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

Let A = (A1, . . . , Ad) be a commuting tuple of Hilbert space

• perators. In this talk, I hope to convince you that:

Philosophy

To study A = (A1, . . . , Ad), we should use ideas from commutative algebra and algebraic geometry.

For monomials in C[z] := C[z1, . . . , zd], write zα = zα1

1 · · · zαd d ,

α ∈ Nd.

Definition

The Drury-Arveson Hilbert space Hd is the completion of z with respect to z z

d d d

We view f Hd as an analytic function on the complex unit ball

d,

f z

d

c z

For monomials in C[z] := C[z1, . . . , zd], write zα = zα1

1 · · · zαd d ,

α ∈ Nd.

Definition

The Drury-Arveson Hilbert space H2

d is the completion of C[z] with

respect to ⟨zα, zβ⟩ = δα,β α1! · · · αd! (α1 + · · · + αd)!, α, β ∈ Nd. We view f Hd as an analytic function on the complex unit ball

d,

f z

d

c z

For monomials in C[z] := C[z1, . . . , zd], write zα = zα1

1 · · · zαd d ,

α ∈ Nd.

Definition

The Drury-Arveson Hilbert space H2

d is the completion of C[z] with

respect to ⟨zα, zβ⟩ = δα,β α1! · · · αd! (α1 + · · · + αd)!, α, β ∈ Nd. We view f ∈ H2

d as an analytic function on the complex unit ball Bd,

f(z) = ∑

α∈Nd

cαzα.

Let Mz = (Mz1, . . . , Mzd) denote the d-tuple of coordinate multiplication operators on C[z], Mzip(z) = zip(z), p ∈ C[z]. This tuple extends to a contractive d-tuple of operators on the Drury-Arveson Hilbert space H2

d.

Let I ◁ C[z] be an ideal. Then I is an invariant subspace for Mz1, . . . , Mzd, so writing H2

d = I⊥ ⊕ I,

Mzi = ( Ai ∗ ∗ ) , 1 ≤ i ≤ d. Note: We view I as the completion of z I.

Observation

The d-tuple A A Ad is the (extension of the) d-tuple of coordinate multiplication operators on z I, Aip z zip z p z

Let I ◁ C[z] be an ideal. Then I is an invariant subspace for Mz1, . . . , Mzd, so writing H2

d = I⊥ ⊕ I,

Mzi = ( Ai ∗ ∗ ) , 1 ≤ i ≤ d. Note: We view I⊥ as the completion of C[z]/I.

Observation

The d-tuple A A Ad is the (extension of the) d-tuple of coordinate multiplication operators on z I, Aip z zip z p z

Let I ◁ C[z] be an ideal. Then I is an invariant subspace for Mz1, . . . , Mzd, so writing H2

d = I⊥ ⊕ I,

Mzi = ( Ai ∗ ∗ ) , 1 ≤ i ≤ d. Note: We view I⊥ as the completion of C[z]/I.

Observation

The d-tuple A = (A1, . . . , Ad) is the (extension of the) d-tuple of coordinate multiplication operators on C[z]/I, Aip(z) = zip(z), p ∈ C[z].

Theorem (Arveson, Müller-Vasilescu)

Every contractive d-tuple of commuting operators A = (A1, . . . , Ad) arises in this way. We may need to consider vector-valued polynomials. But many interesting problems reduce to the scalar case.

Theorem (Arveson, Müller-Vasilescu)

Every contractive d-tuple of commuting operators A = (A1, . . . , Ad) arises in this way. We may need to consider vector-valued polynomials. But many interesting problems reduce to the scalar case.

Summary: Let I ◁ C[z] be an ideal. The d-tuple of coordinate multiplication operators A = (A1, . . . , Ad) on the quotient C[z]/I extend to bounded linear operators on HI = C[z]/I.

Philosophy

To understand arbitrary commuting tuples of operators, we should try to understand A A Ad .

Summary: Let I ◁ C[z] be an ideal. The d-tuple of coordinate multiplication operators A = (A1, . . . , Ad) on the quotient C[z]/I extend to bounded linear operators on HI = C[z]/I.

Philosophy

To understand arbitrary commuting tuples of operators, we should try to understand A = (A1, . . . , Ad).

The Arveson-Douglas essential normality conjecture

Let I ◁ C[z] be an ideal and let A = (A1, . . . , Ad) be the d-tuple of coordinate multiplication operators arising as above on HI = C[z]/I.

Arveson-Douglas Conjecture

We should expect connections between the structure of A = (A1, . . . , Ad) and the geometric structure of the variety V(I) = {λ ∈ Cd | p(λ) = 0 ∀p ∈ I}. .

Let I ◁ C[z] and A = (A1, . . . , Ad) be as above.

Definition

The ideal I is essentially normal (resp. p-essentially normal) if the self-commutators Ai Aj AjAi 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 = (A1, . . . , Ad) be as above.

Definition

The ideal I is essentially normal (resp. p-essentially normal) if the self-commutators A∗

i Aj − AjA∗ i ,

1 ≤ 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 = (A1, . . . , Ad) be as above.

Definition

The ideal I is essentially normal (resp. p-essentially normal) if the self-commutators A∗

i Aj − AjA∗ i ,

1 ≤ 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).

Consequence: A positive solution to the Arveson-Douglas conjecture would imply the sequence 0 − → K(HI) − → C∗(A1, . . . , Ad) + K (HI) − → C(V(I) ∩ ∂Bd) − → 0 is exact. The C*-algebra C∗(A1, . . . , Ad) gives rise to an invariant of V(I), conjectured to be the fundamental class of V(I) ∩ ∂Bd.

Known Results

Let I ◁ C[z] be a homogeneous ideal.

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)

The conjecture is true for the trivial ideal I = 0.

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)

The conjecture is true for the trivial ideal I = 0.

Theorem (Arveson, 2003)

The conjecture is true for ideals generated by monomials, i.e. elements of the form zα for α ∈ Nd.

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 = ⟨z2

1 + z2 2 − z2 3, z2 4 + z2 5 − z2 6, , z2 7 + z2 8 − z2 9⟩ ◁ C[z1, . . . , z9].

Proof relates the conjecture to the Hilbert space geometry of ideals.

Theorem (K, 2012)

Let I ◁ C[z] be an ideal. The conjecture is true for I if it can be decomposed as I = I1 + · · · + In, where I1, . . . , In are ideals satisfying the conjecture with positive 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.

Proof relates the conjecture to the Hilbert space geometry of ideals.

Theorem (K, 2012)

Let I ◁ C[z] be an ideal. The conjecture is true for I if it can be decomposed as I = I1 + · · · + In, where I1, . . . , In are ideals satisfying the conjecture with positive 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.

Proof relates the conjecture to the Hilbert space geometry of ideals.

Theorem (K, 2012)

Let I ◁ C[z] be an ideal. The conjecture is true for I if it can be decomposed as I = I1 + · · · + In, where I1, . . . , In are ideals satisfying the conjecture with positive 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 ≤ 2 (using Gröbner basis techniques due to Orr Shalit). Plus some other cases.

More recently, we consider the geometry of the variety V = V(I).

Theorem (K-Shalit, 2012)

Let V and W be homogeneous varities in

d with isomorphic

coordinate rings. The conjecture is true for V if and only if it is true for W.

Theorem (K-Shalit, 2012)

The conjecture is true for homogeneous varieties in

d that are

unions of finitely many subspaces. We say the conjecture is true for a variety V if it is true for the corresponding ideal I I V .

More recently, we consider the geometry of the variety V = V(I).

Theorem (K-Shalit, 2012)

Let V and W be homogeneous varities in Cd with isomorphic coordinate rings. The conjecture is true for V if and only if it is true for W.

Theorem (K-Shalit, 2012)

The conjecture is true for homogeneous varieties in

d that are

unions of finitely many subspaces. We say the conjecture is true for a variety V if it is true for the corresponding ideal I I V .

More recently, we consider the geometry of the variety V = V(I).

Theorem (K-Shalit, 2012)

Let V and W be homogeneous varities in Cd with isomorphic coordinate rings. The conjecture is true for V if and only if it is true for W.

Theorem (K-Shalit, 2012)

The conjecture is true for homogeneous varieties in Cd that are unions of finitely many subspaces. We say the conjecture is true for a variety V if it is true for the corresponding ideal I I V .

More recently, we consider the geometry of the variety V = V(I).

Theorem (K-Shalit, 2012)

Let V and W be homogeneous varities in Cd with isomorphic coordinate rings. The conjecture is true for V if and only if it is true for W.

Theorem (K-Shalit, 2012)

The conjecture is true for homogeneous varieties in Cd that are unions of finitely many subspaces. We say the conjecture is true for a variety V if it is true for the corresponding ideal I = I(V).

The Arveson-Douglas conjecture is basically a quantitative statement about finite-dimensional matrices. Experiments suggest it is true.

Definition (Arveson, 2011)

A d-tuple of operators A = (A1, . . . , Ad) is hyperrigid if, the restriction of every unital *-representation π of C∗(A1, . . . , Ad) to span{I, A1, . . . , Ad} has a unique unital completely positive extension (namely, π). In other words, if

• n span I A

Ad , then

• n

C A Ad .

Definition (Arveson, 2011)

A d-tuple of operators A = (A1, . . . , Ad) is hyperrigid if, the restriction of every unital *-representation π of C∗(A1, . . . , Ad) to span{I, A1, . . . , Ad} has a unique unital completely positive extension (namely, π). In other words, if Φ = π on span{I, A1, . . . , Ad}, then Φ = π on C∗(A1, . . . , Ad).

More recently, we proved the following qualitative reformulation of the conjecture (for p = ∞).

Theorem (Davidson-K-Shalit, 2013)

Let I z and A A Ad be as above. The conjecture is true for I if and only if A Ad is hyperrigid.

More recently, we proved the following qualitative reformulation of the conjecture (for p = ∞).

Theorem (Davidson-K-Shalit, 2013)

Let I ◁ C[z] and A = (A1, . . . , Ad) be as above. The conjecture is true for I if and only if {A1, . . . , Ad} is hyperrigid.

Thanks!