An update on the classification program of (maximal) ideals of algebras of operators on Banach spaces: the cases of Tsirelson and Schreier spaces . Tomasz Kania Academy of Sciences of the Czech Republic, Praha Madrid, 12.09.2019 joint work with K. Beanland & N. J. Laustsen 1
Overview B ( X ) the Banach algebra of all bdd ops on a B. space X . Goal: to understand the lattice of closed ideals ( ∼ = representations) of B ( X ) . This is an isomorphic problem due to Eidelheit’s thm (1940). X ∼ ⇒ B ( X ) ∼ = Y as B. spaces ⇐ = B ( Y ) as B. algebras. 2
Overview B ( X ) the Banach algebra of all bdd ops on a B. space X . Goal: to understand the lattice of closed ideals ( ∼ = representations) of B ( X ) . This is an isomorphic problem due to Eidelheit’s thm (1940). X ∼ ⇒ B ( X ) ∼ = Y as B. spaces ⇐ = B ( Y ) as B. algebras. Full classification exists for: ◮ 0 ֒ → K ( ℓ 2 ) ֒ → B ( ℓ 2 ) (Calkin, 1940). 2
Overview B ( X ) the Banach algebra of all bdd ops on a B. space X . Goal: to understand the lattice of closed ideals ( ∼ = representations) of B ( X ) . This is an isomorphic problem due to Eidelheit’s thm (1940). X ∼ ⇒ B ( X ) ∼ = Y as B. spaces ⇐ = B ( Y ) as B. algebras. Full classification exists for: ◮ 0 ֒ → K ( ℓ 2 ) ֒ → B ( ℓ 2 ) (Calkin, 1940). ◮ other classical spaces: 2
Overview B ( X ) the Banach algebra of all bdd ops on a B. space X . Goal: to understand the lattice of closed ideals ( ∼ = representations) of B ( X ) . This is an isomorphic problem due to Eidelheit’s thm (1940). X ∼ ⇒ B ( X ) ∼ = Y as B. spaces ⇐ = B ( Y ) as B. algebras. Full classification exists for: ◮ 0 ֒ → K ( ℓ 2 ) ֒ → B ( ℓ 2 ) (Calkin, 1940). ◮ other classical spaces: ◮ 0 ֒ → K ( X ) ֒ → B ( X ) , where X = c 0 or X = ℓ p for p ∈ [ 1 , ∞ ) . 2
Overview B ( X ) the Banach algebra of all bdd ops on a B. space X . Goal: to understand the lattice of closed ideals ( ∼ = representations) of B ( X ) . This is an isomorphic problem due to Eidelheit’s thm (1940). X ∼ ⇒ B ( X ) ∼ = Y as B. spaces ⇐ = B ( Y ) as B. algebras. Full classification exists for: ◮ 0 ֒ → K ( ℓ 2 ) ֒ → B ( ℓ 2 ) (Calkin, 1940). ◮ other classical spaces: ◮ 0 ֒ → K ( X ) ֒ → B ( X ) , where X = c 0 or X = ℓ p for p ∈ [ 1 , ∞ ) . ◮ 0 ֒ → X ℵ 0 ( X ) ֒ → X ℵ 1 ( X ) ֒ → K ( X ) ֒ → . . . ֒ → B ( X ) , where X = c 0 (Γ) or X = ℓ p (Γ) for p ∈ [ 1 , ∞ ) and any set Γ ; X λ ( X ) ideal of ops having range of density at most λ . 2
Overview B ( X ) the Banach algebra of all bdd ops on a B. space X . Goal: to understand the lattice of closed ideals ( ∼ = representations) of B ( X ) . This is an isomorphic problem due to Eidelheit’s thm (1940). X ∼ ⇒ B ( X ) ∼ = Y as B. spaces ⇐ = B ( Y ) as B. algebras. Full classification exists for: ◮ 0 ֒ → K ( ℓ 2 ) ֒ → B ( ℓ 2 ) (Calkin, 1940). ◮ other classical spaces: ◮ 0 ֒ → K ( X ) ֒ → B ( X ) , where X = c 0 or X = ℓ p for p ∈ [ 1 , ∞ ) . ◮ 0 ֒ → X ℵ 0 ( X ) ֒ → X ℵ 1 ( X ) ֒ → K ( X ) ֒ → . . . ֒ → B ( X ) , where X = c 0 (Γ) or X = ℓ p (Γ) for p ∈ [ 1 , ∞ ) and any set Γ ; X λ ( X ) ideal of ops having range of density at most λ . ◮ c 0 - and ℓ 1 -sums of ℓ n 2 as n → ∞ (Laustsen–Loy–Read, Laustsen–Schlumprecht–Zsák). 2
Overview B ( X ) the Banach algebra of all bdd ops on a B. space X . Goal: to understand the lattice of closed ideals ( ∼ = representations) of B ( X ) . This is an isomorphic problem due to Eidelheit’s thm (1940). X ∼ ⇒ B ( X ) ∼ = Y as B. spaces ⇐ = B ( Y ) as B. algebras. Full classification exists for: ◮ 0 ֒ → K ( ℓ 2 ) ֒ → B ( ℓ 2 ) (Calkin, 1940). ◮ other classical spaces: ◮ 0 ֒ → K ( X ) ֒ → B ( X ) , where X = c 0 or X = ℓ p for p ∈ [ 1 , ∞ ) . ◮ 0 ֒ → X ℵ 0 ( X ) ֒ → X ℵ 1 ( X ) ֒ → K ( X ) ֒ → . . . ֒ → B ( X ) , where X = c 0 (Γ) or X = ℓ p (Γ) for p ∈ [ 1 , ∞ ) and any set Γ ; X λ ( X ) ideal of ops having range of density at most λ . ◮ c 0 - and ℓ 1 -sums of ℓ n 2 as n → ∞ (Laustsen–Loy–Read, Laustsen–Schlumprecht–Zsák). ◮ Koszmider’s C ( K ) -space from an AD family that exists under CH mentioned by Jesús on Tuesday. 2
Overview B ( X ) the Banach algebra of all bdd ops on a B. space X . Goal: to understand the lattice of closed ideals ( ∼ = representations) of B ( X ) . This is an isomorphic problem due to Eidelheit’s thm (1940). X ∼ ⇒ B ( X ) ∼ = Y as B. spaces ⇐ = B ( Y ) as B. algebras. Full classification exists for: ◮ 0 ֒ → K ( ℓ 2 ) ֒ → B ( ℓ 2 ) (Calkin, 1940). ◮ other classical spaces: ◮ 0 ֒ → K ( X ) ֒ → B ( X ) , where X = c 0 or X = ℓ p for p ∈ [ 1 , ∞ ) . ◮ 0 ֒ → X ℵ 0 ( X ) ֒ → X ℵ 1 ( X ) ֒ → K ( X ) ֒ → . . . ֒ → B ( X ) , where X = c 0 (Γ) or X = ℓ p (Γ) for p ∈ [ 1 , ∞ ) and any set Γ ; X λ ( X ) ideal of ops having range of density at most λ . ◮ c 0 - and ℓ 1 -sums of ℓ n 2 as n → ∞ (Laustsen–Loy–Read, Laustsen–Schlumprecht–Zsák). ◮ Koszmider’s C ( K ) -space from an AD family that exists under CH mentioned by Jesús on Tuesday. ◮ Argyros–Haydon’s scalar-plus-compact space, sums of finitely many incomparable copies thereof, some variants due to Tarbard and further variants (Motakis–Puglisi–Zisimopoulou). 2
Overview B ( X ) the Banach algebra of all bdd ops on a B. space X . Goal: to understand the lattice of closed ideals ( ∼ = representations) of B ( X ) . This is an isomorphic problem due to Eidelheit’s thm (1940). X ∼ ⇒ B ( X ) ∼ = Y as B. spaces ⇐ = B ( Y ) as B. algebras. Full classification exists for: ◮ 0 ֒ → K ( ℓ 2 ) ֒ → B ( ℓ 2 ) (Calkin, 1940). ◮ other classical spaces: ◮ 0 ֒ → K ( X ) ֒ → B ( X ) , where X = c 0 or X = ℓ p for p ∈ [ 1 , ∞ ) . ◮ 0 ֒ → X ℵ 0 ( X ) ֒ → X ℵ 1 ( X ) ֒ → K ( X ) ֒ → . . . ֒ → B ( X ) , where X = c 0 (Γ) or X = ℓ p (Γ) for p ∈ [ 1 , ∞ ) and any set Γ ; X λ ( X ) ideal of ops having range of density at most λ . ◮ c 0 - and ℓ 1 -sums of ℓ n 2 as n → ∞ (Laustsen–Loy–Read, Laustsen–Schlumprecht–Zsák). ◮ Koszmider’s C ( K ) -space from an AD family that exists under CH mentioned by Jesús on Tuesday. ◮ Argyros–Haydon’s scalar-plus-compact space, sums of finitely many incomparable copies thereof, some variants due to Tarbard and further variants (Motakis–Puglisi–Zisimopoulou). ◮ Z = X AH ⊕ suitably constructed subspace (K.–Laustsen). 2
Maximal ideals A perspective . 3
Maximal ideals A perspective . B ( Z ) has precisely two maximal ideals. ֒ → → M 1 ֒ 0 ֒ → K ( Z ) ֒ → E ( Z ) B ( Z ) → ֒ → ֒ M 2 3
Maximal ideals A perspective . B ( Z ) has precisely two maximal ideals. ֒ → → M 1 ֒ 0 ֒ → K ( Z ) ֒ → E ( Z ) B ( Z ) → ֒ → ֒ M 2 This behaviour is rather rare. M X = { T ∈ B ( X ): I X � = ATB ( A , B ∈ B ( X )) } is the unique maximal ideal of B ( X ) ⇐ ⇒ M X closed under addition. 3
Maximal ideals A perspective . B ( Z ) has precisely two maximal ideals. ֒ → → M 1 ֒ 0 ֒ → K ( Z ) ֒ → E ( Z ) B ( Z ) → ֒ → ֒ M 2 This behaviour is rather rare. M X = { T ∈ B ( X ): I X � = ATB ( A , B ∈ B ( X )) } is the unique maximal ideal of B ( X ) ⇐ ⇒ M X closed under addition. ◮ c 0 , ℓ p (here p = ∞ is included, btw. ℓ ∞ ∼ = L ∞ ); ◮ L p [ 0 , 1 ] for p ∈ [ 1 , ∞ ] . ◮ c 0 (Γ) , ℓ p (Γ) for p ∈ [ 1 , ∞ ) ◮ ℓ ∞ / c 0 , ℓ c ∞ (Γ) for any set Γ (but not every L ∞ ( µ ) is in this class!) ◮ c 0 - and ℓ p -sums of ℓ n 2 s or ℓ n ∞ s as well as more general sums. ◮ Lorentz sequence spaces determined by a decreasing, non-summable sequence and p ∈ [ 1 , ∞ ) . ◮ certain Orlicz spaces. ◮ C [ 0 , 1 ] , C [ 0 , ω ω ] , C [ 0 , ω 1 ] , and the list goes on. 3
Tsirelson space revisited (Figiel–Johnson) Put a norm on c 00 : � � � x � ℓ ∞ , 1 � � x � T = max 2 sup � N i x � T i where the sup runs over j ∈ N and all finite sequences of sets N 1 < · · · < N j in N with j � min N 1 . 4
Tsirelson space revisited (Figiel–Johnson) Put a norm on c 00 : � � � x � ℓ ∞ , 1 � � x � T = max 2 sup � N i x � T i where the sup runs over j ∈ N and all finite sequences of sets N 1 < · · · < N j in N with j � min N 1 . ◮ The standard u.v.b. ( t n ) ∞ n = 1 of T is 1-unconditional. 4
Tsirelson space revisited (Figiel–Johnson) Put a norm on c 00 : � � � x � ℓ ∞ , 1 � � x � T = max 2 sup � N i x � T i where the sup runs over j ∈ N and all finite sequences of sets N 1 < · · · < N j in N with j � min N 1 . ◮ The standard u.v.b. ( t n ) ∞ n = 1 of T is 1-unconditional. ◮ For a space with an unconditional basis and N ⊂ N we call the ideal � P N � generated by the associated basis projection P N spatial 4
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