cl (Prim( G )) and Cl ( b G ), respectively. Theorem 3.6.
Let G be second countable locally compact group. Assume thatthere exists a closed normal subgroup L of G such that G/L is abelian, and forevery
P ∈
Prim( L ) there is T ∈ b G such that supp ( T | L ) = G · P . Then there is a homeomorphism of quasi-orbit spaces (Prim( L ) /G ) ∼ → (Prim( G ) / b A ) ∼ . (3.1) Proof.
We prove in fact that the map S : (Prim( L ) /G ) ≈ → (Prim( G ) / b A ) ≈ , G · Ker π supp (Ind GL ( π )) . (3.2)is a well-defined homeomorphism.We first show that the image of S is contained in (Prim( G ) / b A ) ≈ . To this end,using Remark 3.2 and Lemma 2.9, it suffices to prove the following: π ∈ b L, T ∈ b G, spec ( T | L ) = G · π = ⇒ spec (Ind GL ( π )) = b A · T ∈ ( b G/ b A ) ≈ . (3.3)To prove (3.3), let λ : A → B ( L ( A )) be the regular representation of A . Since A isa locally compact abelian group, the Fourier transform L ( A ) → L ( b A ) is a unitaryequivalence between λ and the unitary representation by multiplication operators, b λ : A → B ( L ( b A )) , ( b λ ( a ) ψ )( χ ) := χ ( a ) ψ ( χ ) . Using the direct integral decomposition b λ = R b A χ d χ with respect to a Haar measureon b A , we obtain a unitary equivalence Z b A χ · T d χ = Z b A ( χ ◦ p ) ⊗ T d χ ≃ Z b A ( χ ◦ p )d χ ⊗ T = ( b λ ◦ p ) ⊗ T. Denoting by τ : L → T the trivial representation of L , we have λ ◦ p = Ind GL ( τ ),hence there is a unitary equivalence b λ ◦ p ≃ Ind GL ( τ ).On the other hand, by [11, Lemma 4.2], there is a unitary equivalenceInd GL ( τ ) ⊗ T ≃ Ind GL ( τ ⊗ T | L ) = Ind GL ( T | L ) . Since the support of the Haar measure of b A is equal to b A , it then follows by [11,Th. 3.1] that the set b A · T = { χ · T | χ ∈ b A } ⊆ b G is weakly equivalent to therepresentation Ind GL ( T | L ). Since T | L is weakly equivalent to Gπ , we obtain (3.3),by using [11, Thm. 4.2] and the same argument as in Lemma 3.3(ii) above. We now construct a continuous inverse of the mapping (3.1). To this end we notethat for arbitrary T ∈ b G and χ ∈ b A , we have ( χ · T ) | L = T | L , hence R (Ker ( χ · T )) = R (Ker T ). Since the mapping R : Prim( G ) → (Prim( L ) /G ) ≈ is continuous andsurjective by Lemma 3.5, while the topological space (Prim( L ) /G ) ≈ is T , it followsby [29, Lemma 6.10] and Lemma 2.3(iv) that there exists a continuous surjectivemapping R ′ : (Prim( G ) / b A ) ≈ → ( b L/G ) ≈ for which the diagramPrim( G ) ι ◦ Q (cid:15) (cid:15) R ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗ (Prim( G ) / b A ) ≈ R ′ / / (Prim( L ) /G ) ≈ (3.4)is commutative, where Q , ι are maps given by Lemma 2.3 for X replaced by Prim( G )and G replaced by b A .To see that the continuous mappings R ′ and (3.1) are inverse to each other, werecall (Lemma 3.3(i)) that for every T ∈ b G there is π ∈ b L such that supp ( T | L ) = G · Ker π. Then, by (3.4), we have R ′ ( b A · Ker T ) = R ′ (( ι ◦ Q )(Ker T )) = R (Ker T )= supp ( T | L ) = G · Ker π. Hence ( S ◦ R ′ )( b A · Ker T )) = S ( G · Ker π ) = supp (Ind GL ( π )) (3.3) = κ G ( b A · T ) = b A · Ker
T , where the last equality holds true since the kernel mapping κ G : b G → Prim( G ) iscontinuous, closed, and equivariant with respect to the action of the automorphismgroup of C ∗ ( G ). (See also Definition 2.7.) On the other hand, for π ∈ b L there is T ∈ b G such that supp ( T | L ) = G · Ker π (Setting 3.1(2)). Then by (3.3) and theproperties of κ G , as above, we have( R ′ ◦ S )( G · Ker π ) = R ′ ( b A · Ker T )= R ′ (( ι ◦ Q )(Ker T )) = R (Ker T ) = supp ( T | L ) = G · π. Thus S ◦ R ′ = id and R ′ ◦ S = id, hence S − = R ′ , which is a continuous mapping.This completes the proof of the fact that S in (3.6) is a homeomorphism. (cid:3) AF-embeddable solvable Lie groups
In the present section we assume that G is a connected simply connected solvableLie group with Lie algebra g . Then the commutator subgroup L := [ G, G ] isnilpotent, and a connected, simply connected, closed normal subgroup of G . Wedenote by Z the center of L . The Lie algebra of L is [ g , g ], and we denote by z theLie algebra of Z .The main result of this section is the following theorem. Theorem 4.1.
Let G be a connected and simply connected solvable Lie group, L = [ G, G ] and let be Z the centre of L . Assume that the action of G on Z has at ROUPS WITH T PRIMITIVE IDEAL SPACE 13 least one purely imaginary weight. Then
Prim( G ) has no non-empty open quasi-compact subsets. Before proving this theorem we give some important consequences. We recallthe following definition (see [1]).
Definition 4.2.
Let G be a Lie group. Then G is said to be of type R providedthe spectra of the operators of the adjoint action of G on its Lie algebra g arecontained in the unit circle. Equivalently, for every X ∈ g the spectrum of theoperator ad g X : g → g should be contained in i R . Corollary 4.3.
Let G be a connected and simply connected solvable Lie group oftype R . Then Prim( G ) has no non-empty open quasi-compact subsets.Proof. The condition that G is of type R implies that the action of G on Z haspurely imaginary weights. Therefore the result is a consequence of Theorem 4.1. (cid:3) We recall that if G is a connected simply connected solvable Lie group, theprimitive ideals of C ∗ ( G ) are maximal, or equivalently, Prim( G ) is T , if and only if G is of type R ; see [22, Thm. 2]. Therefore the previous corollary may be re-writtenas follows. Corollary 4.4.
Let G be a connected and simply connected solvable Lie group suchthat Prim( G ) is T . Then Prim( G ) has no non-empty open quasi-compact subsets. Corollary 4.5.
Let G be a connected and simply connected solvable Lie group, L = [ G, G ] and let be Z the centre of L . Assume that the action of G on Z has atleast one purely imaginary weight. Then C ∗ ( G ) is AF-embeddable.Proof. The result is a consequence of Theorem 4.1 and [12, Cor. B]. (cid:3)
Corollary 4.6.
Let G be a connected and simply connected solvable Lie group suchthat Prim( G ) is T . Then C ∗ ( G ) is AF-embeddable.Proof. The result is a consequence of Corollary 4.4 and [12, Cor. B]. (cid:3)
To prove Theorem 4.1 we need the following technical tool.
Proposition 4.7.
Let G be a connected and simply connected solvable Lie group, L = [ G, G ] and let be Z the centre of L . Then there is a continuous, surjective andopen map Φ : Prim( G ) → ( b Z/G ) ∼ . Proof.
We show first that the group G and its closed normal subgroup L satisfythe conditions is Theorem 3.6. We note that the Lie group L is nilpotent of type I (even liminary). The quotient A = G/L is abelian since L = [ G, G ], so it remainsto show that for every π ∈ b L there exists T ∈ b G such thatsupp ( T | L ) = G · Ker π. (4.1)Since G is a connected simply connected solvable Lie group, for every π ∈ b L there isa closed normal subgroup K of G , L ⊆ K , a representation ρ ∈ b K with ρ | L = π such that T = Ind GK ρ is a factor representation. (See [22, p. 83–83] and the referencestherein.) Let T ∈ b G be such that T ≈ T . By [11, Thm. 4.5] we have that T | K ≈ T | K ≈ G · ρ ⊆ b K. Hence T | L = ( T | K ) | L ≈ G · ρ | L = G · π, and thus (4.1) holds.By Theorem 3.6 and the fact that L is of type I, we obtain a homeomorphism R ′ : (Prim( G ) / b A ) ∼ → ( b L/G ) ∼ . Denote by Q : Prim( G ) → (Prim( G ) / b A ) ∼ the quasi-orbit map (see Lemma 2.3); itis a continuous, open and surjective map. Thus, if we set R := R ′ ◦ Q we obtain acontinuous, open and surjective map R ′ : Prim( G ) → ( b L/G ) ∼ . On the other hand, by Lemma 2.11 there is a continuous, surjective and openmap R L : b L → b Z . It is easy to see that R L is also G -equivariant, hence it gives riseto a continuous, surjective and open mapping R L : b L/G → b Z/G .Let r G : b L/G → ( b L/G ) ∼ be the canonical continuous mapping from b L/G ontoits T -ization (in the sense of Definition 2.2), and let r Z : b Z/G → ( b Z/G ) ∼ be itsanalogous map for the group Z . Then r Z ◦ R L : b L/G → ( b Z/G ) ∼ is continuous.Hence there is a continuous mapping f R L : ( b L/G ) ∼ → ( b Z/G ) ∼ with f R L ◦ r L = r Z ◦ R L , by [29, Lemma 6.10].We thus obtain the commutative diagram b L q G (cid:15) (cid:15) R L / / κ L b Z q Z (cid:15) (cid:15) κ Z { { b L/G r L (cid:15) (cid:15) R L / / b Z/G r Z (cid:15) (cid:15) ( b L/G ) ∼ f R L / / ( b Z/G ) ∼ where the surjective mappings κ L := r L ◦ q L and κ Z := r Z ◦ q Z are continuousand open by Lemma 2.3(iii). Thus κ Z ◦ R L = f R L ◦ κ G , where κ Z ◦ R L and κ L are surjective open mappings. Therefore, by [4, Ch. 3, §
4, Props. 2–3] again, thecontinuous mapping f R L is open.The map Φ := f R L ◦ R : Prim( G ) → ( b Z/G ) ∼ is a continuous, open and surjective map, thus it satisfies the properties in thestatement. (cid:3) Proof of Theorem 4.1.
The closed, normal subgroup L is connected and simplyconnected, thus the quotient A := G/L is an abelian Lie group, connected andsimply connected, and its Lie algebra is a = g / [ g , g ]. ROUPS WITH T PRIMITIVE IDEAL SPACE 15
The action of G on Z , α : G → End( Z ), is trivial on L , therefore there is agroup homomorphism α Z : A → End( Z ) such that α Z ◦ Q = α , where Q is thequotient map Q : G → A . Then α Z induces a natural action of A on Z ∗ ≃ b G , and( Z ∗ /G ) ∼ = ( Z ∗ /A ) ∼ .Since the action of G on Z has at least one purely imaginary weight, the actiond α Z : a → End( Z ) has at least one purely imaginary weights. Then the assertionin the statement follows from Proposition 4.7 and Lemma 2.15. (cid:3) AF-embeddability of simply connected Lie groups with T primitiveideal spaces We prove here the main result of the paper, Theorem 5.3. Along with the resultsof the preceding sections, its proof requires the following two lemmas.
Lemma 5.1.
Let be K a compact group that acts on a topological space X bya continuous action K × X → X , and let q : X → X/K be the correspondingquotient map. Then if for C ⊆ X/K , we have that C is quasi-compact if and onlyif q − ( X/K ) is quasi-compact.Proof. Assume first that q − ( C ) is quasi-compact. Since q is surjective and contin-uous, C = q ( q − ( C )) is quasi-compact.For the direct implication, let { x j } j ∈ J be any net in q − ( C ). Then { q ( x j ) } isa net in C . Since C is quasi-compact, selecting a suitable subnet, we may assumethat there is c ∈ X/C such that q ( x J ) → c in X/C . Hence there exist k j , j ∈ J ,and x ∈ q − ( c ) such that k j · x j → x . The group K is compact, therefore, againby selecting a suitable subnet, we may assume that there is k ∈ K such that k j → k , and thus k − j → k − in K . By the continuity of the action we have then x j = k − j · ( k j · x j ) → k − x ∈ X . On the other hand, q ( k − · x ) = q ( x ) = c , hence k − · x ∈ q − ( c ) ⊆ q − ( C ).Thus every net in q − ( C ) has a subnet that converges to some point in q − ( C ),hence q − ( C ) is quasi-compact. (cid:3) Lemma 5.2.
Let G a locally compact group, K a compact group that acts continu-ously on G and consider G = K ⋉ G . Assume that Prim( G ) is T . If Prim( G ) has no non-empty open quasi-compact subsets, then Prim( G ) has no non-emptyopen quasi-compact subsets.Proof. We have that C ∗ ( G ) = K ⋉ C ∗ ( G ) ([29, Prop. 3.11]). Then the restrictionof ideals gives a mapping R : Prim( G ) → (Prim( G ) /K ) ∼ , which is continuous,surjective and open. (See [15, Thm. 4.8].) Since the action of K on Prim( G ) iscontinuous, K is compact, and Prim( G ) is T it follows by [17, Cor., p. 213] thatthe orbits of K in Prim( G ) are closed, hence (Prim( G ) /K ) ∼ = Prim( G ) /K .Thus we get that R : Prim( G ) → Prim( G ) /K is continuous, surjective and open.The quotient map q : Prim( G ) → Prim( G ) /K is continuous, open and surjective,as well.Assume now that there is C an open quasi-compact subset of Prim( G ). Then R ( C ) is quasi-compact and open in Prim( G ) /K , and by Lemma 5.1, q − ( R ( C )) is a quasi-compact and open subset of Prim( G ). By hypothesis this implies that q − ( R ( C )) = ∅ , hence C = ∅ . This completes the proof of the lemma. (cid:3) Theorem 5.3.
Let G be a simply connected Lie group such that Prim( G ) is T .Then both C ∗ ( G ) and the reduced C ∗ -algebra C ∗ r ( G ) are AF-embeddable.Proof. We can write G = S × G , where both S and G are simply connectedLie groups, S is semisimple, and G has no semisimple factors. (See [23, Proof ofThm. 2, p. 47].)Since S is liminary, C ∗ ( S ) and C ∗ r ( S ) are AF-embeddable (see Remark 6.2).On the other hand, C ∗ ( S ) is nuclear by [6, Prop. 2.7.4], since it is liminary hencetype I. Therefore we have that C ∗ ( G ) ≃ C ∗ ( S ) ⊗ C ∗ ( G ) , (5.1) C ∗ r ( G ) ≃ C ∗ r ( S ) ⊗ C ∗ r ( G ) . (5.2)Thus it suffices to prove that C ∗ ( G ) and C ∗ r ( G ) are AF-embeddable.From (5.1) it follows that there is a homeomorphismPrim( G ) ≃ Prim( S ) × Prim( G ) . (5.3)It is straightforward to check that if the product of two topological spaces is T ,then each of them is T . Hence, by (5.3), Prim( G ) is T .Since G has no semisimple factor, we then have by [23, Prop. 3, p. 47] that G = K ⋉ G , where K and G are simply connected Lie groups, K is compact,while G is solvable of type R . It follows that G is amenable (see [18, Prop. 11.13]),hence C ∗ r ( G ) = C ∗ ( G ). By Theorem 4.3, we have that either G = { } or thereare no non-empty quasi-compact open subsets of Prim( G ). Using Lemma 5.2 weget that either G = K , or Prim( G ) has no non-empty quasi-compact and opensubsets. Thus by [12, Cor. B] C ∗ r ( G ) = C ∗ ( G ) is AF-embeddable. (cid:3) On primitively AF-embeddable C ∗ -algebras Definition 6.1. A C ∗ -algebra A is called primitively AF-embeddable if its primitivequotients A / P for arbitrary P ∈
Prim( A ) are AF-embeddable C ∗ -algebras. Remark 6.2.
We recall that for a type I, separable C ∗ -algebra A , if A is primitivelyAF-embeddable, then it is AF-embeddable. Indeed, since A is primitively AF-embeddable, it follows that for every irreducible ∗ -representation π : A → B ( H ) the C ∗ -algebra π ( A ) ≃ A / Ker π is AF-embeddable, hence π ( A ) is stably finite by [28,Lemma 1.3]. Therefore A is residually finite by [28, Prop. 3.2], and [28, Th. 3.6]implies that A is AF-embeddable.It is not clear to what extent the above Remark 6.2 carries over beyond thetype I C ∗ -algebras, that is, if every primitively AF-embeddable C ∗ -algebra is AF-embeddable. (See also [8].)) We prove however that this is the case for all C ∗ -algebras of connected, simply connected solvable Lie groups, irrespectively of whetherthey are type I or not. More specifically, we prove that if A is the C ∗ -algebra of ROUPS WITH T PRIMITIVE IDEAL SPACE 17 a connected, simply connected solvable Lie group, then the following implicationshold true: A is primitively AF-embeddable ⇐⇒ A is strongly quasi-diagonal ⇐⇒ Prim( A ) is T = ⇒ A is AF-embeddableThe following fact is related to [2, Th. 1.1] and is applicable to Lie groups thatneed not be simply connected or solvable. Lemma 6.3.
Let G be a connected Lie group. If Prim( G ) is T , then the followingassertions hold: (i) The C ∗ -algebra C ∗ ( G ) is primitively AF-embeddable. (ii) The Lie group G is type I if and only if it is liminary.Proof. (i) The hypothesis that Prim( G ) is T is equivalent to the fact that everyprimitive ideal of C ∗ ( G ) is maximal, which is further equivalent to the fact thatevery primitive quotient of C ∗ ( G ) is a simple C ∗ -algebra. Then, by [20, Thm. 2,p. 161], for every P ∈
Prim( G ) we have that C ∗ ( G ) / P ≃ A P ⊗ K ( H P ), where A P is a simple C ∗ -algebra which is either 1-dimensional or a noncommutative torus,and H P is a suitable Hilbert space. To conclude the proof, we must show that ifa noncommutative torus is simple, then it is AF-embeddable. One way to obtainthat conclusion is to combine the fact that if a noncommutative torus is a simple C ∗ -algebra, then it is an approximately homogeneous C ∗ -algebra by [19, Th. 3.8],and on the other hand every approximately sub-homogeneous C ∗ -algebra is AF-embeddable by [25, Prop. 4.1]. Alternatively, one can reason as follows in order toobtain the even stronger property that A P embeds into a unital simple AF-algebra:We recall that if A P is a noncommutative torus and if A P is moreover a simple C ∗ -algebra, then A P has a tracial state τ . (See for instance [19, Th. 1.9].) Since τ is a tracial state, the set N τ := { a ∈ A P | τ ( a ∗ a ) = 0 } is a closed 2-sided idealof A P , which is a simple C ∗ -algebra, hence N τ = { } , that is, the tracial state τ isfaithful. Moreover, A P is separable, nuclear, and satisfies the Universal CoefficientTheorem of [26]. (See for instance the proof of [19, Th. 3.8].) On the other hand,every trace on a nuclear C ∗ -algebra is amenable by [6, Prop. 6.3.4], hence τ is afaithful amenable trace on A P . Since A P is nuclear, hence exact, it then follows by[27, Th. A] that A P embeds into a simple AF-algebra. We thus see that C ∗ ( G ) / P is AF-embeddable for arbitrary P ∈
Prim( G ).(ii) The group G is type I if and only if every primitive quotient is type I, andthen the assertion follows as a by-product of the above reasoning, since no simplenoncommutative torus is type I, as seen for instance by using [2, Lemma 4.2(ii)] forthe aforementioned tracial state τ . (cid:3) The following result is essentially contained in [17, proof of Th. 4 and Cor. 3,page 212] at least in the special case of connected groups. However we give here amore direct proof that does not use the Lie theoretic characterization [17, Th. 1]of connected locally compact groups whose primitive ideal space is T . We recallthat a topological group G is called almost connected if the quotient group G/G iscompact, where G ⊆ G is the connected component that contains the unit element ∈ G . Lemma 6.4.
Let G be an almost connected, locally compact group, and denote by L ( G ) the set of all compact normal subgroups H ⊆ G for which G/H is a Lie group.Then
Prim( G ) is T if and only if Prim(
G/H ) is T for every H ∈ L ( G ) .Proof. For every H ∈ L ( G ) we denote by j H : G → G/H its corresponding quotientmap. We define c j H : [ G/H → b G, [ σ ] [ σ ◦ j H ]and ( j H ) ∗ : Prim( G/H ) → Prim( G ) , Ker σ Ker ( σ ◦ j H ) . By [5, Prop. 8.C.8], there exists a surjective ∗ -morphism( j H ) ∗ : C ∗ ( G ) → C ∗ ( G/H )satisfying σ ◦ ( j H ) ∗ = σ ◦ j H for every [ σ ] ∈ [ G/H ≃ \ C ∗ ( G/H ), where [ σ ◦ j H ] ∈ b G ≃ \ C ∗ ( G ). Therefore( j H ) ∗ ( P ) = (( j H ) ∗ ) − ( P ) for every P ∈
Prim( C ∗ ( G/H )) . (6.1)The mapping c j H is a homeomorphism of [ G/H onto an open-closed subset of b G by[16, Th. 5.4]. Similarly, the mapping ( j H ) ∗ is a homeomorphism of Prim( G/H )onto a closed subset of Prim( G ) by (6.1) and [10, Prop. 3.2.1].Since the maps κ G : b G → Prim( G ), [ π ] Ker π and κ G/H : [ G/H → Prim(
G/H ),[ σ ] Ker σ are open, continuous, and surjective, while the diagram [ G/H κ G/H (cid:15) (cid:15) c j H / / b G κ G (cid:15) (cid:15) Prim(
G/H ) ( j H ) ∗ / / Prim( G )is commutative, we obtain that the image of the mapping ( j H ) ∗ is also open, henceis an open-closed subset of Prim( G ).Finally, one has b G = [ H ∈L ( G ) c j H ( [ G/H )by [16, Th. 5.4], hence alsoPrim( G ) = [ H ∈L ( G ) ( j H ) ∗ (Prim( G/H ))and then the assertion follows at once since we have already seen that every set( j H ) ∗ (Prim( G/H )) is an open-closed subset of Prim( G ) for H ∈ L ( G ). (cid:3) Theorem 6.5.
Let G be a connected, locally compact group. If Prim( G ) is T , then C ∗ ( G ) is primitively AF-embeddable.Proof. We use the notation of Lemma 6.4 and its proof. Let π : G → B ( H ) be anarbitrary unitary irreducible representation. It follows by [16, Th. 3.1] that thereexist H ∈ L ( G ) and an unitary irreducible representation π : G/H → B ( H ) with ROUPS WITH T PRIMITIVE IDEAL SPACE 19 π = π ◦ j H . Hence, when we extend π and π to irreducible ∗ -representations of C ∗ ( G ) and C ∗ ( G/H ), respectively, we have π = π ◦ ( j H ) ∗ , where ( j H ) ∗ : C ∗ ( G ) → C ∗ ( G/H ) is a surjective ∗ -morphism. (See [5, Prop. 8.C.8].)Since Prim( G ) is T , it follows by Lemma 6.4 that Prim( G/H ) is T . On theother hand, G is connected, hence G/H is a connected Lie group, hence C ∗ ( G/H ) isprimitively AF-embeddable by Lemma 6.3(i). In particular π ( C ∗ ( G/H )) is an AF-embeddable C ∗ -algebra. Since π ( C ∗ ( G )) = π (( j H ) ∗ ( C ∗ ( G ))) = π ( C ∗ ( G/H )), itthe follows that π ( C ∗ ( G )) is AF-embeddable. This completes the proof. (cid:3) Corollary 6.6.
Let G be a connected, locally compact group. If Prim( G ) is T ,then C ∗ ( G ) is strongly quasi-diagonal.Proof. Use Theorem 6.5 and the fact that every AF-embeddable C ∗ -algebra isquasi-diagonal. (cid:3) In the case of connected and simply connected solvable Lie groups the aboveresults go in the reverse direction, as well; we have the following corollary.
Corollary 6.7.
Let G be a connected simply connected solvable Lie group. Thenthe following assertions are equivalent. (i) G is of type R . (ii) C ∗ ( G ) is primitively AF-embeddable. (iii) C ∗ ( G ) is strongly quasi-diagonal.Proof. The implication (i) ⇒ (ii) follows from Lemma 6.3 and [22, Thm. 2, p. 161].On the other hand, (ii) clearly implies (iii).We now prove that (iii) ⇒ (i), by showing that if G is connected simply connectedsolvable and not of the type R , then it cannot be strongly quasi-diagonal. To thisend we show that there exists a closed 2-sided ideal J ⊆ C ∗ ( G ) such that thequotient C ∗ ( G ) / J is not strongly quasi-diagonal.It follows by [1, Prop. 2.2, Ch. V] that if G is a connected simply connectedsolvable Lie group and G is not of type R , then there exists a connected simplyconnected closed normal subgroup H ⊆ G for which the quotient Lie group G/H is isomorphic to one of the following Lie groups:(1) S := R ⋊ R , the connected real ax + b -group, defined via α : ( R , +) → Aut( R , +) , α ( t ) s = e t s. (2) S σ := R ⋊ α σ R , defined for σ ∈ R \ { } via α σ : ( R , +) → Aut( R , +) , α σ ( t ) = e σt (cid:18) cos t sin t − sin t cos t (cid:19) . (3) S := R ⋊ β R , defined via β : ( R , +) → Aut( R , +) , β ( t, s ) = e t (cid:18) cos s sin s − sin s cos s (cid:19) . We get thus short exact sequence of amenable locally compact groups → H → G → G/H → that leads to a short exact sequence of C ∗ -algebras0 → J → C ∗ ( G ) → C ∗ ( G/H ) → , for a suitable closed 2-sided ideal J of C ∗ ( G ). Therefore, in order to show that C ∗ ( G ) is not strongly quasi-diagonal, it suffices to check that the C ∗ -algebra of theabove groups S , S σ with σ ∈ R \ { } , and S are not strongly quasi-diagonal, andthis fact was established in the proof of [2, Thm. 1.1, (i) ⇒ (iii)]. (cid:3) References [1]
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