In this section we illustrate the results of the previous sections by considering various (mostly well-known) classes of C*-algebras. Some of them are wider than but similar to the classes of CCR-and GCR-algebras CCR-and the other are generated by the real rank zero, AF, nuclear CCR-and exact C*-algebras. Many results of this section are well known. However, our main purpose is to show that they arise naturally from the theory of relations in lattices developed in the previous sections.
5.1 C*-properties that consist of simple algebras
In what follows by S we denote any C*-property that consists of simple C*-algebras. As concrete examples one can have in mind the class of all simple algebras, or the class C of all algebras isomorphic to the algebras C(H) of all compact operators on separable Hilbert spaces H with dim H ≤ ∞, or the class of Cuntz algebras On, or uniformly hyperfinite algebras, etc.
Using S and R(S) as ”basic” C*-properties, we construct wider C*-properties using Definitions 4.2 and 4.14. It follows from Definitions 4.2 and 4.14 that a C*-algebra A is
1) a G(R(S))-algebra (GS-algebra) if each its non-zero quotient has a non-zero R(S)-ideal (S-ideal);
2) an NG(R(S))-algebra (NGS-algebra) if it has no R(S)-ideals (S-ideals);
3) a dG(R(S)-algebra (dGS-algebra) if each ideal of A has a R(S)-quotient (S-quotient);
4) a dNG(R(S)-algebra (dNGS-algebra) if A has no R(S)-quotients (S-quotients).
As in (1.2), for each C*-property and A ∈ A, we consider the corresponding relations. Since each C*-property S is lower and upper stable, Theorems 3.2 and 4.26 and Corollary 4.27 yield Corollary 5.1 (i) The C*-property R(S) is lower and upper stable, and
S ⊆ R(S) = R(R(S) ⊆ dGS = dGR(S) and GS ∪ R(S) ⊆ G(R(S)) ∩ R(GS). (5.1) (ii) The relations ≪S and ≪R(S) are H- and dual H-relations in each IdA.
(iii) ≪⊳S= ≪dGS= ≪dGR(S)= ≪⊳R(S) is a dual R-order, so that p⊳S = pdGS = pdGR(S) = p⊳R(S).
(iv) A C∗-algebra A is a dGS-algebra if and only if, for each {0} 6= I ∈ IdA, there is π′ ∈ Π(I) such that π′(I) ∈ S.
We write GR(S) for G(R(S)), NGR(S) for NG(R(S)), dGR(S) for dG(R(S)) and dNGR(S) for dNG(R(S)). For S = C, the C*-property R(C) is usually denoted by CCR,
GR(C) by GCR, NGR(C) by N GCR, dGR(C) by dGCR, dNGR(C) by dN GCR. (5.2) The C*-properties of GSR- and NGSR-algebras. Combining some previous results yields Theorem 5.2 (i) GR(S) is a lower and upper stable C*-property; NGR(S) is lower stable.
(ii) ≪GR(S)= ≪⊲R(S) is an R-order, ≪NGR(S)= ←−−
≪P is a dual R-order and rGR(S) = r⊲R(S) = pNGR(S)
in IdA for each A ∈ A.
(iii) If r⊲R(S) * I 6= A then there is J ∈ IdA such that J/I is a R(S)-algebra.
(iv) If r⊲R(S)+ I = A then A/I is a GR(S)-algebra.
Proof. (i) follows from Lemma 4.3 and Theorem 4.26. Part (ii) follows from Theorem 4.5.
Part (iii) follows from Corollary 3.3 and (iv) from Proposition 3.7(ii).
The reflexive relation ≪R(S), generally, is not transitive. For example, if S = C then {0} ≪CCR C(H) ≪CCR C(H) +C1, while {0} 6≪CCR C(H) +C1,
if dim H = ∞. Theorem 4.6 and Proposition 4.7 give the following extension of well-known results for GCR- and N GCR-algebras ([D]).
Theorem 5.3 (i) The radical r⊲R(S) is the largest GR(S)-ideal of A and the smallest ideal with N GR(S)-quotient. There is an ascending transfinite ≪R(S)-series of ideals of A from {0} to r⊲R(S).
(ii) A is a GR(S)-algebra if and only if A = r⊲R(S); it is a NGR(S)-algebra if and only if r⊲R(S) = {0}.
(iii) If some I ∈ IdA and A/I are GR(S)-algebras then A is a GR(S)-algebra.
(iv) If some I ∈ IdA and A/I are NGR(S)-algebras then A is a NGR(S)-algebra.
The C*-properties of dGSR- and dNGSR-algebras. Clearly, C(H)+C1 is a dGCR-algebra, and B(H) is a dN GCR-algebra, if dim H = ∞.
Lemma 5.4 (i) A ∈ dGR(S) if and only if, for each I ∈ IdA, there is π′ ∈ Π(I) such that π′(I) ∈ S.
(ii) If there are representations {πλ}λ∈Λ in Π(A) such that ∩λ∈Λker πλ = {0} and πλ(A) ∈ S for each λ ∈ Λ, then A is a dGR(S)-algebra.
(iii) A is a dNGR(S)-algebra if and only if π(A) /∈ S for all π ∈ Π(A).
Proof. (i) If A ∈ dGR(S) and {0} 6= I ∈ IdA, then I/J is a R(S)-algebra for some J $ I. Hence π(I/J) ∈ S for all π ∈ Π(I/J). By (4.9), each π extends to π′ ∈ Π(I) with π′(I) = π(I/J) ∈ S.
Conversely, if π′(I) ∈ S for each I ∈ IdA and some π′ ∈ Π(I), then I/ ker π′ ∈ S ⊆ R(S). So A ∈ dGR(S).
(ii) Let {0} 6= I ∈ IdA. As S consists of simple algebras, either πλ(I) = πλ(A) ∈ S, or I ⊆ ker πλ for each λ ∈ Λ. As ∩λ∈Λker πλ = {0}, πλ(I) ∈ S for some λ ∈ Λ. So A is a dGR(S)-algebra by (i).
(iii) Let A be a dNGR(S)-algebra. If π(A) ∈ S ⊂ R(S) for some π ∈ Π(A), then A/ ker π ≈ π(A) is a R(S)-algebra, a contradiction (Definition 4.14).
Conversely, let π(A) /∈ S for all π ∈ Π(A). If A is not dNGalgebra then A/I is a R(S)-algebra for some I ∈ IdA. Then π(A/I) ∈ S for all π ∈ Π(A/I). By (4.10), each π extends to π′ ∈ Π(A) such that π′(A) = π(A/I) ∈ S, a contradiction.
If all dim πλ < ∞ in Lemma 5.4(ii) then A is called residually finite-dimensional (RFD) (see Example 4.16). Hence, by Lemma 5.4, RFD-algebras are dGCR-algebras.
Example 5.5 1) The group C*-algebra C∗(F2) of the free group F2 on 2 generators has finite dimensional representations {πk}k≥1 with ∩k≥1ker πk = {0} (Proposition VII.6.1 [Da]). Thus C∗(F2) is an RFD-algebra. So it is a dGCR-algebra.
2) The C*-algebra M of all bounded sequences (a1, ..., an, ...), an∈ Mk(C), is an RFD-algebra.Thus it is a dGCR-algebra.
3) The GCR-algebra A∞ in Proposition 5.13 is a dN GCR-algebra.
Let A ∈ A. As R(S) is a lower and upper stable, ≪R(S) is an H- and a dual H-relation in IdA. Let p⊳R
(S) be the dual ≪⊳R
(S)-radical in IdA. Corollary 3.3, Lemma 4.15 and Theorem 4.17 yield Theorem 5.6 (i) dGR(S) is a lower and dNGR(S) is an upper stable C*-properties.
(ii) ≪dGR(S)= ≪⊳R(S) is a dual R-order, ≪dNGR(S)= ←−−−≪R(S) is an R-order, pdNR(S) = p⊳R(S) = r⊲dNGR(S) in IdA for each A ∈ A.
(iii) If {0} 6= I * p⊳R(S) then there is J ∈ IdA such that I/J is a R(S)-algebra.
(iv) If p⊳R(S)∩ I = {0} for I ∈ IdA, then I is a dGR(S)-algebra.
Remark 5.7 1) The C*-properties dGCR and RFD (of all RFD-algebras) are not upper stable.
The group C*-algebra C∗(F2) of the free group F2 is a RFD-algebra and, therefore, a dGCR-algebra (Example 5.5). Let C∗r(F2) = π(l1(F2)) be the reduced group C*-algebra of F2. It is the norm closure of the image of the representation π of the algebra l1(F2) on the Hilbert space l2(F2).
Then C∗r(F2) ≈ C∗(F2)/I for some ideal I of C∗(F2). It is a simple algebra (Corollary VII.7.5 and Theorem VII.8.6 [Da]) which is not isomorphic to C(H) for dim H ≤ ∞. Thus C∗r(F2) is neither an RFD- nor a dGCR-algebra. So the quotient C∗(F2)/I is neither an RFD- nor a dGCR-algebra.
Thus the C*-properties RFD and dGCR are not upper stable.
2) The C*-property dNGR(S) is not lower stable. For example, not each ideal of a dN GCR-algebra is a dN GCR-GCR-algebra. Indeed, B(H) is a dN GCR-GCR-algebra, but its ideal C(H) is not.
Using Theorem 4.19 and Proposition 4.20, we have
Theorem 5.8 (i) The dual radical p⊳R(S) is the largest dNGR(S)-ideal and the smallest ideal with dGR(S)-quotient. There is a descending transfinite ≪R(S)-series of ideals from A to p⊳R(S).
(ii) A is a dGR(S)-algebra if and only if p⊳R(S) = {0}; it is a dNGR(S)-algebra iff p⊳R(S) = A.
(iii) If some I ∈ IdA and A/I are dGR(S)-algebras then A is a dGR(S)-algebra.
(iv) If some I ∈ IdA and A/I are dNGR(S)-algebras then A is a dNGR(S)-algebra.
Summarizing the results as in (4.8), we get that, for each A ∈ A, rGR(S) = r⊲R(S) = pNGR(S) ∈ GR(S) and
pGR(S) ⊆ p⊳S = pdGS = pdGR(S) = p⊳R(S) = rdNGR(S) ∈ dNGR(S). (5.3) The classes of GS-, NGS-, dGS- and dNGS-algebras. In the previous subsections we took the class of R(S)-algebras as the ”basic” class and ”constructed” wider classes of GR(S), NGR(S), dGR(S) and dNGR(S)-algebras.
Take now S as the ”basic” class and consider the C*-properties of GS, NGS, dGS and dNGS-algebras. They define the corresponding relations in IdAfor all A ∈ A. Since S is a lower and upper stable C*-property, all the results of Theorems 5.2, 5.3, 5.6, 5.8 hold with R(S) replaced by S.
Unlike the dual ≪⊳S- and the dual ≪⊳R
(S)-radicals p⊳S and p⊳R
(S) which are always equal, the ≪⊲S -and ≪⊲R(S)-radicals r⊲S and r⊲R(S) may differ for some algebras A.
Corollary 5.9 (i) r⊲S ⊆ r⊲R(S) in each A ∈ A.
(ii) If A is a dGS-, or an R(S)-algebra, then p⊳S = {0} and there is a descending transfinite
≪S-series of ideals from A to {0}.
Proof. (i) As S ⊆ R(S), we have r⊲S ⊆ r⊲R(S) by (4.4) for all C*-algebras A.
(ii) If A ∈ dGS then pdGS = {0}. If A ∈ R(S) then, as R(S) ⊆ dGS by (5.1), pdGS = {0}. By (5.3), p⊳S = pdGS. So p⊳S = {0} and the transfinite series exists by Theorem 4.19.
For S = C, Corollary 5.9(i) gives a well-known result that each CCR-algebra A has a descending transfinite ≪C-series of ideals from A to {0}.
For many C*-algebras A, the radicals r⊲S and r⊲R(S) differ in IdA. For example, let S = C. Clearly A = C(0, 1) is a CCR-algebra, so that r⊲CCR = A, while r⊲C = {0}, as A has no ideals isomorphic to C(H). On the other hand, for C*-algebras A with separable conjugate space, r⊲C = r⊲CCR in IdA
(see Corollary 5.12).
Denote by Asep the C*-property that consists of all separable C*-algebras, and by A∗sep the C*-property that consists of all C*-algebras with separable conjugate space.
Lemma 5.10 The C*-properties Asep and A∗sep are lower and upper stable.
Proof. Let A ∈ A, I ∈ IdA and p: A → A/I be the standard epimorphism. If A ∈ Asep then, clearly, I and A/I are seprable. So Asep is a lower and upper stable C*-property.
Now let A ∈ A∗sep. Set B = A/I. Each functional g ∈ B∗ extends to a functional g∗ ∈ A∗ by g∗(x) = g(p(x)) for x ∈ A. The map g → g∗ is an isometric isomorphism from B∗ onto the closed subspace I⊥= {f ∈ A∗: f |I = 0} of A∗ ([DS] II.4.18(b)). As A∗ is separable, I⊥ is also separable.
So B∗ is separable. Thus A/I ∈ A∗sep.
We also have that I∗ ≈ A∗/I⊥. As A∗ is separable, I∗ is separable. Thus I ∈ A∗sep. Hence A∗sep is a lower and upper stable C*-property.
It follows from Theorem 4.30, from (5.1) and Example 4.28 that
GC$ GCR ⊆ GΠC. (5.4)
However, in Asep the C*-properties GCR and GΠC coincide, and in A∗sep the C*-properties GC, GCR and GΠC coincide.
Proposition 5.11 (i) GCR ∩ Asep= GΠC ∩ Asep.
(ii) A∗sep⊆ GC, so that GC ∩ A∗sep= GCR ∩ A∗sep= GΠC ∩ A∗sep= A∗sep.
Proof. (i) Let A ∈ GΠC ∩ Asep. By Definition 4.29, π(A) ⊇ C(Hπ) for each π ∈ Π(A).
As A is separable, it follows from Theorem 9.1 [D] that A is a GCR-algebra. Hence, by (5.4), GCR ∩ Asep⊆ GΠC ∩ Asep⊆ GCR ∩ Asep. So GCR ∩ Asep= GΠC ∩ Asep.
(ii) Let A ∈ A∗sep. By Lemma 5.10, for each I ∈ IdA, the quotient A/I ∈ A∗sep. Tomiyama [To]
proved that each algebra in A∗sep has an ideal isomorphic to C(H) for a separable H. Hence A/I contains an ideal isomorphic to C(H). Therefore A is a GC-algebra. Thus A∗sep ⊆ GC. From this and from (5.4) follows the rest of (ii).
Corollary 5.12 Each A ∈ A∗sep is a GC-algebra and a GCR-algebra, r⊲C = r⊲CCR = A and ≪GC = ≪GCR = ⊆ in IdA.
The algebra A has a countable ascending transfinite ≪C-series (Iλ)1≤λ≤γ of ideals such that I1 = {0}, Iγ = A and Iλ+1/Iλ≈ C(Hλ) for some separable Hλ.
Proof. By Proposition 5.11, if A ∈ A∗sep, we have A is a GC-algebra and a GCR-algebra. It follows from Theorems 4.6(ii) and 5.3(ii) that r⊲C = r⊲CCR = A.
As GC ⊆ GCR, we have I ≪GC J ⇒ I ≪GCR J ⇒ I ⊆ J in IdA. If I ⊆ J then J/I ∈ A∗sep by Lemma 5.10. Hence, by Proposition 5.11, J/I ∈ GC. So I ≪GC J. Thus ≪GC= ≪GCR= ⊆ in IdA.
As r⊲C = A, we have from Theorem 4.6(i) that there is an ascending transfinite ≪C-series of ideals from {0} to A. As A∗ is separable, A is separable. So, by 4.3.8 [D], the series is countable.
We will now construct a GCR-algebra which is a dN GCR-algebra.
Let R = C(K ⊗ H), where dim H = dim K = ∞, and A be a C∗-algebra in B(K). Then R ∩ (A ⊗ 1H) = {0}, B = R + A ⊗ 1H is a C*-algebra in B(K ⊗ H) by Corollary 1.8.4 [D], and
IdB = {{0}, R + I ⊗ 1H: I ∈ IdA}. (5.5) Indeed, if {0} 6= J ∈ IdB, J ∩ R is either {0} or R. If J ∩ R = {0} then JR = {0}, so J = {0}.
Thus R ⊆ J and J = R + I ⊗ 1H where I ∈ IdA. Conversely, R + I ⊗ 1H ∈ IdB for I ∈ IdA. Denote by Hn the tensor product of n copies of H. Let A0 = {0}, A1= C(H) and
An= C(Hn) + An−1⊗ 1H = C(Hn) +
n−1X
i=1
C(Hn−i) ⊗ 1Hi, (5.6) where 1Hi is the tensor product of i copies of 1H. As above, An are C*-algebras and, by (5.5), IdAn= {Ik}nk=0= {{0} = I0 ⊂ I1 ⊂ ... ⊂ In−1⊂ In= An} where
Ik= C(Hn) +
k−1X
i=1
C(Hn−i) ⊗ 1Hi ⊂ Ik+1 = Ik+ C(Hn−k) ⊗ 1Hk for 1 ≤ k ≤ n − 1. (5.7) For an isometry U from H onto H ⊗ H, set Un = U ⊗ 1Hn−1, n ≥ 2. Then Un is an isometry from Hn onto Hn+1. Thus θn: B → UnBUn−1 is an isomorphism of B(Hn) onto B(Hn+1) and θn(C(Hn)) = C(Hn+1). As Un= Un−i⊗ 1Hi for all i < n, we have θn= θn−i⊗ 1Hi, so that
θn(C(Hn−i) ⊗ 1Hi) = θn−i(C(Hn−i)) ⊗ 1Hi = C(Hn+1−i) ⊗ 1Hi. (5.8) Then IdAn+1 = {Jk}n+1k=0, where J0 = {0}, Jn+1= An+1 and
Jk(5.7)
= C(Hn+1) +
k−1X
i=1
C(Hn+1−i) ⊗ 1Hi (5.8)
= θn(Ik) for 0 ≤ k ≤ n.
Thus ϕkn= θn...θk is an isomorphism of Ak onto the ideal Jk. Identifying Ak and Jk, we have IdAn+1 = {Jk}n+1k=0 = {{0}, A1, A2, ..., An, An+1}. (5.9) The union ∪nAn with relation: a ∈ Ak ∼ b ∈ An+1, if ϕkn(a) = b, is a *-algebra with the C*-norm kak = kakA
k, if a ∈ Ak. Its completion A∞= ∪nAn – the inductive limit of {An}∞n=0 – is a C*-algebra and An can be considered as ideals of A∞: {0} = A0 ⊂ A1⊂ A2⊂ ... ⊂ A∞.
Proposition 5.13 Let A∞= ∪nAn. Then IdA∞ = {An}∞n=0 and {An}∞n=0 is a countable ascending
≪C-series of ideals from {0} to A∞: An+1/An≈ C(H).
The algebra A∞ is a GC-, GCR- and dN GCR-algebra and r⊲C = r⊲CCR = p⊳CCR = p⊳C = A∞..
Proof. Let J ∈ IdA∞. By Lemma III.4.1 [Da], J = ∪n(J ∩ An). Each J ∩ An is an ideal in An. By (5.9), J ∩ Ancoincides with some Akn in IdAn, kn≤ n. Thus J = ∪nAkn. If the sequence {kn} is unbounded then J = A∞. If it is bounded and k = max{kn} then J = Ak. Thus IdA∞= {An}∞n=0. By (5.6), Ak+1 = θk(Ak) + C(H) ⊗ 1Hk,so that Ak+1/Ak ≈ Ak+1/θk(Ak) ≈ C(H). Hence {An}∞n=0 is a countable ascending ≪C-series of ideals from {0} to A∞.
As the ascending ≪C-series of ideals from {0} to A exists, r⊲C = A∞. As r⊲C ⊆ r⊲CCR by Corollary 5.9, r⊲C = r⊲CCR = A∞. Hence, by Theorems 4.6 and 5.3, A∞ is a GC-, GCR-algebra.
By (5.3), p⊳CCR = p⊳C. As A∞ has no ideal I such that A∞/I ≈ C(H), we have p⊳C = A∞ by Corollary 3.3(ii). Hence p⊳CCR = p⊳C = A∞. Thus it is a dN GCR-algebra by Theorem 5.8.
5.2 C*-algebras with continuous trace and dual C∗-algebras Continuous trace C*-property. Let A be a GCR-algebra. Set
BA= {a ∈ A+: the function Ta(π) = Tr π(a) is finite and continuous on Π(A)},
NA= {a ∈ A: aa∗∈ BA} and MA= NA2. (5.10)
By Lemma 4.5.1 [D], the subspaces NA, MA are (non-closed) *-ideals of A,
MA= span(BA), BA= MA∩ A+ and MA= NA. (5.11) A C*-algebra A is called a continuous trace (c.t.) algebra, if MA = A. Each c.t. algebra is a CCR-algebra (Proposition 4.5.3 [D]) and C(H) is a c.t. algebra. Denote by Pc.t. the class of all c.t. algebras in A. It is a C*-property. Let ≪Pc.t. be the corresponding relation in IdA for A ∈ A.
Proposition 5.14 Pc.t. is a lower and upper stable C*-property; ≪Pc.t. is an and a dual H-relation.
Proof. Let A be a c. t. algebra, I ∈ IdA and p: A → A/I. Then, by (5.11),
p(MA) = p(MA) = p(A) = A/I. (5.12)
By Proposition 3.2.1 [D], θ: π → π ◦ p, π ∈ Π(A/I), is a homeomorphism from Π(A/I) onto the closed subset Π(A)I = {σ ∈ Π(A): σ(I) = 0} of Π(A). Let a ∈ BA. As Ta(σ) = Tr σ(a), σ ∈ Π(A), is a finite and continuous function on Π(A), it is finite and continuous on Π(A)I. As θ is a homeomorphism from Π(A/I) onto Π(A)I, the function Tp(a) on Π(A/I) defined by
Tp(a)(π) = Tr(π(p(a)) = Tr((π ◦ p)(a)) = Ta(π ◦ p) = Ta(θ(π)) for π ∈ Π(A/I), is finite and continuous on Π(A/I). Thus p(a) ∈ BA/I, so that p(BA) ⊆ BA/I. Hence
p(MA)(5.11)
= p(span(BA)) = span(p(BA)) ⊆ span(BA/I) = MA/I.
So, by (5.12), A/I = p(MA) ⊆ MA/I ⊆ A/I. Thus MA/I = A/I, so that A/I is a c. t. algebra.
Hence Pc.t. is upper stable.
As MA= NA= A by (5.11), the ideal NA has a bounded a. i. {aλ}λ∈Λ of A. Thus NA∩ I ∋ aλx → x for each x ∈ I. Hence NA ∩ I is a dense *-ideal of I. Set bλ,x = aλx(aλx)∗. Then bλ,x∈ BA∩ I by (5.10), and bλ,x→ xx∗. So BΛ,I = {bλ,x: λ ∈ Λ, x ∈ I} ⊆ BA∩ I is dense in I+.
Each π ∈ Π(I) extends to π′ ∈ Π(A) and ϕ: π → π′ is a homeomorphism from Π(I) to the open set Π(A)I = {σ ∈ Π(A): σ(I) 6= 0} in Π(A) (Proposition 3.2.1 [D]). As bλ,x ∈ BA, the function Tbλ,x: σ → Tr σ(bλ,x), σ ∈ Π(A), is finite and continuous on Π(A). So it is continuous on Π(A)I and the function bTbλ,x on Π(I) defined by
Tbbλ,x(π) = Tr(π(bλ,x)) = Tr(π′(bλ,x)) = Tbλ,x(ϕ(π)), π ∈ Π(I),
is finite and continuous on Π(I). Thus bλ,x∈ BI, so that BΛ,I ⊆ BI. As BΛ,I is dense in I+, BI is dense in I+. So MI = span(BI) is dense in I. Thus I is a c. t. algebra. Hence Pc.t. is lower stable.
It follows from Theorem 3.2 that ≪Pc.t. is an H- and a dual H-relation.
Let r⊲
Pc.t. be the ≪⊲
Pc.t.-radical and r⊲CCRbe the ≪⊲CCR-radical in IdA. We generalize now Theorems 4.5.5 and 4.7.12 a), b) [D].
Proposition 5.15 (i) GCR = GPc.t., and the equality r⊲CCR = r⊲
Pc.t. holds in IdA for all A ∈ A.
(ii) A is a GCR-algebra if and only if A = r⊲
Pc.t., i.e., A has an ascending transfinite ≪Pc.t.-series (Iλ)1≤λ≤γ of ideals, I1= {0}, Iγ= A and all Iλ+1/Iλ are continuous trace algebras.
(iii) If r⊲Pc.t. " I 6= A then J/I is a continuous trace algebra for some J ∈ IdA, I$ J.
Proof. (i) By Proposition 4.5.3 [D], Pc.t. ⊆ CCR. Let B ∈ CCR and I ∈ IdB. Then B/I ∈ CCR. By Lemma 4.4.4 [D], each GCR-algebra has a c.t.-ideal. Hence B/I has such an ideal. Then, by Definition 4.2, B ∈GPc.t.. So Pc.t. ⊆ CCR ⊆ GPc.t.. The rest follows from Proposition 4.10.
(ii) By (i) and Theorem 5.3, A is a GCR-algebra if and only if A = r⊲
Pc.t.. The rest of (ii) and (iii) follow from Corollary 3.3(i).
Let p⊳
Pc.t. be the dual ≪⊳
Pc.t.-radical and p⊳CCR be the dual ≪⊳CCR-radical of A. Recall (Definition 4.14) that A is a dGCR-algebra, if each I ∈ IdA has a CCR quotient. For dual radicals we obtain results similar to the results of Proposition 5.15.
Proposition 5.16 (i) dGCR = dGPc.t., and the equality p⊳CCR = p⊳
Pc.t. holds in IdA for all A ∈ A.
(ii) A is a dGCR-algebra if and only if p⊳
Pc.t. = {0}, i.e., A has a descending transfinite ≪Pc.t. -series (Iλ)1≤λ≤γ of ideals, I1 = A, Iγ = {0} and all Iλ/Iλ+1 are continuous trace algebras.
(iii) If {0} 6= I " p⊳Pc.t. then I/J is a continuous trace algebra for some J ∈ IdA, J $ I.
Proof. (i) By Proposition 4.5.3 [D], Pc.t. ⊆ CCR. Let B ∈ CCR and I ∈ IdB. Then I ∈ CCR.
For π ∈ Π(I), π(I) ∈ C by Definition 4.25. Hence I/ ker π ≈ π(I) ∈ C ⊆ Pc.t.. Thus, by Definition 4.14, B ∈ dGPc.t.. So Pc.t. ⊆ CCR ⊆ dGPc.t.. The rest follows from Proposition 4.18.
(ii) By (i) and Theorem 4.19, A is a GCR-algebra if and only if p⊳
Pc.t. = {0}. The rest of (ii) and (iii) follow from Corollary 3.3(ii).
The C*-property of dual algebras. The left and right annihilators of a subset E of an algebra A are defined by lan(E) = {a ∈ A: aE = {0}} and ran(E) = {a ∈ A: Ea = {0}}.
A C*-algebra A is dual (see 4.7.20 [D]) if, for each closed left ideal L and right ideal R, lan(ran(L)) = L and ran(lan(R)) = R.
If I ∈ IdAthen lan(I) = ran(I) are closed ideals (Lemma 32.4 [BD]). Set an(I) := lan(I) = ran(I).
Kaplansky [Ka] proved that A ∈ A is dual if and only if it is a C*(∞)-sum of the algebras C(Hλ) on some Hilbert spaces (restricted product of C*-algebras [D, 1.9.14]), i.e.,
A = {a = (aλ)λ∈Λ: aλ∈ C(Hλ); for any ε > 0, the set {λ ∈ Λ: kaλk ≥ ε} is finite}. (5.13) We denote by D the C*-property that consists of all dual C*-algebras. It can be also charac-terized ([AW]) as the class of all compact C*-algebras (a Banach algebra A is compact if, for each a ∈ A, the map x 7→ axa is a compact operator on A).
Proposition 5.17 The C*-property D is lower and upper stable.
Proof. If A ∈ D and I ∈ IdA. Then, for each a ∈ I, the operator x 7→ axa is compact on I. So I is a compact algebra whence I ∈ D. Thus D is a lower stable C*-property.
By Lemma 32.4 [BD], I⊕an(I) = A. Hence A/I ≈ an(I). As an(I) ∈ D by the above, A/I is dual. Thus D is upper stable.
For A ∈ A, define the relation ≪D on IdAas in (1.2). It is not transitive, as {0} ≪D C(H) ≪D C(H) +C1, if dim H = ∞, while C(H) + C1 is not dual. By Theorem 3.2 and Proposition 5.17,
≪D is an H- and a dual H-relation. Let r⊲D be the ≪⊲D-radical and p⊳D the dual ≪⊳D-radical in A.
Recall that each A ∈ A has the ≪⊲C-radical r⊲C and the dual ≪⊳C-radical p⊳C.
Proposition 5.18 (i) Each A ∈ D is a GC- and a dGC-algebra, so that D ⊆ GC ∩ dGC.
(ii) For A ∈ A, r⊲D = r⊲C and p⊳D = p⊳C. If A is dual then r⊲C = A and p⊳C = {0}.
Proof. (i) Let A ∈ D and I ∈ IdA. By Proposition 5.17, I and A/I are dual. By (5.13), dual C*-algebras have ideals isomorphic to C(H). So A/I has such an ideal. Thus A is a GC-algebra.
As I is dual, it has a quotient isomorphic to C(H) by (5.13). So A is a dGC-algebra.
(ii) If B ∈ C then B ≈ C(H). So B is dual. Thus C ⊆ D. By (i), C ⊆ D ⊆ GC and C ⊆ D ⊆ dGC. Hence, by Propositions 4.10 and 4.18, r⊲D = r⊲C and p⊳D = p⊳C for all A ∈ A.
Combining the above results with (5.3), we have for each A ∈ A, p⊳NG
C = r⊲G
C = r⊲C = r⊲D ⊆ r⊲CCR = r⊲
Pc.t. = r⊲GCR = p⊳N GCR, p⊳GCR ⊆ p⊳CCR = p⊳C = p⊳
Pc.t. = p⊳D = p⊳dGCR = r⊲d .
Corollary 5.19 (i) GD = Sc - the C*-property of scattered algebras (see Example 4.4).
(ii) dGD = dGC ⊃ RF D ∪ CCR.
(iii) R(D) = CCR.
Proof. (i) Since C ⊂ D ⊂ GC by Proposition 5.18, GD = GC by Proposition 4.10. So GD = Sc, since GC = Sc by Example 4.4.
(ii) Similarly, dGD = dGC. The inclusion dGC ⊃ CCR follows from Theorem 4.26 for P = C.
If A ∈ RF D then ∩{ker π: π ∈ Π(A), dim π < ∞} = {0} by (4.5). Hence, for each J ∈ IdA, there is π ∈ Π(A), dim π < ∞, with π(J) 6= 0. So the algebra J/(J ∩ ker(π)) ∈ C. Thus A ∈ dGC.
(iii) Let A ∈ R(D). Then, for each π ∈ Π(A), the algebra π(A) ∈ D. As π(A) is irreducible, its center is trivial. So it follows from (5.13) that π(A) ≈ C(H). Thus π(A) ∈ C. Hence R(D) ⊂ R(C) = CCR. The converse inclusion is evident.
5.3 Real rank zero, AF, nuclear and exact C*-algebras
A unital C*-algebra A has real rank zero ([BP], [Da]) if its invertible selfadjoint elements are dense in the set of all selfadjoint elements of A. A non-unital algebra is real rank zero if its unitization is real rank zero. Denote by R0 the class of all real rank zero algebras. Clearly, R0 is a C*-property.
Let I ∈ IdA. By Theorem 3.14 [BP], A is a R0-algebra if and only if I and A/I are R0-algebras and all projections in A/I lift to projections in A. In Remark 3.17 [BP] Brown and Pedersen write that ”split extensions of real rank zero algebras by other real rank zero algebras produce algebras of real rank zero. For general extensions this is no longer true: Every Bunce-Deddens algebra ABD has a one-dimensional extension bABD (determined by a nonliftable projection in the corona), with real rank one.” Combining this with Theorem 3.14 [BP] yields
Corollary 5.20 The C*-property R0 is lower and upper stable, but not extension stable.
By Theorem 3.2, Proposition 3.5 and Corollary 5.20, the relation ≪R0 in IdA, A ∈ A, (see (1.2)) is an H- and a dual H-relation, but not necessarily transitive. Let r⊲
R0 be the ≪⊲
R0 is the largest GR0-ideal of A and contains all R0-ideals. It is also the smallest ideal with NGR0-quotient.
(iii) A is a GR0-algebra if and only if A = r⊲
R0; it is a NGR0-algebra if and only if r⊲
R0 = {0}.
(iv) p⊳
R0 is the largest dNGR0-ideal of A. It is the smallest ideal with dGR0-quotient.
(v) A is a dGR0-algebra if and only if p⊳
R0 = {0}; it is a dNGR0-algebra if and only if p⊳
R0 = A.
Proof. (i) If A ∈ C, A ≈ C(H) for some separable H. So A ∈ R0. Thus C ⊂ R0. So (i) follows from Propositions 4.10 and 4.18. Parts (ii) – (v) follow from Theorems 4.6 and 4.19.
Corollary 3.3 further yields Theorem 5.22 (i) If r⊲
R0 * I 6= A then there is an ideal I $ J such that J/I is a R0-algebra.
(ii) If I ⊆ r⊲
R0 then there is an ascending transfinite ≪R0-series of ideals from I to r⊲
R0. (iii) If {0} 6= I * p⊳R0 then there is an ideal J $ I such that I/J is R0-algebra.
(iv) If p⊳
R0 ⊆ I then there is a descending transfinite ≪R0-series of ideals from I to p⊳
R0.
A C*-algebra A is approximately finite-dimensional (AF -algebra) if it is the closure of an increasing union of finite-dimensional *-subalgebras. It is nuclear if, for each C*-algebra B, the norms k·kmax and k·kmin on the algebraic tensor product A ⊙ B coincide, so that there is only one C*-norm on A ⊙ B. Furthermore, A is exact if, for each C*-algebra B and every I ∈ IdB,
0 → A ⊗minI → A ⊗minB → A ⊗minB/I is an exact sequence.
Denote by AF, N U and EX the corresponding C*-properties of C*-algebras. Finite-dimensional and commutative C*-algebras, C(H), all AF and all C*-algebras of type I are nuclear. Nuclear algebras are exact. So
AF ⊆ N U ⊆ EX. (5.14)
The next theorem follows from the results in [Da], [K], Corollaries 2.5, 9.3 [W], Corollary XV,3.4 [T], [Br].
Theorem 5.23 (i) The C*-properties AF , N U and EX are lower stable, upper stable and closed under inductive limits;
(ii) The C*-properties AF and N U are extension stable.
By Theorems 3.2 and 5.23, the C*-properties AF , N U , EX define H- and dual H-relations
≪AF, ≪N U, ≪EX on IdA for each A ∈ A.
Corollary 5.24 Theorems 5.21 and 5.22 hold with ”R0” being replaced by AF, N U and EX, respectively.
For nuclear and AF algebras, Theorem 5.23 yields (cf. Theorem 2.22, Corollary 2.23 [ST]):
Corollary 5.25 (i) For each A ∈ A, the relations ≪AF, ≪N U are R-orders in IdA, so that
≪AF = ≪⊲AF and ≪N U = ≪⊲N U . (5.15)
(ii) For each A ∈ A, the ≪AF-radical rAF is the largest AF-ideal in A and the ≪N U-radical rN U is the largest nuclear ideal in A.
(ii) G(AF ) = AF and G(N U ) = N U.
Proof. (i) By Theorem 5.23, AF and N U are upper stable and extension stable (Definition 3.4) C*-properties; the inductive limit of AF -algebras is an AF -algebra and of N U -algebras is a N U -algebra. Hence, by Corollary 4.12, ≪AF, ≪N U are R-order in IdA and (5.15) holds.
(ii) From (i) and from Theorem 4.6 it follows that rAF = r⊲AF (the ≪⊲AF-radical) is the largest AF ideal in A. Similarly, rN U = r⊲N U (the ≪⊲N U-radical) is the largest nuclear ideal in A.
(iii) follows from (5.15) and Corollary 4.23.
From (4.4), (5.14) and Corollary 5.25 it follows that rAF ⊆ rN U ⊆ r⊲EX, so that A/rAF has no AF ideals, A/rN U has no nuclear ideals and A/r⊲EX has no exact ideals by Corollary 3.3.