The Arveson-Douglas essential normality conjecture Matthew Kennedy - PowerPoint PPT Presentation

The Arveson-Douglas essential normality conjecture Matthew Kennedy Carleton University Aug. 3, 2013 1

Commuting tuples of operators

Philosophy operators. In this talk, I hope to convince you that: algebra and algebraic geometry. Let A = ( A 1 , . . . , A d ) be a commuting tuple of Hilbert space To study A = ( A 1 , . . . , A d ) , we should use ideas from commutative

The Drury-Arveson Hilbert space H d is the completion of H d as an analytic function on the complex unit ball c z d d f z d , We view f d d z z respect to z with Definition For monomials in C [ z ] := C [ z 1 , . . . , z d ] , write z α = z α 1 1 · · · z α d α ∈ N d . d ,

H d as an analytic function on the complex unit ball c z respect to d f z d , Definition We view f For monomials in C [ z ] := C [ z 1 , . . . , z d ] , write z α = z α 1 1 · · · z α d α ∈ N d . d , The Drury-Arveson Hilbert space H 2 d is the completion of C [ z ] with α 1 ! · · · α d ! ⟨ z α , z β ⟩ = δ α,β α, β ∈ N d . ( α 1 + · · · + α d )! ,

respect to Definition For monomials in C [ z ] := C [ z 1 , . . . , z d ] , write z α = z α 1 1 · · · z α d α ∈ N d . d , The Drury-Arveson Hilbert space H 2 d is the completion of C [ z ] with α 1 ! · · · α d ! ⟨ z α , z β ⟩ = δ α,β α, β ∈ N d . ( α 1 + · · · + α d )! , We view f ∈ H 2 d as an analytic function on the complex unit ball B d , ∑ c α z α . f ( z ) = α ∈ N d

d . This tuple extends to a contractive d -tuple of operators on the Let M z = ( M z 1 , . . . , M z d ) denote the d -tuple of coordinate multiplication operators on C [ z ] , M z i p ( z ) = z i p ( z ) , p ∈ C [ z ] . Drury-Arveson Hilbert space H 2

A d is the (extension of the) d -tuple of z as the completion of p z i p z A i p z z I , coordinate multiplication operators on A The d -tuple A Observation z I . Note: We view I A i Let I ◁ C [ z ] be an ideal. Then I is an invariant subspace for d = I ⊥ ⊕ I , M z 1 , . . . , M z d , so writing H 2 ( ) 0 M z i = 1 ≤ i ≤ d . , ∗ ∗

A d is the (extension of the) d -tuple of z Observation p z i p z A i p z A i z I , coordinate multiplication operators on A The d -tuple A Let I ◁ C [ z ] be an ideal. Then I is an invariant subspace for d = I ⊥ ⊕ I , M z 1 , . . . , M z d , so writing H 2 ( ) 0 M z i = 1 ≤ i ≤ d . , ∗ ∗ Note: We view I ⊥ as the completion of C [ z ]/ I .

Observation A i Let I ◁ C [ z ] be an ideal. Then I is an invariant subspace for d = I ⊥ ⊕ I , M z 1 , . . . , M z d , so writing H 2 ( ) 0 M z i = 1 ≤ i ≤ d . , ∗ ∗ Note: We view I ⊥ as the completion of C [ z ]/ I . The d -tuple A = ( A 1 , . . . , A d ) is the (extension of the) d -tuple of coordinate multiplication operators on C [ z ]/ I , A i p ( z ) = z i p ( z ) , p ∈ C [ z ] .

Theorem (Arveson, Müller-Vasilescu) arises in this way. We may need to consider vector-valued polynomials. But many interesting problems reduce to the scalar case. Every contractive d-tuple of commuting operators A = ( A 1 , . . . , A d )

Theorem (Arveson, Müller-Vasilescu) arises in this way. We may need to consider vector-valued polynomials. But many interesting problems reduce to the scalar case. Every contractive d-tuple of commuting operators A = ( A 1 , . . . , A d )

A d . Philosophy To understand arbitrary commuting tuples of operators, we should try to understand A A Summary: Let I ◁ C [ z ] be an ideal. The d -tuple of coordinate multiplication operators A = ( A 1 , . . . , A d ) on the quotient C [ z ]/ I extend to bounded linear operators on H I = C [ z ]/ I .

Philosophy To understand arbitrary commuting tuples of operators, we should try Summary: Let I ◁ C [ z ] be an ideal. The d -tuple of coordinate multiplication operators A = ( A 1 , . . . , A d ) on the quotient C [ z ]/ I extend to bounded linear operators on H I = C [ z ]/ I . to understand A = ( A 1 , . . . , A d ) .

The Arveson-Douglas essential normality conjecture

Arveson-Douglas Conjecture We should expect connections between the structure of . Let I ◁ C [ z ] be an ideal and let A = ( A 1 , . . . , A d ) be the d -tuple of coordinate multiplication operators arising as above on H I = C [ z ]/ I . A = ( A 1 , . . . , A d ) and the geometric structure of the variety V ( I ) = { λ ∈ C d | p ( λ ) = 0 ∀ p ∈ I } .

A i A j Definition The ideal I is essentially normal (resp. p -essentially normal ) if the self-commutators A j A i i j d are compact (resp. contained in the Schatten p -class). Arveson-Douglas Conjecture Suppose I is homogeneous (i.e. generated by homogeneous polynomials). Then I is p -essentially normal for every p dim V I . Let I ◁ C [ z ] and A = ( A 1 , . . . , A d ) be as above.

Definition The ideal I is essentially normal (resp. p -essentially normal ) if the self-commutators are compact (resp. contained in the Schatten p -class). Arveson-Douglas Conjecture Suppose I is homogeneous (i.e. generated by homogeneous polynomials). Then I is p -essentially normal for every p dim V I . Let I ◁ C [ z ] and A = ( A 1 , . . . , A d ) be as above. A ∗ i A j − A j A ∗ 1 ≤ i , j ≤ d i ,

Definition The ideal I is essentially normal (resp. p -essentially normal ) if the self-commutators are compact (resp. contained in the Schatten p -class). Arveson-Douglas Conjecture Suppose I is homogeneous (i.e. generated by homogeneous Let I ◁ C [ z ] and A = ( A 1 , . . . , A d ) be as above. A ∗ i A j − A j A ∗ 1 ≤ i , j ≤ d i , polynomials). Then I is p -essentially normal for every p > dim V ( I ) .

Consequence: A positive solution to the Arveson-Douglas conjecture would imply the sequence → C ∗ ( A 1 , . . . , A d ) + K ( H I ) − 0 − → K ( H I ) − → C ( V ( I ) ∩ ∂ B d ) − → 0 is exact. The C*-algebra C ∗ ( A 1 , . . . , A d ) gives rise to an invariant of V ( I ) , conjectured to be the fundamental class of V ( I ) ∩ ∂ B d .

Known Results

Theorem (Arveson, 1998) The conjecture is true for the trivial ideal I . Theorem (Arveson, 2003) The conjecture is true for ideals generated by monomials, i.e. elements of the form z for d . Let I ◁ C [ z ] be a homogeneous ideal.

Theorem (Arveson, 1998) Theorem (Arveson, 2003) The conjecture is true for ideals generated by monomials, i.e. elements of the form z for d . Let I ◁ C [ z ] be a homogeneous ideal. The conjecture is true for the trivial ideal I = 0 .

Theorem (Arveson, 1998) Theorem (Arveson, 2003) The conjecture is true for ideals generated by monomials, i.e. Let I ◁ C [ z ] be a homogeneous ideal. The conjecture is true for the trivial ideal I = 0 . elements of the form z α for α ∈ N d .

The best known results are due to Guo and Wang. Theorem (Guo-Wang, 2007) The conjecture is true for ideals generated by a single homogeneous polynomial. Theorem (Guo-Wang, 2007) The conjecture is true for d .

The best known results are due to Guo and Wang. Theorem (Guo-Wang, 2007) The conjecture is true for ideals generated by a single homogeneous polynomial. Theorem (Guo-Wang, 2007) The conjecture is true for d ≤ 3 .

Theorem (K, 2012) The conjecture is true for ideals generated by homogeneous polynomials in mutually disjoint variables. For example, the conjecture is true for the ideal I z z z z z z z z z z z

Theorem (K, 2012) The conjecture is true for ideals generated by homogeneous polynomials in mutually disjoint variables. For example, the conjecture is true for the ideal I = ⟨ z 2 1 + z 2 2 − z 2 3 , z 2 4 + z 2 5 − z 2 6 , , z 2 7 + z 2 8 − z 2 9 ⟩ ◁ C [ z 1 , . . . , z 9 ] .

Proof relates the conjecture to the Hilbert space geometry of ideals. Theorem (K, 2012) decomposed as angles (in the sense of Friedrichs) between them. Q: When can we obtain such a decomposition? A (so far): If I is generated by monomials, if I is generated by polynomials in mutually disjoint variables, or if d (using Gröbner basis techniques due to Orr Shalit). Plus some other cases. Let I ◁ C [ z ] be an ideal. The conjecture is true for I if it can be I = I 1 + · · · + I n , where I 1 , . . . , I n are ideals satisfying the conjecture with positive

Proof relates the conjecture to the Hilbert space geometry of ideals. Theorem (K, 2012) decomposed as angles (in the sense of Friedrichs) between them. Q: When can we obtain such a decomposition? A (so far): If I is generated by monomials, if I is generated by polynomials in mutually disjoint variables, or if d (using Gröbner basis techniques due to Orr Shalit). Plus some other cases. Let I ◁ C [ z ] be an ideal. The conjecture is true for I if it can be I = I 1 + · · · + I n , where I 1 , . . . , I n are ideals satisfying the conjecture with positive

Proof relates the conjecture to the Hilbert space geometry of ideals. Theorem (K, 2012) decomposed as angles (in the sense of Friedrichs) between them. Q: When can we obtain such a decomposition? A (so far): If I is generated by monomials, if I is generated by basis techniques due to Orr Shalit). Plus some other cases. Let I ◁ C [ z ] be an ideal. The conjecture is true for I if it can be I = I 1 + · · · + I n , where I 1 , . . . , I n are ideals satisfying the conjecture with positive polynomials in mutually disjoint variables, or if d ≤ 2 (using Gröbner