Maximal Ideals of Triangular Operator Algebras

Transcription

1 Maximal Ideals of Triangular Operator Algebras John Lindsay Orr University of Nebraska Lincoln and Lancaster University May 17, 2007 John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

2 jorr/presentations John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

3 Ideals of upper triangular operators Statement of the problem Let H := l 2 (N) and let {e k } k=1 be the standard basis. Let T be the algebra of all (bounded) operators which are upper triangular with respect to {e k }. Question What are the maximal two-sided ideals of T? John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

4 Ideals of upper triangular operators Statement of the problem Let H := l 2 (N) and let {e k } k=1 be the standard basis. Let T be the algebra of all (bounded) operators which are upper triangular with respect to {e k }. Question What are the maximal two-sided ideals of T? All ideals are assumed two-sided. John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

5 Ideals of upper triangular operators Statement of the problem What would I like the answer to be? John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

6 Ideals of upper triangular operators Statement of the problem What would I like the answer to be? Observe that D, the set of diagonal operators w.r.t. {e k } is *-isomorphic to l (N), so we identify them. Write S for the set of strictly upper triangular operators w.r.t. {e k }. Fact Let M be a maximal ideal of l (N) and let J := M + S. Then J is a maximal ideal of T. John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

7 Ideals of upper triangular operators Statement of the problem What would I like the answer to be? Observe that D, the set of diagonal operators w.r.t. {e k } is *-isomorphic to l (N), so we identify them. Write S for the set of strictly upper triangular operators w.r.t. {e k }. Fact Let M be a maximal ideal of l (N) and let J := M + S. Then J is a maximal ideal of T. Proof. Write (T ) for the diagonal part of T. Suppose T J. John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

8 Ideals of upper triangular operators Statement of the problem What would I like the answer to be? Observe that D, the set of diagonal operators w.r.t. {e k } is *-isomorphic to l (N), so we identify them. Write S for the set of strictly upper triangular operators w.r.t. {e k }. Fact Let M be a maximal ideal of l (N) and let J := M + S. Then J is a maximal ideal of T. Proof. Write (T ) for the diagonal part of T. Suppose T J. T (T ) = J S J and so (T ) J, hence (T ) M. Thus D (T ) + M = I and so D(T J) + M = I T, J. John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

9 Ideals of upper triangular operators Statement of the problem The maximal ideals of l (N) are points in βn, the Stone-Cech compactification of N, so this would give a good description of the maximal ideals of T. Question Are all the maximal ideals of T of the form M + S where M is a maximal ideal of l (N)? John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

10 Ideals of upper triangular operators Re-statement of the problem Proposition TFAE: 1 All the maximal ideals of T are of the form M + S. 2 All the maximal ideals of T contain S. 3 No proper ideal of T contains an operator I + S, (S S). John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

11 Ideals of upper triangular operators Re-statement of the problem Proposition TFAE: 1 All the maximal ideals of T are of the form M + S. 2 All the maximal ideals of T contain S. 3 No proper ideal of T contains an operator I + S, (S S). Proof. (1) (2) (3): Obvious. John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

12 Ideals of upper triangular operators Re-statement of the problem Proposition TFAE: 1 All the maximal ideals of T are of the form M + S. 2 All the maximal ideals of T contain S. 3 No proper ideal of T contains an operator I + S, (S S). Proof. (1) (2) (3): Obvious. (3) (2): Contrapositive. Suppose J S is a maximal ideal of T. Then J + S = T and so I = J S. John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

13 Ideals of upper triangular operators Re-statement of the problem Proposition TFAE: 1 All the maximal ideals of T are of the form M + S. 2 All the maximal ideals of T contain S. 3 No proper ideal of T contains an operator I + S, (S S). Proof. (1) (2) (3): Obvious. (3) (2): Contrapositive. Suppose J S is a maximal ideal of T. Then J + S = T and so I = J S. (2) (1): Let J be a maximal ideal of T. Since J S, then also J (J ). But (J ) D so let M (J ) be a maximal ideal of D and we saw M + S is a maximal ideal of T that contains J. John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

14 Ideals of upper triangular operators Re-statement of the problem Proposition TFAE: 1 All the maximal ideals of T are of the form M + S. 2 All the maximal ideals of T contain S. 3 No proper ideal of T contains an operator I + S, (S S). Proof. (1) (2) (3): Obvious. (3) (2): Contrapositive. Suppose J S is a maximal ideal of T. Then J + S = T and so I = J S. (2) (1): Let J be a maximal ideal of T. Since J S, then also J (J ). But (J ) D so let M (J ) be a maximal ideal of D and we saw M + S is a maximal ideal of T that contains J. John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

15 Ideals of upper triangular operators Re-statement of the problem Question Is it possible for an operator of the form I + S (S strictly upper triangular) to lie in a proper ideal of T? John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

16 Ideals of upper triangular operators Re-statement of the problem Question Is it possible for an operator of the form I + S (S strictly upper triangular) to lie in a proper ideal of T? Just to be clear, an operator X fails to belong to a proper ideal of T iff we can find A 1,..., A n and B 1,..., B n such that A 1 XB A n XB n = I John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

17 Ideals of upper triangular operators Operators of the form I + S In finite dimensions, all operators I + S are invertible. John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

18 Ideals of upper triangular operators Operators of the form I + S In finite dimensions, all operators I + S are invertible. Not so in infinite dimensions Let be the unilateral backward shift Then I U = is not invertible John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

19 Ideals of upper triangular operators Operators of the form I + S Nevertheless this isn t a counterexample. It s easy to see that I U doesn t lie in any proper ideal of T : John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

20 Ideals of upper triangular operators Operators of the form I + S Nevertheless this isn t a counterexample. It s easy to see that I U doesn t lie in any proper ideal of T : Let σ N and let P σ := Proj (span{e k Note UP 2N = P 2N 1 U and UP 2N 1 = P 2N U : k σ}) John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

21 Ideals of upper triangular operators Operators of the form I + S Nevertheless this isn t a counterexample. It s easy to see that I U doesn t lie in any proper ideal of T : Let σ N and let P σ := Proj (span{e k : k σ}) Note UP 2N = P 2N 1 U and UP 2N 1 = P 2N U Thus P 2N (I U)P 2N + P 2N 1 (I U)P 2N 1 = I John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

22 Ideals of upper triangular operators Operators of the form I + S Nevertheless this isn t a counterexample. It s easy to see that I U doesn t lie in any proper ideal of T : Let σ N and let P σ := Proj (span{e k : k σ}) Note UP 2N = P 2N 1 U and UP 2N 1 = P 2N U Thus P 2N (I U)P 2N + P 2N 1 (I U)P 2N 1 = I This simple observation connects us to a famous open problem known as The Kadison-Singer problem or The Paving Problem. John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

23 The Kadison-Singer Problem The paving problem Let the standard atomic masa, D, and the projections, P σ, be as defined before. Definition Say that X B(H) can be paved if, given any ɛ > 0, there are pwd sets σ 1,... σ n N such that σ 1 σ n = N and n (X ) P σk XP σk < ɛ k=1 John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

24 The Kadison-Singer Problem The paving problem Let the standard atomic masa, D, and the projections, P σ, be as defined before. Definition Say that X B(H) can be paved if, given any ɛ > 0, there are pwd sets σ 1,... σ n N such that σ 1 σ n = N and n (X ) P σk XP σk < ɛ k=1 Question (Paving Problem) Can every operator in B(H) be paved? John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

25 The Kadison-Singer Problem Application to maximal ideals of T Proposition If every operator can be paved, then no operator of the form I + S (S S) can belong to a proper ideal of T. Proof. I + S can be paved by projections in D. So n I P σi (I + S)P σi < 1 k=1 and n k=1 P σ i (I + S)P σi is invertible in T. John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

26 The Kadison-Singer Problem Extensions of pure states In [KS59] Kadison and Singer studied Extensions of Pure States. Let B A be C algebras. If φ is a pure state of B then it extends to a state on A. Are such extensions unique? John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

27 The Kadison-Singer Problem Extensions of pure states In [KS59] Kadison and Singer studied Extensions of Pure States. Let B A be C algebras. If φ is a pure state of B then it extends to a state on A. Are such extensions unique? Question (Kadison-Singer) Let D be an atomic masa in B(H). Does every pure state of D have a unique extension to a state of B(H)? John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

28 The Kadison-Singer Problem Extensions of pure states If M is a non-atomic masa in B(H) (i.e. L (0, 1)) then it has pure states with non-unique extensions [KS59]. (In fact no pure states on L (0, 1) extend uniquely [And79a].) If D is an atomic masa in B(H) (i.e. l (N)) and φ is a pure state on D, then φ is a state on B(H). (Anderson [And79b] showed it is a pure state.) Is φ the only extension of φ to a state of B(H)? John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

29 The Kadison-Singer Problem Extensions of pure states Proposition TFAE 1 Every operator in B(H) can be paved. 2 Every pure state of D has a unique state extension to B(H). John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

30 The Kadison-Singer Problem Extensions of pure states Proposition TFAE 1 Every operator in B(H) can be paved. 2 Every pure state of D has a unique state extension to B(H). Proof. (1) (2): Let ˆφ be a state extension of φ. Then ˆφ is a D-bimodule map. Thus by paving X we can arrange ( n ) n φ (X ) = ˆφ (X ) ɛ ˆφ P σi XP σi = φ(p σi ) 2 ˆφ(X ) = ˆφ(X ) k=1 k=1 John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

31 The Kadison-Singer Problem Extensions of pure states Proposition TFAE 1 Every operator in B(H) can be paved. 2 Every pure state of D has a unique state extension to B(H). Proof. (1) (2): Let ˆφ be a state extension of φ. Then ˆφ is a D-bimodule map. Thus by paving X we can arrange ( n ) n φ (X ) = ˆφ (X ) ɛ ˆφ P σi XP σi = φ(p σi ) 2 ˆφ(X ) = ˆφ(X ) k=1 k=1 John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

32 The Kadison-Singer Problem Extensions of pure states Lemma ˆφ is a D-bimodule map. Proof. Let p D be a projection. Then ˆφ(p) = φ(p) = φ(p) 2 = 0, 1. If φ(p) = 0 then by Cauchy-Schwartz, ˆφ(px) = 0 = ˆφ(p) ˆφ(x) If φ(p) = 1 then, again by Cauchy-Schwartz, ˆφ(px) = ˆφ(x) ˆφ(p x) = ˆφ(x) = ˆφ(p) ˆφ(x) (Extend to arbitrary a D by spectral theory.) John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

33 The Kadison-Singer Problem Progress on the problem Reid; [Rei71] Anderson; [And79a, And79b] Berman, Halpern, Kaftal, Weiss; [BHKW88] Bourgain, Tzafriri; [BT91] Weaver; [Wea04, Wea03] Casazza, Christensen, Lindner, Vershynin; [CCLV05] Casazza, Tremain The paving conjecture is equivalent to the paving conjecture for triangular matrices ; [CT] John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

34 More upper triangular ideals One-term interpolation Return to X = I + S T (S S). John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

35 More upper triangular ideals One-term interpolation Return to X = I + S T (S S). We want to find A i, B i such that A 1 XB A n XB n = I. John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

36 More upper triangular ideals One-term interpolation Return to X = I + S T (S S). We want to find A i, B i such that A 1 XB A n XB n = I. How about solving AXB = I for A, B T? John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

37 More upper triangular ideals One-term interpolation Return to X = I + S T (S S). We want to find A i, B i such that A 1 XB A n XB n = I. How about solving AXB = I for A, B T? Unfortunately... Proposition Let X T. There are A, B T with AXB = I iff X is an invertible operator. John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

38 More upper triangular ideals One-term interpolation Return to X = I + S T (S S). We want to find A i, B i such that A 1 XB A n XB n = I. How about solving AXB = I for A, B T? Unfortunately... Proposition Let X T. There are A, B T with AXB = I iff X is an invertible operator. Proof. If AXB = I let P n := P {1,...,n} and note P n = (P n AP n ) (P n XP n ) (P n BP n ) = (P n BAP n ) P n XP n since P n BP n is the (two-sided) inverse of P n AXP n in P n H. Taking WOT-limits we see BAX = I and similarly XBA = I. John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

39 More upper triangular ideals One-term interpolation Return to X = I + S T (S S). We want to find A i, B i such that A 1 XB A n XB n = I. How about solving AXB = I for A, B T? Unfortunately... Proposition Let X T. There are A, B T with AXB = I iff X is an invertible operator. So how about solving AXB + CXD = I? John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

40 More upper triangular ideals Two-term interpolation First express as a finite dimensional problem: Question Given an n n matrix X = I + S (S strictly upper triangular), can we find upper triangular matrices A,..., D such that AXB + CXD = I John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

41 More upper triangular ideals Two-term interpolation First express as a finite dimensional problem: Question Given an n n matrix X = I + S (S strictly upper triangular), can we find upper triangular matrices A,..., D such that AXB + CXD = I where the max{ A,..., D } is bounded in terms of X but independently of n? John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

42 More upper triangular ideals Two-term interpolation Lemma Let X = I + S M n (C) where S is strictly upper triangular. Then there are A,..., D M n (C) such that AXB + CXD = I and max{ A,..., D } X. John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

43 More upper triangular ideals Two-term interpolation Lemma Let X = I + S M n (C) where S is strictly upper triangular. Then there are A,..., D M n (C) such that AXB + CXD = I and max{ A,..., D } X. Proof. Assume for simplicity n is even. Let s 1 s 2 s n be the singular values of X. John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

44 More upper triangular ideals Two-term interpolation Lemma Let X = I + S M n (C) where S is strictly upper triangular. Then there are A,..., D M n (C) such that AXB + CXD = I and max{ A,..., D } X. Proof. Assume for simplicity n is even. Let s 1 s 2 s n be the singular values of X. Since all s i X and n i=1 s i = det X = det X = 1, we cannot have n/2 of the s i satisfying s i < 1/ X. John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

45 More upper triangular ideals Two-term interpolation Lemma Let X = I + S M n (C) where S is strictly upper triangular. Then there are A,..., D M n (C) such that AXB + CXD = I and max{ A,..., D } X. Proof. Assume for simplicity n is even. Let s 1 s 2 s n be the singular values of X. Since all s i X and n i=1 s i = det X = det X = 1, we cannot have n/2 of the s i satisfying s i < 1/ X. For in that case 1 = det X < X n/2 / X n/2 1. Thus the first n/2 of the s i are at least X 1. John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

46 More upper triangular ideals Two-term interpolation Lemma Let X = I + S M n (C) where S is strictly upper triangular. Then there are A,..., D M n (C) such that AXB + CXD = I and max{ A,..., D } X. Proof. Assume for simplicity n is even. Let s 1 s 2 s n be the singular values of X. Since all s i X and n i=1 s i = det X = det X = 1, we cannot have n/2 of the s i satisfying s i < 1/ X. For in that case 1 = det X < X n/2 / X n/2 1. Thus the first n/2 of the s i are at least X 1. There are o.n. bases u i, v i (1 i n) such that Xu i = s i v i. Let A, B be matrices mapping v i (1/s i )e i and e i u i for 1 i n/2. Then AXB is the projection onto span{e 1,... e n } and A, B s 1 n X. 2 2 John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

47 More upper triangular ideals Two-term interpolation Lemma Let X = I + S M n (C) where S is strictly upper triangular. Then there are A,..., D M n (C) such that AXB + CXD = I and max{ A,..., D } X. Proof. Assume for simplicity n is even. Let s 1 s 2 s n be the singular values of X. Since all s i X and n i=1 s i = det X = det X = 1, we cannot have n/2 of the s i satisfying s i < 1/ X. For in that case 1 = det X < X n/2 / X n/2 1. Thus the first n/2 of the s i are at least X 1. There are o.n. bases u i, v i (1 i n) such that Xu i = s i v i. Let A, B be matrices mapping v i (1/s i )e i and e i u i for 1 i n/2. Then AXB is the projection onto span{e 1,... e n } and A, B s 1 n X. Likewise get CXD as 2 2 the projection onto span{e n 2 +1,... e n } with norm control. John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

48 More upper triangular ideals Two-term interpolation But although we used the fact X is upper triangular we lost all control on triangularity of A,..., D. John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

49 More upper triangular ideals Two-term interpolation But although we used the fact X is upper triangular we lost all control on triangularity of A,..., D. At least we see there is no spectral obstruction to a two-term decomposition. Might there be other obstructions? Index perhaps? Question Given X = I + S (S S), are there A,..., D T such that AXB + CXD = I? John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

50 More upper triangular ideals Filters Suppose now that there is a maximal ideal J of T that contains X = I + S (S S) and deduce some consequences. Let Σ = {σ N : I P σ J } John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

51 More upper triangular ideals Filters Proposition Let Then 1 Σ is a filter. Σ = {σ N : I P σ J } 2 Σ contains all cofinite subset of N. 3 σ Σ σ + 1 Σ. 4 Σ is not an ultrafilter. John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

52 More upper triangular ideals Filters Proposition Let Then 1 Σ is a filter. Σ = {σ N : I P σ J } 2 Σ contains all cofinite subset of N. 3 σ Σ σ + 1 Σ. 4 Σ is not an ultrafilter. Proof. If σ Σ and τ σ then P τ c = P τ c P σ c J. If σ 1, σ 2 Σ then P σ 1 σ 2 = P σ c 1 σ c 2 = P σ c 1 + P σ c 2 P σ c 1 P σ c 2. John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

53 More upper triangular ideals Filters Proposition Let Then 1 Σ is a filter. Σ = {σ N : I P σ J } 2 Σ contains all cofinite subset of N. 3 σ Σ σ + 1 Σ. 4 Σ is not an ultrafilter. Proof. For each k, P {k} = P {k} XP {k} J so {k} c Σ, a filter. John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

54 More upper triangular ideals Filters Proposition Let Then 1 Σ is a filter. Σ = {σ N : I P σ J } 2 Σ contains all cofinite subset of N. 3 σ Σ σ + 1 Σ. 4 Σ is not an ultrafilter. Proof. J S and so S + J = T. Let U be the backward shift. Then UT = T U = S and so U is invertible (mod)j. But UP σ+1 = P σ U so P σ = I (mod)j iff P σ+1 = I (mod)j. John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

55 More upper triangular ideals Filters Proposition Let Then 1 Σ is a filter. Σ = {σ N : I P σ J } 2 Σ contains all cofinite subset of N. 3 σ Σ σ + 1 Σ. 4 Σ is not an ultrafilter. Proof. Neither 2N nor 2N 1 can be in Σ for then its complement is in Σ also. John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

56 More upper triangular ideals Filters Proposition Let Then 1 Σ is a filter. Σ = {σ N : I P σ J } 2 Σ contains all cofinite subset of N. 3 σ Σ σ + 1 Σ. 4 Σ is not an ultrafilter. John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

57 Nest Algebras Definitions Nest algebras Definition (Ringrose, [Rin65]) Let H be a Hilbert space and N a complete chain of subspaces containing 0 and H. This is called a nest. Define the nest algebra, Alg(N ), for a given nest N to be Alg(N ) := {X B(H) : XN N N N } See Davidson, Nest Algebras, [Dav88]. John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

58 Nest Algebras Definitions Nest algebras Definition (Ringrose, [Rin65]) Let H be a Hilbert space and N a complete chain of subspaces containing 0 and H. This is called a nest. Define the nest algebra, Alg(N ), for a given nest N to be Alg(N ) := {X B(H) : XN N N N } See Davidson, Nest Algebras, [Dav88]. John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

59 Nest Algebras Examples Example Let e 1,..., e n be the standard basis for C n. Let N i := span{e 1,..., e i } and N := {0, N i : 1 i n}. Then Alg(N ) = T n (C). John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

60 Nest Algebras Examples Example Let e i (i N) be the standard basis for H = l 2 (N). Let N i := span{e 1,..., e i } and N := {0, N i, H : i N}. Then Alg(N ) is the algebra of all bounded operators which are upper triangular w.r.t. {e i }. In other words, Alg(N ) = T John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

61 Nest Algebras Examples The Volterra Nest Example Let H = L 2 (0, 1). For each t [0, 1] let N t := {f L 2 (0, 1) : f is supported a.e. on [0,t]} In other words, P(N t ) is multiplication by χ [0,t]. Clearly N := {N t : t [0, 1]} is a nest. John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

62 Nest Algebras Examples The Volterra Nest Example Let H = L 2 (0, 1). For each t [0, 1] let N t := {f L 2 (0, 1) : f is supported a.e. on [0,t]} In other words, P(N t ) is multiplication by χ [0,t]. Clearly N := {N t : t [0, 1]} is a nest. Remark Alg(N ) contains the Volterra integral operator, f 1 x f (t) dt John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

63 Nest Algebras Classification and similarity Classification of nest algebras Theorem (Ringrose, [Rin66]) Let φ : Alg(N 1 ) Alg(N 2 ) be an algebraic isomorphsim. Then there is an invertible operator S B(H 1, H 2 ) such that φ(t ) = STS 1 = Ad S (T ) for all T Alg(N 1 ) Now φ = Ad S iff {SN : N N 1 } = N 2. So classifying nest algebras up to isomorphism means classifying nests up to similarity. John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

64 Nest Algebras Classification and similarity Theorem (Erdos, [Erd67]) Nests are completely classified up to unitary equivalence by An order type A measure class, and A multiplicity function C.f. Unitary invariants for bounded selfadjoint operators (spectrum, measure class, mutliplicity function). Question Any similarity transform preserves order type. Must it also preserve multiplicity and/or measure class? John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

65 Nest Algebras Classification and similarity Let N be the Volterra nest on H = L 2 (0, 1). I.e. N = {N t : t [0, 1]} where N t = {f : f (x) = 0 a.e. x [0, t]} Example The map N t N t N t preserves order type and measure class, but not spectral multiplicity. Example Let f : [0, 1] [0, 1] be increasing, bjijective, not absolutely continuous. The map N t N f (t) preserves order type and multiplicity, but not measure class. John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

66 Nest Algebras Classification and similarity Theorem (Davidson, [Dav84]) Let N 1, N 2 be nests and θ : N 1 N 2 be and order isomorphism. There is an invertible operator S such that iff θ is dimension-preserving, i.e. if θ(n) = SN for all N N 1 dim θ(n) θ(m) = dim N M for all M < N in N 1 John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

67 Nest Algebras Classification and similarity Corollary Both of the previous two examples are implemented by invertibles! Corollary Nest algebras are classified up to isomorphism by order-dimension type. Proof uses Voiculescu s notion of approximate unitary equivalence. Based on N. T. Andersen s study of unitary equivalence of quasi-triangular algebras Slightly earlier result of D. Larson [Lar85] showed all continuous nests are similar. John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

68 Nest Algebras Algebraic implications of Similarity Theory Proposition The commutator ideal of a continuous nest is the whole algebra. John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

69 Nest Algebras Algebraic implications of Similarity Theory Proposition The commutator ideal of a continuous nest is the whole algebra. Proof. By the Similarity Theorem, Alg(N ) = Alg(N N ) = M 2 (Alg(N )) and so [( ) ( ) ( ) ( )] 2 = I John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

70 Nest Algebras Algebraic implications of Similarity Theory Corollary Let N be the Volterra nest. Then there is no ideal S Alg(N ) such that Alg(N ) = D(N ) S. John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

71 Nest Algebras Algebraic implications of Similarity Theory Corollary Let N be the Volterra nest. Then there is no ideal S Alg(N ) such that Alg(N ) = D(N ) S. Proof. D(N ) = N = N is abelian so S would contain the commutator ideal. John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

72 Nest Algebras Algebraic implications of Similarity Theory Proposition Alg(N ) has non-zero idempotents which are zero on the diagonal, i.e. P(N bi N ai ) Q P(N bi N ai ) = 0 where P(N bi N ai ) = I i John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

73 Nest Algebras Algebraic implications of Similarity Theory Proposition Alg(N ) has non-zero idempotents which are zero on the diagonal, i.e. P(N bi N ai ) Q P(N bi N ai ) = 0 where P(N bi N ai ) = I i Proof. Write the Cantor middle- 1 3 set as K = [0, 1] \ i=1 (a i, b i ). Let f : [0, 1] [0, 1] map K to a non-null set. By the Similarity Theorem, SN t = N f (t). Let P = M χf (K) and Q = SPS 1. John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

74 Continuous nest algebras Interpolation Theorem Let N be the Volterra nest. For a Borel set S [0, 1] write E(S) = M χs. Define the diagonal seminorm i x (T ) := inf{ P(N x N t )TP(N x N t ) : t < x} Theorem (Interpolation Theorem, [Orr95]) Let T Alg(N ), a > 0, and S := {x : i x (T ) a} Then there are A, B Alg(N ) such that ATB = E(S). John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

75 Continuous nest algebras Proof uses: Larson-Pitts [LP91] classification of idempotent equivalence Construction of zero-diagonal idempotents which sum to an idempotent that is equivalent to E(S) Factorization of zero-diagonal idempotents through T John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

76 Continuous nest algebras Ideals of continuous algebras Corollary Let N be a continuous nest and X Alg(N ). TFAE: 1 There are A 1..., A n and B 1,..., B n in Alg(N ) such that A 1 XB A N XB n = I. I.e. X does not belong to any proper ideal of Alg(N ). 2 There are A, B Alg(N ) such that AXB = I. 3 i t (X ) a > 0 for all 0 t 1. I.e. inf{ P(N t N s )TP(N t N s ) : 0 s < t I } > 0 John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

77 Continuous nest algebras Ideals of continuous algebras Corollary Let N be a continuous nest and X Alg(N ). TFAE: 1 There are A 1..., A n and B 1,..., B n in Alg(N ) such that A 1 XB A N XB n = I. I.e. X does not belong to any proper ideal of Alg(N ). 2 There are A, B Alg(N ) such that AXB = I. 3 i t (X ) a > 0 for all 0 t 1. I.e. inf{ P(N t N s )TP(N t N s ) : 0 s < t I } > 0 Compare this with T where: 3. is analgous to X = I + S We saw We could not settle whether a version of 2. with two terms might be possible. John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

78 Continuous nest algebras Ideals of continuous algebras Consequences of the Interpolation Theorem include: Identification of maximal off-diagonal ideals and constructions of maximal triangular algebras [Orr95] Classification of the maximal ideals of continuous nest algebra and the lattice they generate [Orr94] The invertibles are connected in many nest algebras [DO95, DOP95] Description of epimorphisms of nest algebras [DHO95] Classification of the automorphism invariant ideals of a continuous nest algebra [Orr01, Orrar] John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

79 Continuous nest algebras Ideals of continuous algebras Consequences of the Interpolation Theorem include: Identification of maximal off-diagonal ideals and constructions of maximal triangular algebras [Orr95] Classification of the maximal ideals of continuous nest algebra and the lattice they generate [Orr94] The invertibles are connected in many nest algebras [DO95, DOP95] Description of epimorphisms of nest algebras [DHO95] Classification of the automorphism invariant ideals of a continuous nest algebra [Orr01, Orrar] John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

80 Continuous nest algebras Ideals of continuous algebras Consequences of the Interpolation Theorem include: Identification of maximal off-diagonal ideals and constructions of maximal triangular algebras [Orr95] Classification of the maximal ideals of continuous nest algebra and the lattice they generate [Orr94] The invertibles are connected in many nest algebras [DO95, DOP95] Description of epimorphisms of nest algebras [DHO95] Classification of the automorphism invariant ideals of a continuous nest algebra [Orr01, Orrar] John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

81 Continuous nest algebras Ideals of continuous algebras Consequences of the Interpolation Theorem include: Identification of maximal off-diagonal ideals and constructions of maximal triangular algebras [Orr95] Classification of the maximal ideals of continuous nest algebra and the lattice they generate [Orr94] The invertibles are connected in many nest algebras [DO95, DOP95] Description of epimorphisms of nest algebras [DHO95] Classification of the automorphism invariant ideals of a continuous nest algebra [Orr01, Orrar] John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

82 Continuous nest algebras Ideals of continuous algebras Consequences of the Interpolation Theorem include: Identification of maximal off-diagonal ideals and constructions of maximal triangular algebras [Orr95] Classification of the maximal ideals of continuous nest algebra and the lattice they generate [Orr94] The invertibles are connected in many nest algebras [DO95, DOP95] Description of epimorphisms of nest algebras [DHO95] Classification of the automorphism invariant ideals of a continuous nest algebra [Orr01, Orrar] John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

83 Continuous nest algebras Ideals of continuous algebras Consequences of the Interpolation Theorem include: Identification of maximal off-diagonal ideals and constructions of maximal triangular algebras [Orr95] Classification of the maximal ideals of continuous nest algebra and the lattice they generate [Orr94] The invertibles are connected in many nest algebras [DO95, DOP95] Description of epimorphisms of nest algebras [DHO95] Classification of the automorphism invariant ideals of a continuous nest algebra [Orr01, Orrar] John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

84 Continuous nest algebras Epimorphisms Davidson-Harrison-Orr, [DHO95] described almost all epimorphisms between nest algebras. Essentially one case was left open: Question Does there exist an epimorphism φ : T B(H)? John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

85 Continuous nest algebras Epimorphisms Davidson-Harrison-Orr, [DHO95] described almost all epimorphisms between nest algebras. Essentially one case was left open: Question Does there exist an epimorphism φ : T B(H)? Fact If so, then ker φ contains an operator I + S (S S). John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

86 Continuous nest algebras Epimorphisms Davidson-Harrison-Orr, [DHO95] described almost all epimorphisms between nest algebras. Essentially one case was left open: Question Does there exist an epimorphism φ : T B(H)? Fact If so, then ker φ contains an operator I + S (S S). Proof. The commutator ideal of T is S and the commutator ideal of B(H) is B(H). Thus φ(s) = I = φ(i ) and so I S ker φ. John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

87 Bibliography and other questions Stable rank Definition The Bass stable rank of an algebra is the smallest n such that whenever (g 1,..., g n+1 ) generate the algebra as a left-ideal then we can find a i such that (g 1 + b 1 g n+1, g 2 + b 2 g n+1,... g n + b n g n+1 ) also generate the algebra as a left ideal. Question What is the Bass stable rank of T? Theorem (Arveson, [Arv75]) G 1,..., G n generate T as a left ideal iff G 1 P k G G n P k G n ap k John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

88 Bibliography and other questions Stable rank jorr/presentations John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

89 Bibliography and other questions Bibliography Joel Anderson. Extensions, restrictions, and representations of states on c -algebras. Transactions of the Amer. Math. Soc., 249(2): , Joel Anderson. Extreme points in sets of positive linear maps on ( ). J. Func. Anal., 31(2): , William B. Arveson. Interpolation problems in nest algebras. J. Func. Anal., 20: , Kenneth Berman, Herbert Halpern, Victor Kaftal, and Gary Weiss. Matrix norm inequalities and the relative dixmier property. Integral Equations Operator Theory, J. Bourgain and L. Tzafriri. On a problem of kadison and singer. J. reine angew. Math., John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

90 Bibliography and other questions Bibliography Peter G. Casazza, Ole Christensen, Alexander M. Lindner, and Roman Vershynin. Frames and the feichtinger conjecture. Proc. Amer. Math. Soc., Peter G. Casazza and Janet C. Tremain. The paving conjecture is equivalent to the paving conjecture for triangular matrices. Kenneth R. Davidson. Similarity and compact perturbations of nest algebras. J. reine angew. Math., 348: , Kenneth R. Davidson. Nest Algebras, volume 191 of Res. Notes Math. Pitman, Boston, Kenneth R. Davidson, Kenneth J. Harrison, and John L. Orr. Epimorphisms of nest algebras. John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

91 Bibliography and other questions Internat. J. Math., Bibliography Kenneth R. Davidson and John L. Orr. The invertibles are connected in infinite multiplicity nest algebras. Bull. London Math. Soc., Kenneth R. Davidson, John L. Orr, and David R. Pitts. Connectedness of the invertibles in certain nest algebras. Canad. Math Bull., John A. Erdos. Unitary invariants for nests. Pacific J. Math., 23: , Richard V. Kadison and I. M. Singer. Extensions of pure states. Amer. J. Math., 81: , David R. Larson. Nest algebras and similarity transformations. Ann. Math., 121: , John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

92 Bibliography and other questions Bibliography David R. Larson and David R. Pitts. Idempotents in nest algebras. J. Func. Anal., 97: , John L. Orr. The maximal ideals of a nest algebra. J. Func. Anal., 124: , John L. Orr. Triangular algebras and ideals of nest algebras. Memoirs of the Amer. Math. Soc., 562(117), John L. Orr. The stable ideals of a continuous nest algebra. J. Operator Theory, 45: , John L. Orr. The stable ideals of a continuous nest algebra II. J. Operator Theory, to appear. G. A. Reid. John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41

93 Bibliography and other questions Bibliography On the calkin representations. Proc. London Math. Soc., 23(3): , John R. Ringrose. On some algebras of operators. Proc. London Math. Soc., 15(3):61 83, John R. Ringrose. On some algebras of operators II. Proc. London Math. Soc., 16(3): , Nik Weaver. A counterexample to a conjecture of akemann and anderson. Bull. London Math. Soc., Nik Weaver. The kadison-singer problem in discrepancy theory. Discrete Math., John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, / 41