Central Sequences in C*-Algebras and Strongly Purely Infinite Algebras

摘要: If A is a separable unital C∗–algebra and if the relative commutant := A′ ∩Aω simple, then either = C · 1A ∼= Mn, or are both simple purely infinite. In particular, A⊗O∞ 6= ·1A. version of this result for non-unital given A/Ann(A,Aω) simple. The converse holds in nuclear case: infinite, separable, unital, (and infinite). We show that Q 1 Calkin algebra L/K, contrast to case. introduce an invariant cov(B) ∈ N ∪ {∞} C∗–algebras B with ≤ cov(C) there *-homomorphism from into B. has no finite-dimensional quotient dr(B) + decomposition rank cov(Z) 2 Jian–Su Z, because dr(Z) 1. It shown (non-simple) strongly infinite sense [12] does not admit non-trivial lower semi-continuous 2-quasi-trace, cov(A/Ann(Aω, A)) < ∞ image C∗((0, 1],M2) generates full hereditary C∗-subalgebra A/Ann(Aω, A)). follows A/Ann(A) contains C∗– unitally such ∞. ⊗ Z A+ admits 2-quasi-trace. case suppose C∗–algebra. Let ω free ultra-filter on N. also denote by related character `∞(N) ω(c0(N)) {0}. Recall limω αn means complex number ω(α1, α2, . .) (α1, `∞(N). Then Aω `∞(A)/cω(A) cω(A) {(a1, a2, `∞(A) ; , ‖an‖ 0}. natural epimorphism onto denoted πω. Sometimes we say (a1, representing sequence b πω(a1, b. consider as C∗–subalgebra diagonal embedding 7→ πω(a, a, (a, cω(A), Date: Aug 31, 2004. 1991 Mathematics Subject Classification. Primary: 46L35; Secondary: 46L80. let ∩ (ω–) central sequences A. (two-sided) annihilator Ann(A) Ann(A,Aω) {b bA {0} Ab} contained but carry much information about below mentioned (or later needed) basic facts proved Section 3 (Appendix). closed ideal A, C∗-algebra. only unital. There ρ : (A/Ann(A))⊗ A→ ρ((d Ann(A)) b) db d ρ(1 (cf. (A.1)). K compact operators `2(N). huge, K/Ann(K) Cω. More generally, p projection naturally isomorphic (pAp) ⊂ (pAp)ω p(Aω)p (A.1) Appendix.). cf. (A.2). To get main Theorem 1.8 section, have improve here (in where simple) some general results Appendix. Remark 1.1. σ-unital JA generated if, K(H) Hilbert space H. 6∼= K(H), K(Hω) Dim(H) =∞). Proof. easy see (with help sequences) b, c (Aω)+ contraction ‖c‖d∗bd ‖b‖c Conversely, Clearly, Suppose any H, i.e. antiliminary. (JA)+ ‖b‖ ‖c‖. Since antiliminary, (A.10) exists *-monomorphism ψ C0((0, 1],K) ↪→ bψ(f) f every 1],K). D subalgebra ψ. non-zero, stable satisfies bg g gb all D. JA. stable, d∗d dd∗ Thus d∗bd c. can take find (d1, d2, Aω. Lemma 1.2. C∗–algebra, properly non-zero ∗-subalgebra AD +DA ∩D 1A. faithful *-representation φ → L(H) over H normal state μ L(H). By assumption, isometries s1, s2 s1s2 0. a1, dense positive contractions c1 ∑ n≥1(s2) s1ans ∗ 1(s 2) (as C∗–algebra) five self-adjoint elements c1, c2 (s s1)/2, c3 − s1)/2i, c4 (s2 s2)/2, c5 s2)/2i norm 1, *-epimorphism h C(F5)→ h(gj) ej Here F5 denotes group 5 generators g1, g5, (F5) ∗–group algebra. l(w) reduced word-length element w F5. (obviously) l(w1w2) l(w1)+ l(w2) one easily R(n) ]{w n} tends n→∞ 10.