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Maximal Ideals of Triangular Operator Algebras John Lindsay Orr jorr@math.unl.edu University of Nebraska Lincoln and Lancaster University May 17, 2007 John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 1 / 41 http://www.math.unl.edu/ jorr/presentations John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 2 / 41 Ideals of <a...
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Maximal Ideals of Triangular Operator Algebras John Lindsay Orr jorr@math.unl.edu University of Nebraska Lincoln and Lancaster University May 17, 2007 John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 1 / 41 http://www.math.unl.edu/ jorr/presentations John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 2 / 41 Ideals of <a href="/keyword/upper-triangular/" >upper triangular</a> operators Statement of the problem Let H := 2 (N) and let {ek } be the standard basis. Let T be the k=1 algebra of all (bounded) operators which are <a href="/keyword/upper-triangular/" >upper triangular</a> with respect to {ek }. Question What are the maximal two-sided ideals of T ? John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 3 / 41 Ideals of <a href="/keyword/upper-triangular/" >upper triangular</a> operators Statement of the problem Let H := 2 (N) and let {ek } be the standard basis. Let T be the k=1 algebra of all (bounded) operators which are <a href="/keyword/upper-triangular/" >upper triangular</a> with respect to {ek }. 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, 2007 3 / 41 Ideals of <a href="/keyword/upper-triangular/" >upper triangular</a> 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, 2007 4 / 41 Ideals of <a href="/keyword/upper-triangular/" >upper triangular</a> operators Statement of the problem What would I like the answer to be? Observe that D, the set of diagonal operators w.r.t. {ek } is *-isomorphic to (N), so we identify them. Write S for the set of strictly <a href="/keyword/upper-triangular/" >upper triangular</a> operators w.r.t. {ek }. Fact Let M be a maximal ideal of maximal ideal of T . (N) and let J := M + S. Then J is a John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 4 / 41 Ideals of <a href="/keyword/upper-triangular/" >upper triangular</a> operators Statement of the problem What would I like the answer to be? Observe that D, the set of diagonal operators w.r.t. {ek } is *-isomorphic to (N), so we identify them. Write S for the set of strictly <a href="/keyword/upper-triangular/" >upper triangular</a> operators w.r.t. {ek }. Fact Let M be a maximal ideal of maximal ideal of T . (N) and let J := M + S. Then J is a Proof. Write (T ) for the diagonal part of T . Suppose T J . John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 4 / 41 Ideals of <a href="/keyword/upper-triangular/" >upper triangular</a> operators Statement of the problem What would I like the answer to be? Observe that D, the set of diagonal operators w.r.t. {ek } is *-isomorphic to (N), so we identify them. Write S for the set of strictly <a href="/keyword/upper-triangular/" >upper triangular</a> operators w.r.t. {ek }. Fact Let M be a maximal ideal of maximal ideal of T . (N) and let J := M + S. Then J is a 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, 2007 4 / 41 Ideals of <a href="/keyword/upper-triangular/" >upper triangular</a> operators Statement of the problem The maximal ideals of (N) are points in N, the Stone-Cech compacti cation 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 (N)? John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 5 / 41 Ideals of <a href="/keyword/upper-triangular/" >upper triangular</a> operators Re-statement of the problem Proposition TFAE: 1 2 3 All the maximal ideals of T are of the form M + S. All the maximal ideals of T contain S. No proper ideal of T contains an operator I + S, (S S). John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 6 / 41 Ideals of <a href="/keyword/upper-triangular/" >upper triangular</a> operators Re-statement of the problem Proposition TFAE: 1 2 3 All the maximal ideals of T are of the form M + S. All the maximal ideals of T contain S. 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, 2007 6 / 41 Ideals of <a href="/keyword/upper-triangular/" >upper triangular</a> operators Re-statement of the problem Proposition TFAE: 1 2 3 All the maximal ideals of T are of the form M + S. All the maximal ideals of T contain S. 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, 2007 6 / 41 Ideals of <a href="/keyword/upper-triangular/" >upper triangular</a> operators Re-statement of the problem Proposition TFAE: 1 2 3 All the maximal ideals of T are of the form M + S. All the maximal ideals of T contain S. 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, 2007 6 / 41 Ideals of <a href="/keyword/upper-triangular/" >upper triangular</a> operators Re-statement of the problem Proposition TFAE: 1 2 3 All the maximal ideals of T are of the form M + S. All the maximal ideals of T contain S. 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, 2007 6 / 41 Ideals of <a href="/keyword/upper-triangular/" >upper triangular</a> operators Re-statement of the problem Question Is it possible for an operator of the form I + S (S strictly <a href="/keyword/upper-triangular/" >upper triangular</a> ) to lie in a proper ideal of T ? John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 7 / 41 Ideals of <a href="/keyword/upper-triangular/" >upper triangular</a> operators Re-statement of the problem Question Is it possible for an operator of the form I + S (S strictly <a href="/keyword/upper-triangular/" >upper triangular</a> ) to lie in a proper ideal of T ? Just to be clear, an operator X fails to belong to a proper ideal of T i we can nd A1 , . . . , An and B1 , . . . , Bn such that A1 XB1 + + An XBn = I John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 7 / 41 Ideals of <a href="/keyword/upper-triangular/" >upper triangular</a> operators Operators of the form I + S In nite dimensions, all operators I + S are invertible. John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 8 / 41 Ideals of <a href="/keyword/upper-triangular/" >upper triangular</a> operators Operators of the form I + S In nite dimensions, all operators I Not so in in nite dimensions. 0 1 0 0 1 0 Let 0 1 0 .. .. .. . . . + S are invertible. be the unilateral backward shift is not invertible . 1 1 0 0 1 1 0 Then I U = 0 1 1 0 .. .. .. . . . .. John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 8 / 41 Ideals of <a href="/keyword/upper-triangular/" >upper triangular</a> 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, 2007 9 / 41 Ideals of <a href="/keyword/upper-triangular/" >upper triangular</a> 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 N and let P := Proj (span{ek : k }) Note UP2N = P2N 1 U and UP2N 1 = P2N U John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 9 / 41 Ideals of <a href="/keyword/upper-triangular/" >upper triangular</a> 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 N and let P := Proj (span{ek : k }) Note UP2N = P2N 1 U and UP2N 1 = P2N U Thus P2N (I U)P2N + P2N 1 (I U)P2N 1 = I John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 9 / 41 Ideals of <a href="/keyword/upper-triangular/" >upper triangular</a> 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 N and let P := Proj (span{ek : k }) Note UP2N = P2N 1 U and UP2N 1 = P2N U Thus P2N (I U)P2N + P2N 1 (I U)P2N 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, 2007 9 / 41 The Kadison-Singer Problem The paving problem Let the standard atomic masa, D, and the projections, P , be as de ned before. De nition Say that X B(H) can be paved if, given any 1 , . . . n N such that 1 n = N and (X ) k=1 n > 0, there are pwd sets P k XP k < John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 10 / 41 The Kadison-Singer Problem The paving problem Let the standard atomic masa, D, and the projections, P , be as de ned before. De nition Say that X B(H) can be paved if, given any 1 , . . . n N such that 1 n = N and (X ) k=1 n > 0, there are pwd sets P k XP k < Question (Paving Problem) Can every operator in B(H) be paved? John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 10 / 41 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 k=1 P i (I + S)P i < 1 and n k=1 P i (I + S)P i is invertible in T . John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 11 / 41 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 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, 2007 12 / 41 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 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, 2007 12 / 41 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. (N)) and is a pure state on D, then is a state on B(H). (Anderson [And79b] showed it is a pure state.) Is the only extension of to a state of B(H)? John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 13 / 41 The Kadison-Singer Problem Extensions of pure states Proposition TFAE 1 2 Every operator in B(H) can be paved. 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, 2007 14 / 41 The Kadison-Singer Problem Extensions of pure states Proposition TFAE 1 2 Every operator in B(H) can be paved. Every pure state of D has a unique state extension to B(H). Proof. (1) (2): Let be a state extension of . Then is a D-bimodule map. Thus by paving X we can arrange n n (X ) = (X ) k=1 P i XP i = k=1 (P i )2 (X ) = (X ) John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 14 / 41 The Kadison-Singer Problem Extensions of pure states Proposition TFAE 1 2 Every operator in B(H) can be paved. Every pure state of D has a unique state extension to B(H). Proof. (1) (2): Let be a state extension of . Then is a D-bimodule map. Thus by paving X we can arrange n n (X ) = (X ) k=1 P i XP i = k=1 (P i )2 (X ) = (X ) John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 14 / 41 The Kadison-Singer Problem Extensions of pure states Lemma is a D-bimodule map. Proof. Let p D be a projection. Then (p) = (p) = (p)2 = 0, 1. If (p) = 0 then by Cauchy-Schwartz, (px) = 0 = (p) (x) If (p) = 1 then, again by Cauchy-Schwartz, (px) = (x) (p x) = (x) = (p) (x) (Extend to arbitrary a D by spectral theory.) John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 15 / 41 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, 2007 16 / 41 More <a href="/keyword/upper-triangular/" >upper triangular</a> ideals One-term interpolation Return to X = I + S T (S S). John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 17 / 41 More <a href="/keyword/upper-triangular/" >upper triangular</a> ideals One-term interpolation Return to X = I + S T (S S). We want to nd Ai , Bi such that A1 XB1 + + An XBn = I . John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 17 / 41 More <a href="/keyword/upper-triangular/" >upper triangular</a> ideals One-term interpolation Return to X = I + S T (S S). We want to nd Ai , Bi such that A1 XB1 + + An XBn = I . How about solving AXB = I for A, B T ? John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 17 / 41 More <a href="/keyword/upper-triangular/" >upper triangular</a> ideals One-term interpolation Return to X = I + S T (S S). We want to nd Ai , Bi such that A1 XB1 + + An XBn = I . How about solving AXB = I for A, B T ? Unfortunately. . . Proposition Let X T . There are A, B T with AXB = I i X is an invertible operator. John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 17 / 41 More <a href="/keyword/upper-triangular/" >upper triangular</a> ideals One-term interpolation Return to X = I + S T (S S). We want to nd Ai , Bi such that A1 XB1 + + An XBn = I . How about solving AXB = I for A, B T ? Unfortunately. . . Proposition Let X T . There are A, B T with AXB = I i X is an invertible operator. Proof. If AXB = I let Pn := P{1,...,n} and note Pn = (Pn APn ) (Pn XPn ) (Pn BPn ) = (Pn BAPn ) Pn XPn since Pn BPn is the (two-sided) inverse of Pn AXPn in Pn H. Taking WOT-limits we see BAX = I and similarly XBA = I . John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 17 / 41 More <a href="/keyword/upper-triangular/" >upper triangular</a> ideals One-term interpolation Return to X = I + S T (S S). We want to nd Ai , Bi such that A1 XB1 + + An XBn = I . How about solving AXB = I for A, B T ? Unfortunately. . . Proposition Let X T . There are A, B T with AXB = I i X is an invertible operator. So how about solving AXB + CXD = I ? John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 17 / 41 More <a href="/keyword/upper-triangular/" >upper triangular</a> ideals Two-term interpolation First express as a nite dimensional problem: Question Given an n n matrix X = I + S (S strictly <a href="/keyword/upper-triangular/" >upper triangular</a> ), can we nd <a href="/keyword/upper-triangular/" >upper triangular</a> matrices A, . . . , D such that AXB + CXD = I John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 18 / 41 More <a href="/keyword/upper-triangular/" >upper triangular</a> ideals Two-term interpolation First express as a nite dimensional problem: Question Given an n n matrix X = I + S (S strictly <a href="/keyword/upper-triangular/" >upper triangular</a> ), can we nd <a href="/keyword/upper-triangular/" >upper triangular</a> 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, 2007 18 / 41 More <a href="/keyword/upper-triangular/" >upper triangular</a> ideals Two-term interpolation Lemma Let X = I + S Mn (C) where S is strictly <a href="/keyword/upper-triangular/" >upper triangular</a> . Then there are A, . . . , D Mn (C) such that AXB + CXD = I and max{ A , . . . , D } X . John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 19 / 41 More <a href="/keyword/upper-triangular/" >upper triangular</a> ideals Two-term interpolation Lemma Let X = I + S Mn (C) where S is strictly <a href="/keyword/upper-triangular/" >upper triangular</a> . Then there are A, . . . , D Mn (C) such that AXB + CXD = I and max{ A , . . . , D } X . Proof. Assume for simplicity n is even. Let s1 s2 sn be the singular values of X . John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 19 / 41 More <a href="/keyword/upper-triangular/" >upper triangular</a> ideals Two-term interpolation Lemma Let X = I + S Mn (C) where S is strictly <a href="/keyword/upper-triangular/" >upper triangular</a> . Then there are A, . . . , D Mn (C) such that AXB + CXD = I and max{ A , . . . , D } X . Proof. Assume for simplicity n is even. Let s1 s2 sn be the singular values of X . Since all si X and n si = det |X | = | det X | = 1, we i=1 cannot have n/2 of the si satisfying si < 1/ X . John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 19 / 41 More <a href="/keyword/upper-triangular/" >upper triangular</a> ideals Two-term interpolation Lemma Let X = I + S Mn (C) where S is strictly <a href="/keyword/upper-triangular/" >upper triangular</a> . Then there are A, . . . , D Mn (C) such that AXB + CXD = I and max{ A , . . . , D } X . Proof. Assume for simplicity n is even. Let s1 s2 sn be the singular values of X . Since all si X and n si = det |X | = | det X | = 1, we i=1 cannot have n/2 of the si satisfying si < 1/ X . For in that case 1 = det X < X n/2 / X n/2 1. Thus the rst n/2 of the si are at least X 1 . John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 19 / 41 More <a href="/keyword/upper-triangular/" >upper triangular</a> ideals Two-term interpolation Lemma Let X = I + S Mn (C) where S is strictly <a href="/keyword/upper-triangular/" >upper triangular</a> . Then there are A, . . . , D Mn (C) such that AXB + CXD = I and max{ A , . . . , D } X . Proof. Assume for simplicity n is even. Let s1 s2 sn be the singular values of X . Since all si X and n si = det |X | = | det X | = 1, we i=1 cannot have n/2 of the si satisfying si < 1/ X . For in that case 1 = det X < X n/2 / X n/2 1. Thus the rst n/2 of the si are at least X 1 . There are o.n. bases ui , vi (1 i n) such that Xui = si vi . Let A, B be matrices mapping vi (1/si )ei and ei ui for 1 i n/2. Then AXB is the projection onto span{e1 , . . . e n } and A , B s 1 X . n 2 2 John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 19 / 41 More <a href="/keyword/upper-triangular/" >upper triangular</a> ideals Two-term interpolation Lemma Let X = I + S Mn (C) where S is strictly <a href="/keyword/upper-triangular/" >upper triangular</a> . Then there are A, . . . , D Mn (C) such that AXB + CXD = I and max{ A , . . . , D } X . Proof. Assume for simplicity n is even. Let s1 s2 sn be the singular values of X . Since all si X and n si = det |X | = | det X | = 1, we i=1 cannot have n/2 of the si satisfying si < 1/ X . For in that case 1 = det X < X n/2 / X n/2 1. Thus the rst n/2 of the si are at least X 1 . There are o.n. bases ui , vi (1 i n) such that Xui = si vi . Let A, B be matrices mapping vi (1/si )ei and ei ui for 1 i n/2. Then AXB is the projection onto span{e1 , . . . e n } and A , B s 1 X . Likewise get CXD as n 2 2 the projection onto span{e n +1 , . . . en } with norm control. 2 John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 19 / 41 More <a href="/keyword/upper-triangular/" >upper triangular</a> ideals Two-term interpolation But although we used the fact X is <a href="/keyword/upper-triangular/" >upper triangular</a> we lost all control on triangularity of A, . . . , D. John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 20 / 41 More <a href="/keyword/upper-triangular/" >upper triangular</a> ideals Two-term interpolation But although we used the fact X is <a href="/keyword/upper-triangular/" >upper triangular</a> 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, 2007 20 / 41 More <a href="/keyword/upper-triangular/" >upper triangular</a> 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 = { N : I P J } John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 21 / 41 More <a href="/keyword/upper-triangular/" >upper triangular</a> ideals Filters Proposition Let = { N : I P J } Then 1 2 3 4 is a lter. contains all co nite subset of N. + 1 . is not an ultra lter. John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 22 / 41 More <a href="/keyword/upper-triangular/" >upper triangular</a> ideals Filters Proposition Let = { N : I P J } Then 1 2 3 4 is a lter. contains all co nite subset of N. + 1 . is not an ultra lter. Proof. If and then P c = P c P c J . c c c c c c If 1 , 2 then P 1 2 = P 1 2 = P 1 + P 2 P 1 P 2 . John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 22 / 41 More <a href="/keyword/upper-triangular/" >upper triangular</a> ideals Filters Proposition Let = { N : I P J } Then 1 2 3 4 is a lter. contains all co nite subset of N. + 1 . is not an ultra lter. Proof. For each k, P{k} = P{k} XP{k} J so {k}c , a lter. John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 22 / 41 More <a href="/keyword/upper-triangular/" >upper triangular</a> ideals Filters Proposition Let = { N : I P J } Then 1 2 3 4 is a lter. contains all co nite subset of N. + 1 . is not an ultra lter. 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 +1 = P U so P = I (mod)J i P +1 = I (mod)J . John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 22 / 41 More <a href="/keyword/upper-triangular/" >upper triangular</a> ideals Filters Proposition Let = { N : I P J } Then 1 2 3 4 is a lter. contains all co nite subset of N. + 1 . is not an ultra lter. Proof. Neither 2N nor 2N 1 can be in for then its complement is in also. John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 22 / 41 More <a href="/keyword/upper-triangular/" >upper triangular</a> ideals Filters Proposition Let = { N : I P J } Then 1 2 3 4 is a lter. contains all co nite subset of N. + 1 . is not an ultra lter. John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 22 / 41 Nest Algebras De nitions Nest algebras De nition (Ringrose, [Rin65]) Let H be a Hilbert space and N a complete chain of subspaces containing 0 and H. This is called a nest. De ne 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, 2007 23 / 41 Nest Algebras De nitions Nest algebras De nition (Ringrose, [Rin65]) Let H be a Hilbert space and N a complete chain of subspaces containing 0 and H. This is called a nest. De ne 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, 2007 23 / 41 Nest Algebras Examples Example Let e1 , . . . , en be the standard basis for Cn . Let Ni := span{e1 , . . . , ei } and N := {0, Ni : 1 i n}. Then Alg(N ) = Tn (C). John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 24 / 41 Nest Algebras Examples Example Let ei (i N) be the standard basis for H = 2 (N). Let Ni := span{e1 , . . . , ei } and N := {0, Ni , H : i N}. Then Alg(N ) is the algebra of all bounded operators which are <a href="/keyword/upper-triangular/" >upper triangular</a> w.r.t. {ei }. In other words, Alg(N ) = T John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 25 / 41 Nest Algebras Examples The Volterra Nest Example Let H = L2 (0, 1). For each t [0, 1] let Nt := {f L2 (0, 1) : f is supported a.e. on [0,t]} In other words, P(Nt ) is multiplication by [0,t] . Clearly N := {Nt : t [0, 1]} is a nest. John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 26 / 41 Nest Algebras Examples The Volterra Nest Example Let H = L2 (0, 1). For each t [0, 1] let Nt := {f L2 (0, 1) : f is supported a.e. on [0,t]} In other words, P(Nt ) is multiplication by [0,t] . Clearly N := {Nt : t [0, 1]} is a nest. Remark Alg(N ) contains the Volterra integral operator, 1 f x f (t) dt John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 26 / 41 Nest Algebras Classi cation and similarity Classi cation of nest algebras Theorem (Ringrose, [Rin66]) Let : Alg(N1 ) Alg(N2 ) be an algebraic isomorphsim. Then there is an invertible operator S B(H1 , H2 ) such that (T ) = STS 1 = AdS (T ) for all T Alg(N1 ) Now = AdS i {SN : N N1 } = N2 . 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, 2007 27 / 41 Nest Algebras Classi cation and similarity Theorem (Erdos, [Erd67]) Nests are completely classi ed 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, 2007 28 / 41 Nest Algebras Classi cation and similarity Let N be the Volterra nest on H = L2 (0, 1). I.e. N = {Nt : t [0, 1]} where Nt = {f : f (x) = 0 a.e. x [0, t]} Example The map Nt Nt Nt 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 Nt Nf (t) preserves order type and multiplicity, but not measure class. John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 29 / 41 Nest Algebras Classi cation and similarity Theorem (Davidson, [Dav84]) Let N1 , N2 be nests and : N1 N2 be and order isomorphism. There is an invertible operator S such that (N) = SN i is dimension-preserving, i.e. if dim (N) (M) = dim N M for all M < N in N1 for all N N1 John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 30 / 41 Nest Algebras Classi cation and similarity Corollary Both of the previous two examples are implemented by invertibles! Corollary Nest algebras are classi ed 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, 2007 31 / 41 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, 2007 32 / 41 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 ) = M2 (Alg(N )) and so = 0 1 1 0 0 0 1 0 0 0 1 0 0 1 1 0 2 =I John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 32 / 41 Nest Algebras Algebraic implications of Similarity Theory Corollary Let N be the Volterra nest. Then there is no ideal S Alg(N ) = D(N ) S. Alg(N ) such that John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 33 / 41 Nest Algebras Algebraic implications of Similarity Theory Corollary Let N be the Volterra nest. Then there is no ideal S Alg(N ) = D(N ) S. Alg(N ) such that 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, 2007 33 / 41 Nest Algebras Algebraic implications of Similarity Theory Proposition Alg(N ) has non-zero idempotents which are zero on the diagonal , i.e. P(Nbi Nai ) Q P(Nbi Nai ) = 0 where i P(Nbi Nai ) = I John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 34 / 41 Nest Algebras Algebraic implications of Similarity Theory Proposition Alg(N ) has non-zero idempotents which are zero on the diagonal , i.e. P(Nbi Nai ) Q P(Nbi Nai ) = 0 where i P(Nbi Nai ) = I Proof. 1 Write the Cantor middle- 3 set as K = [0, 1] \ (ai , bi ). Let i=1 f : [0, 1] [0, 1] map K to a non-null set. By the Similarity Theorem, SNt = Nf (t) . Let P = M f (K ) and Q = SPS 1 . John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 34 / 41 Continuous nest algebras Interpolation Theorem Let N be the Volterra nest. For a Borel set S [0, 1] write E (S) = M S . De ne the diagonal seminorm ix (T ) := inf{ P(Nx Nt )TP(Nx Nt ) : t < x} Theorem (Interpolation Theorem, [Orr95]) Let T Alg(N ), a > 0, and S := {x : ix (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, 2007 35 / 41 Continuous nest algebras Proof uses: Larson-Pitts [LP91] classi cation 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, 2007 36 / 41 Continuous nest algebras Ideals of continuous algebras Corollary Let N be a continuous nest and X Alg(N ). TFAE: 1 There are A1 . . . , An and B1 , . . . , Bn in Alg(N ) such that A1 XB1 + + AN XBn = I . I.e. X does not belong to any proper ideal of Alg(N ). 2 3 There are A, B Alg(N ) such that AXB = I . it (X ) a > 0 for all 0 t 1. I.e. inf{ P(Nt Ns )TP(Nt Ns ) : 0 s < t I} > 0 John L. Orr (Univ. Nebr.) Maximal Ideals of Triangular Algebras May 17, 2007 37 / 41 Continuous nest algebras Ideals of continuous algebras Corollary Let N be a continuous nest and X Alg(N ). TFAE: 1 There are A1 . . . , An and B1 , . . . , Bn in Alg(N ) such that A1 XB1 + + AN XBn = I . I.e. X does not belong to any proper ideal of Alg(N ). 2 3 There are A, B Alg(N ) such that AXB = I . it (X ) a > 0 for all 0 t 1. I.e. inf{ P(Nt Ns )TP(Nt Ns ) : 0 s < t I} > 0 Compare this with T where: 3. is analgous to X = I + S We saw 1. 2. 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, 2007 37 / 41 Continuous nest algebras Ideals of continuous algebras Consequences of the Interpolation Theorem include: Identi cation of maximal o -diagonal ideals and constructions of maximal triangular algebras [Orr95] Classi cation of the maximal ideals of continuous nest algebra and the lattice they generate [Orr94] The invertibles are connec...
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