Ideals in Multiplier Algebras.

Transcription

1 Ideals in Multiplier Algebras. Victor Kaftal University of Cincinnati ECOAS U. Lousiana, October 8, 2017 Joint work with P.W. Ng (U. Louisiana) & Shuang Zhang (U. Cincinnati) Victor Kaftal Ideals in Multiplier Algebras. 1 / 22

2 Some questions about ideals in M(A) C*-algebra A: σ-unital non-unital K simple stably finite. M(A) multiplier algebra of A I min := {J M(A), A J } Victor Kaftal Ideals in Multiplier Algebras. 2 / 22

3 Some questions about ideals in M(A) C*-algebra A: σ-unital non-unital K simple stably finite. M(A) multiplier algebra of A I min := {J M(A), A J } Relation between various characterizations of I min When is A I min? When is M(A)/A purely infinite? When does M(A) have finitely many ideals? Strict comparison for M(A)? Victor Kaftal Ideals in Multiplier Algebras. 2 / 22

4 Notations: the tracial simplex T (A) is the Choquet simplex of densely defined, lower semicontinuous tracial weights on A, normalized on a nonzero positive element of the Pedersen ideal Ped(A), with topology of pointwise convergence on Ped(A). E.g., if A = A o K for a unital algebra A o, then T (A) is identified with the tracial states on A o. Traces can be extended to A and hence to M(A). The extremal boundary e (T (A)) is the collection of extremal traces of T (A). I τ := span{x M(A) + τ(x ) < } Victor Kaftal Ideals in Multiplier Algebras. 3 / 22

5 Notations: the evaluation & dimension functions Given 0 A M(A) +, the evaluation function: T (A) τ Â(τ) := τ(a). Â LAff(T (A)) ++ σ, is the pointwise limit of an increasing sequence of strictly positive continuous affine functions. the dimension function: d τ (A) = lim n τ(a 1/n ) = τ(r A ) d τ (A) LAff(T (A)) ++ σ. S := 1 M(A) the scale of A. f LAff(T (A)) ++ σ is complemented in S if g LAff(T (A)) ++ σ {0}, s.t. f + g = S. Victor Kaftal Ideals in Multiplier Algebras. 4 / 22

6 Projection-surjectivity/injectivity Definition A is 1-projection-surjective if for every f LAff(T (A)) ++ σ complemented in S there is a projection P M(A) \ A such that ˆP = f. A is 1-projection-injective if for projections P, Q M(A) \ A, ˆP = ˆQ P Q. A is projection-surjective (resp., projection-injective) if M (A) is 1-projection-surjective (resp., 1-projection-injective) Victor Kaftal Ideals in Multiplier Algebras. 5 / 22

7 Algebras that are projection-surjective/injective Perera (2001): Projection-surjectivity and injectivity for simple σ-unital nonunital nonelementary C*-algebras with RR0, stable rank one, and weakly unperforated K 0 (A) ( strict comparison of projections by quasitraces): V (M(A)) V (A) W d σ (S u ) Victor Kaftal Ideals in Multiplier Algebras. 6 / 22

8 Algebras that are projection-surjective/injective Perera (2001): Projection-surjectivity and injectivity for simple σ-unital nonunital nonelementary C*-algebras with RR0, stable rank one, and weakly unperforated K 0 (A) ( strict comparison of projections by quasitraces): V (M(A)) V (A) W d σ (S u ) Lin&Ng: Projection-surjectivity and injectivity for A K if A is unital separable simple exact stable finite and Z-stable. These include a large class of algebras... Victor Kaftal Ideals in Multiplier Algebras. 6 / 22

9 Stable algebras with finitely many extremal traces Elliott (1974) stable matroid algebras Lin (1988) simple stable AF algebras Rørdam (1991) simple stable algebras with finite e (T (A)) and strict comparison by traces M(A) has finitely many ideals, all arising from traces I min = I τ. τ e(t (A)) Victor Kaftal Ideals in Multiplier Algebras. 7 / 22

10 Other approaches to I min Lin&Zhang (1992), studied separable algebras with property (SP) and approximate identity of projections and proved the existence of thin sequences (aka l 1 sequences) of projections: 0 p A + N m > n N m p j p. n The ideal generated by any thin sequence is I min. Perera (2001) studied real rank zero stable rank one algebras with strict comparison of projections by quasi-traces and defined I min in terms of continuity of the evaluation functions (see the ideal I cont defined more in general later). Victor Kaftal Ideals in Multiplier Algebras. 8 / 22

11 Lin s approach (1991, 2004) Cuntz subequivalence: a, b A +, a b if x n A such that a x n bx n 0. A simple σ-unital w. approx. identity {e n } (with e n+1 e n = e n ). K o ({e n }) := {X M(A) + 0 a A + N N m > n N (e m e n )X (e m e n ) a}. K o ({e n }) is a positive hereditary cone in M(A). Definition&Theorem, Lin (1991) I min := spank o ({e n }) = {X M(A) X X K o ({e n })} Theorem (Lin (1991, 2004)) I min = M(A) (i.e., M(A)/A is simple) A has continuous scale M(A)/A is simple and purely infinite. Victor Kaftal Ideals in Multiplier Algebras. 9 / 22

12 Questions about I min Theorem If I min A then I min /A is p.i. & simple, hence has real rank zero. Theorem A I min in the following cases (i) A is separable; (ii) the Cuntz semigroup is order separable ( {b n } 1 A + s.t. 0 a A +, b n a some n); (iii) A has the (SP) property and its dimension semigroup D(A) of Murray-von Neumann equivalence classes of projections is order separable; (iv) A has strict comparison of positive elements by traces Question Is A I min always? Victor Kaftal Ideals in Multiplier Algebras. 10 / 22

13 Strict comparison and I cont Definition (Strict comparison in a simple algebra) A has strict comparison of positive elements by traces if a b whenever a, b A + with d τ (a) < d τ (b) for all those τ T (A) for which d τ (b) < (here d τ (a) = lim n τ(a 1/n ) = τ(r a )). I cont = span{a M(A) + Â continuous} A I cont I min Theorem if A is simple σ-unital nonunital nonelementary and with strict comparison by traces, then I min = I cont Victor Kaftal Ideals in Multiplier Algebras. 11 / 22

14 Example where I min I cont Villadsen constructed AH algebras where strict comparison does not hold (the dimension growth is not slow). We specialize his construction and obtain Theorem I min I cont = I τ. We can say more: Theorem If A has flat dimension growth, then I cont is the largest proper ideal of M(A K). Victor Kaftal Ideals in Multiplier Algebras. 12 / 22

15 The Villadsen-type algebras For AH algebras, slow dimension growth strict comparison of positive elements. To avoid it, choose large dimensions and special connecting maps X i = D n 0 CP n1 CP n i Φ i,i+1 : C(X i ) K C(X i+1 ) K with n i i(i!) special connecting maps. p i = Φ 0,i (θ) θ trivial line bundle over X 0 A i = p i (C(X i ) K)p i for i 0; A = lim (A i, Φ i,i+1 ). A is an AH algebra with unique tracial weight and no strict comparison. Victor Kaftal Ideals in Multiplier Algebras. 13 / 22

16 I min I cont Construction: q i := Φ i, (θ i ) e ii A K with θ i trivial line bundles. Q := k=1 q k. Then τ(q) <, hence Q I cont. r i := Φ i, (ρ i ) e ii A K with ρ i pullbacks of universal line bundles. The sequence {r i } is thin R := k=1 r k I min \ A K. The manifolds corresponding to q i (resp., r i ), have all Euler class zero (resp., nonzero). Theorem Q I (R), hence I min I cont. Theorem If A has flat dimension growth, then I cont is the largest proper ideal of M(A K). Victor Kaftal Ideals in Multiplier Algebras. 14 / 22

17 Strict comparison in the multiplier algebra Definition M(A) has strict comparison of positive elements by traces, if A B whenever (i) A, B M(A) + with d τ (A) < d τ (B) for all those τ T (A) for which d τ (B) < and (ii) A belongs to the ideal I (B) generated by B. Theorem If A has strict comparison, then I cont (= I min ) has strict comparison In particular if A has continuous scale then M(A) (= I cont ), has strict comparison. When else does M(A) have strict comparison? Victor Kaftal Ideals in Multiplier Algebras. 15 / 22

18 Quasicontinuous scale Definition (Kucerovsky&Perera (2011) in RR=0 case) The scale of A, S = 1 M(A), is said to be quasicontinuous if (i) the set F := {τ e (T (A)) S(τ) = } is finite (possibly empty); Victor Kaftal Ideals in Multiplier Algebras. 16 / 22

19 Quasicontinuous scale Definition (Kucerovsky&Perera (2011) in RR=0 case) The scale of A, S = 1 M(A), is said to be quasicontinuous if (i) the set F := {τ e (T (A)) S(τ) = } is finite (possibly empty); (ii) the complementary face F of co(f ) is closed (possibly empty) (F is the union of the faces of T (A) disjoint from co(f )); Victor Kaftal Ideals in Multiplier Algebras. 16 / 22

20 Quasicontinuous scale Definition (Kucerovsky&Perera (2011) in RR=0 case) The scale of A, S = 1 M(A), is said to be quasicontinuous if (i) the set F := {τ e (T (A)) S(τ) = } is finite (possibly empty); (ii) the complementary face F of co(f ) is closed (possibly empty) (F is the union of the faces of T (A) disjoint from co(f )); (iii) the restriction S F is continuous. Victor Kaftal Ideals in Multiplier Algebras. 16 / 22

21 Quasicontinuous scale Definition (Kucerovsky&Perera (2011) in RR=0 case) The scale of A, S = 1 M(A), is said to be quasicontinuous if (i) the set F := {τ e (T (A)) S(τ) = } is finite (possibly empty); (ii) the complementary face F of co(f ) is closed (possibly empty) (F is the union of the faces of T (A) disjoint from co(f )); (iii) the restriction S F is continuous. If A is stable then it has quasicontinuous scale if and only if e (T (A)) <. Victor Kaftal Ideals in Multiplier Algebras. 16 / 22

22 Quasicontinuous scale Definition (Kucerovsky&Perera (2011) in RR=0 case) The scale of A, S = 1 M(A), is said to be quasicontinuous if (i) the set F := {τ e (T (A)) S(τ) = } is finite (possibly empty); (ii) the complementary face F of co(f ) is closed (possibly empty) (F is the union of the faces of T (A) disjoint from co(f )); (iii) the restriction S F is continuous. If A is stable then it has quasicontinuous scale if and only if e (T (A)) <. Theorem If A is σ-unital simple, has strict comparison and has quasicontinuous scale then M(A) has strict comparison. Victor Kaftal Ideals in Multiplier Algebras. 16 / 22

23 Beyond I min Definition I b := {I τ τ T (A)} I fin := {I τ τ e (T (A))} I min I cont I b I fin All these inclusions can be proper. If A has strict comparison and e (T (A)) < or if A has continuous scale, then all these ideals coincide. Victor Kaftal Ideals in Multiplier Algebras. 17 / 22

24 Beyond I min Definition I b := {I τ τ T (A)} I fin := {I τ τ e (T (A))} I min I cont I b I fin All these inclusions can be proper. If A has strict comparison and e (T (A)) < or if A has continuous scale, then all these ideals coincide. The following theorem extends work of Kucherovsky-Perera and Kucherovsky-Ng-Perera which assumes RR0 and for some implications requires that T (A) is a Bauer simplex. Victor Kaftal Ideals in Multiplier Algebras. 17 / 22

25 Main theorem Theorem Let A be a non-unital, σ-unital, simple C*-algebra with metrizable T (A), projection-injectivity, projection-surjectivity and strict comparison of positive elements by traces for M n (A) for all n. Then TFAE. (i) A has quasicontinuous scale; (ii) M(M n (A)) has strict comparison of positive elements by traces for all n; (iii) M(A)/A is purely infinite; (iv) M(A)/I min is purely infinite; (v) M(A) has finitely many ideals; (vi) I min = I fin. Victor Kaftal Ideals in Multiplier Algebras. 18 / 22

26 Not all the hypotheses are required (i) QC scale (ii) M(M n (A)) str. comp. (iii) M(A)/A p.i. (iv) M(A)/I cont p.i. (v) M(A) fin. many ideals (vi) I cont = I fin Hypotheses codes: (p) projection surjectivity/injectivity+ metrizability of T (A) (sc) strict comparison of positive elements by traces for A Implications (i) (ii) (iii) (iv) (sc) (sc) (sc) (v) (vi) (i) (p) (p) Victor Kaftal Ideals in Multiplier Algebras. 19 / 22

27 Chains of ideals& more With the same standing hypotheses on A. Proposition If A is stable, then A has QC scale (equivalently, e (T (A)) < ) if and only if I min = I b. Proposition if P I fin \ I b is a projection (resp., P I b \ I min ), then there is an infinite decreasing chain of principal ideals I (P n ) in I fin \ I b (resp., I b \ I min ). Victor Kaftal Ideals in Multiplier Algebras. 20 / 22

28 Some questions Question Can we find increasing chains of principal ideals? Yes if A is stable and e (T (A)) = ℵ 0. Question Can we replace I min = I b with I b = I fin? Yes if T (A) is a Bauer simplex. Victor Kaftal Ideals in Multiplier Algebras. 21 / 22

29 THANK YOU Victor Kaftal Ideals in Multiplier Algebras. 22 / 22