Positive extensions of Schur multipliers Ying-Fen Lin Queens - - PowerPoint PPT Presentation

Motivation Generalisations to ℓ2(X) Generalisations to measure spaces

Positive extensions of Schur multipliers

Ying-Fen Lin

Queen’s University Belfast (joint work with Rupert Levene and Ivan Todorov)

SOAR November 13, 2015

Ying-Fen Lin Positive extensions

Motivation Generalisations to ℓ2(X) Generalisations to measure spaces

An n × n matrix is called partially defined if only some of its entries are specified with the unspecified entries treated as complex variables. A completion of a partially defined matrix is simply a specification

• f the unspecified entries.

Question: determine whether or not a completion of a partially defined matrix exists which has some property, for example, contraction, positive...

Ying-Fen Lin Positive extensions

Motivation Generalisations to ℓ2(X) Generalisations to measure spaces

An n × n matrix is called partially defined if only some of its entries are specified with the unspecified entries treated as complex variables. A completion of a partially defined matrix is simply a specification

• f the unspecified entries.

Question: determine whether or not a completion of a partially defined matrix exists which has some property, for example, contraction, positive...

Ying-Fen Lin Positive extensions

Motivation Generalisations to ℓ2(X) Generalisations to measure spaces

An n × n matrix is called partially defined if only some of its entries are specified with the unspecified entries treated as complex variables. A completion of a partially defined matrix is simply a specification

• f the unspecified entries.

Question: determine whether or not a completion of a partially defined matrix exists which has some property, for example, contraction, positive...

Ying-Fen Lin Positive extensions

Motivation Generalisations to ℓ2(X) Generalisations to measure spaces

Known results

Dym and Gohberg (1981) If T = (tij) is a partially defined n × n matrix with tij defined only for |i − j| ≤ k, 0 < k < n − 1, which has the property that all its fully defined k × k principal submatrices are positive semi-definite, then T can be completed to a positive semi-definite matrix. Grone, Johnson, Sa and Wolkowitz (1984) A characterisation is given of those symmetric patterns J such that every partially positive matrix with pattern J has a positive completion.

Ying-Fen Lin Positive extensions

Motivation Generalisations to ℓ2(X) Generalisations to measure spaces

Known results

Dym and Gohberg (1981) If T = (tij) is a partially defined n × n matrix with tij defined only for |i − j| ≤ k, 0 < k < n − 1, which has the property that all its fully defined k × k principal submatrices are positive semi-definite, then T can be completed to a positive semi-definite matrix. Grone, Johnson, Sa and Wolkowitz (1984) A characterisation is given of those symmetric patterns J such that every partially positive matrix with pattern J has a positive completion.

Ying-Fen Lin Positive extensions

Motivation Generalisations to ℓ2(X) Generalisations to measure spaces

Some definitions

A subset J ⊆ {1, . . . , n} × {1, . . . , n} is called a pattern. A partially defined n × n matrix T = (tij) is said to have pattern J if tij is specified if and only if (i, j) ∈ J. If (i, i) ∈ J for all i and if (i, j) ∈ J then (j, i) ∈ J, we call J a symmetric pattern. To each pattern J, we can associate a subspace SJ of Mn by SJ = {(aij) ∈ Mn : aij = 0 if (i, j) ∈ J}. Note that SJ is an operator system if and only if J is symmetric.

Ying-Fen Lin Positive extensions

Motivation Generalisations to ℓ2(X) Generalisations to measure spaces

Some definitions

A subset J ⊆ {1, . . . , n} × {1, . . . , n} is called a pattern. A partially defined n × n matrix T = (tij) is said to have pattern J if tij is specified if and only if (i, j) ∈ J. If (i, i) ∈ J for all i and if (i, j) ∈ J then (j, i) ∈ J, we call J a symmetric pattern. To each pattern J, we can associate a subspace SJ of Mn by SJ = {(aij) ∈ Mn : aij = 0 if (i, j) ∈ J}. Note that SJ is an operator system if and only if J is symmetric.

Ying-Fen Lin Positive extensions

Motivation Generalisations to ℓ2(X) Generalisations to measure spaces

Some definitions

A subset J ⊆ {1, . . . , n} × {1, . . . , n} is called a pattern. A partially defined n × n matrix T = (tij) is said to have pattern J if tij is specified if and only if (i, j) ∈ J. If (i, i) ∈ J for all i and if (i, j) ∈ J then (j, i) ∈ J, we call J a symmetric pattern. To each pattern J, we can associate a subspace SJ of Mn by SJ = {(aij) ∈ Mn : aij = 0 if (i, j) ∈ J}. Note that SJ is an operator system if and only if J is symmetric.

Ying-Fen Lin Positive extensions

Motivation Generalisations to ℓ2(X) Generalisations to measure spaces

Some definitions

A subset J ⊆ {1, . . . , n} × {1, . . . , n} is called a pattern. A partially defined n × n matrix T = (tij) is said to have pattern J if tij is specified if and only if (i, j) ∈ J. If (i, i) ∈ J for all i and if (i, j) ∈ J then (j, i) ∈ J, we call J a symmetric pattern. To each pattern J, we can associate a subspace SJ of Mn by SJ = {(aij) ∈ Mn : aij = 0 if (i, j) ∈ J}. Note that SJ is an operator system if and only if J is symmetric.

Ying-Fen Lin Positive extensions

Motivation Generalisations to ℓ2(X) Generalisations to measure spaces

Some definitions

A subset J ⊆ {1, . . . , n} × {1, . . . , n} is called a pattern. A partially defined n × n matrix T = (tij) is said to have pattern J if tij is specified if and only if (i, j) ∈ J. If (i, i) ∈ J for all i and if (i, j) ∈ J then (j, i) ∈ J, we call J a symmetric pattern. To each pattern J, we can associate a subspace SJ of Mn by SJ = {(aij) ∈ Mn : aij = 0 if (i, j) ∈ J}. Note that SJ is an operator system if and only if J is symmetric.

Ying-Fen Lin Positive extensions

Motivation Generalisations to ℓ2(X) Generalisations to measure spaces

A generalisation of previous results in n × n matrices

Theorem (Paulsen, Power and Smith (1989)) Let J be a symmetric pattern and let T = (tij) be a partially defined matrix with pattern J. Then the following are equivalent:

1 T has a positive completion, 2 φT : SJ → Mn defined by φT((aij)) = (aijtij) is positive, 3 ΨT : SJ → C defined by ΨT((aij)) =

ij aijtij is positive.

Ying-Fen Lin Positive extensions

Motivation Generalisations to ℓ2(X) Generalisations to measure spaces

A generalisation of previous results in n × n matrices

Theorem (Paulsen, Power and Smith (1989)) Let J be a symmetric pattern and let T = (tij) be a partially defined matrix with pattern J. Then the following are equivalent:

1 T has a positive completion, 2 φT : SJ → Mn defined by φT((aij)) = (aijtij) is positive, 3 ΨT : SJ → C defined by ΨT((aij)) =

ij aijtij is positive.

Ying-Fen Lin Positive extensions

Motivation Generalisations to ℓ2(X) Generalisations to measure spaces

A generalisation of previous results in n × n matrices

Theorem (Paulsen, Power and Smith (1989)) Let J be a symmetric pattern and let T = (tij) be a partially defined matrix with pattern J. Then the following are equivalent:

1 T has a positive completion, 2 φT : SJ → Mn defined by φT((aij)) = (aijtij) is positive, 3 ΨT : SJ → C defined by ΨT((aij)) =

ij aijtij is positive.

Ying-Fen Lin Positive extensions

Motivation Generalisations to ℓ2(X) Generalisations to measure spaces

Let J be a symmetric pattern and T = (tij) be a partially defined matrix with pattern J. The matrix T is called partially positive if it is symmetric and every m × m submatrix B = (bk,l) of T with bk,l = tik,il, where (ik, il) ∈ J for 1 ≤ k, l ≤ m, is positive. Note: T is partially positive if and only if φT(P) is positive for every rank one positive P in SJ. Theorem (Paulsen, Power and Smith (1989)) Let J be a symmetric pattern, then the following are equivalent:

1 every partially positive matrix with pattern J has a positive

completion,

2 every positive P ∈ SJ is a sum of rank one positives in SJ, 3 the graph GJ is chordal. Ying-Fen Lin Positive extensions

Motivation Generalisations to ℓ2(X) Generalisations to measure spaces

Let J be a symmetric pattern and T = (tij) be a partially defined matrix with pattern J. The matrix T is called partially positive if it is symmetric and every m × m submatrix B = (bk,l) of T with bk,l = tik,il, where (ik, il) ∈ J for 1 ≤ k, l ≤ m, is positive. Note: T is partially positive if and only if φT(P) is positive for every rank one positive P in SJ. Theorem (Paulsen, Power and Smith (1989)) Let J be a symmetric pattern, then the following are equivalent:

1 every partially positive matrix with pattern J has a positive

completion,

2 every positive P ∈ SJ is a sum of rank one positives in SJ, 3 the graph GJ is chordal. Ying-Fen Lin Positive extensions

Motivation Generalisations to ℓ2(X) Generalisations to measure spaces

Let J be a symmetric pattern and T = (tij) be a partially defined matrix with pattern J. The matrix T is called partially positive if it is symmetric and every m × m submatrix B = (bk,l) of T with bk,l = tik,il, where (ik, il) ∈ J for 1 ≤ k, l ≤ m, is positive. Note: T is partially positive if and only if φT(P) is positive for every rank one positive P in SJ. Theorem (Paulsen, Power and Smith (1989)) Let J be a symmetric pattern, then the following are equivalent:

1 every partially positive matrix with pattern J has a positive

completion,

2 every positive P ∈ SJ is a sum of rank one positives in SJ, 3 the graph GJ is chordal. Ying-Fen Lin Positive extensions

Motivation Generalisations to ℓ2(X) Generalisations to measure spaces

Let J be a symmetric pattern and T = (tij) be a partially defined matrix with pattern J. The matrix T is called partially positive if it is symmetric and every m × m submatrix B = (bk,l) of T with bk,l = tik,il, where (ik, il) ∈ J for 1 ≤ k, l ≤ m, is positive. Note: T is partially positive if and only if φT(P) is positive for every rank one positive P in SJ. Theorem (Paulsen, Power and Smith (1989)) Let J be a symmetric pattern, then the following are equivalent:

1 every partially positive matrix with pattern J has a positive

completion,

2 every positive P ∈ SJ is a sum of rank one positives in SJ, 3 the graph GJ is chordal. Ying-Fen Lin Positive extensions

Motivation Generalisations to ℓ2(X) Generalisations to measure spaces

Let J be a symmetric pattern and T = (tij) be a partially defined matrix with pattern J. The matrix T is called partially positive if it is symmetric and every m × m submatrix B = (bk,l) of T with bk,l = tik,il, where (ik, il) ∈ J for 1 ≤ k, l ≤ m, is positive. Note: T is partially positive if and only if φT(P) is positive for every rank one positive P in SJ. Theorem (Paulsen, Power and Smith (1989)) Let J be a symmetric pattern, then the following are equivalent:

1 every partially positive matrix with pattern J has a positive

completion,

2 every positive P ∈ SJ is a sum of rank one positives in SJ, 3 the graph GJ is chordal. Ying-Fen Lin Positive extensions

Motivation Generalisations to ℓ2(X) Generalisations to measure spaces

On the setting of ℓ2(X)

Let X be a set and H = ℓ2(X) with the canonical orthonormal basis (ex)x∈X. For x, y ∈ X, denote by Ex,y the corresponding matrix unit in B(H). For κ ⊆ X × X, define S(κ) := span{Ex,y : (x, y) ∈ κ}

w∗

Note: T ∈ B(H) is in S(κ) if and only if the matrix (tx,y), tx,y = (Tey, ex), has tx,y = 0 whenever (x, y) ∈ κc. the subspace S(κ) is an operator system if and only if

1

κ is symmetric,

2

κ contains the diagonal of X × X.

Note that each such κ gives rise to a graph on X.

Ying-Fen Lin Positive extensions

Motivation Generalisations to ℓ2(X) Generalisations to measure spaces

On the setting of ℓ2(X)

Let X be a set and H = ℓ2(X) with the canonical orthonormal basis (ex)x∈X. For x, y ∈ X, denote by Ex,y the corresponding matrix unit in B(H). For κ ⊆ X × X, define S(κ) := span{Ex,y : (x, y) ∈ κ}

w∗

Note: T ∈ B(H) is in S(κ) if and only if the matrix (tx,y), tx,y = (Tey, ex), has tx,y = 0 whenever (x, y) ∈ κc. the subspace S(κ) is an operator system if and only if

1

κ is symmetric,

2

κ contains the diagonal of X × X.

Note that each such κ gives rise to a graph on X.

Ying-Fen Lin Positive extensions

Motivation Generalisations to ℓ2(X) Generalisations to measure spaces

On the setting of ℓ2(X)

Let X be a set and H = ℓ2(X) with the canonical orthonormal basis (ex)x∈X. For x, y ∈ X, denote by Ex,y the corresponding matrix unit in B(H). For κ ⊆ X × X, define S(κ) := span{Ex,y : (x, y) ∈ κ}

w∗

Note: T ∈ B(H) is in S(κ) if and only if the matrix (tx,y), tx,y = (Tey, ex), has tx,y = 0 whenever (x, y) ∈ κc. the subspace S(κ) is an operator system if and only if

1

κ is symmetric,

2

κ contains the diagonal of X × X.

Note that each such κ gives rise to a graph on X.

Ying-Fen Lin Positive extensions

Motivation Generalisations to ℓ2(X) Generalisations to measure spaces

On the setting of ℓ2(X)

Let X be a set and H = ℓ2(X) with the canonical orthonormal basis (ex)x∈X. For x, y ∈ X, denote by Ex,y the corresponding matrix unit in B(H). For κ ⊆ X × X, define S(κ) := span{Ex,y : (x, y) ∈ κ}

w∗

Note: T ∈ B(H) is in S(κ) if and only if the matrix (tx,y), tx,y = (Tey, ex), has tx,y = 0 whenever (x, y) ∈ κc. the subspace S(κ) is an operator system if and only if

1

κ is symmetric,

2

κ contains the diagonal of X × X.

Note that each such κ gives rise to a graph on X.

Ying-Fen Lin Positive extensions

Motivation Generalisations to ℓ2(X) Generalisations to measure spaces

On the setting of ℓ2(X)

Let X be a set and H = ℓ2(X) with the canonical orthonormal basis (ex)x∈X. For x, y ∈ X, denote by Ex,y the corresponding matrix unit in B(H). For κ ⊆ X × X, define S(κ) := span{Ex,y : (x, y) ∈ κ}

w∗

Note: T ∈ B(H) is in S(κ) if and only if the matrix (tx,y), tx,y = (Tey, ex), has tx,y = 0 whenever (x, y) ∈ κc. the subspace S(κ) is an operator system if and only if

1

κ is symmetric,

2

κ contains the diagonal of X × X.

Note that each such κ gives rise to a graph on X.

Ying-Fen Lin Positive extensions

Motivation Generalisations to ℓ2(X) Generalisations to measure spaces

On the setting of ℓ2(X)

Let X be a set and H = ℓ2(X) with the canonical orthonormal basis (ex)x∈X. For x, y ∈ X, denote by Ex,y the corresponding matrix unit in B(H). For κ ⊆ X × X, define S(κ) := span{Ex,y : (x, y) ∈ κ}

w∗

Note: T ∈ B(H) is in S(κ) if and only if the matrix (tx,y), tx,y = (Tey, ex), has tx,y = 0 whenever (x, y) ∈ κc. the subspace S(κ) is an operator system if and only if

1

κ is symmetric,

2

κ contains the diagonal of X × X.

Note that each such κ gives rise to a graph on X.

Ying-Fen Lin Positive extensions

Motivation Generalisations to ℓ2(X) Generalisations to measure spaces

Schur multipliers and positive extensions

Definition A function ψ : κ → B(K) is called an (operator-valued) Schur multiplier if Sψ(tx,y) := (tx,yψ(x, y))x,y∈X ∈ B(H ⊗ K) for every (tx,y)x,y∈X ∈ S(κ). A Schur multiplier φ : X × X → B(K) is called positive if Sφ is positive, i.e., for every positive T ∈ B(H), the operator Sφ(T) ∈ B(H ⊗ K) is positive. Let ψ : κ → B(K) be a Schur multiplier. We say that ψ is partially positive if for α ⊆ X with α × α ⊆ κ, the Schur multiplier ψ|α×α is positive.

Ying-Fen Lin Positive extensions

Motivation Generalisations to ℓ2(X) Generalisations to measure spaces

Schur multipliers and positive extensions

Definition A function ψ : κ → B(K) is called an (operator-valued) Schur multiplier if Sψ(tx,y) := (tx,yψ(x, y))x,y∈X ∈ B(H ⊗ K) for every (tx,y)x,y∈X ∈ S(κ). A Schur multiplier φ : X × X → B(K) is called positive if Sφ is positive, i.e., for every positive T ∈ B(H), the operator Sφ(T) ∈ B(H ⊗ K) is positive. Let ψ : κ → B(K) be a Schur multiplier. We say that ψ is partially positive if for α ⊆ X with α × α ⊆ κ, the Schur multiplier ψ|α×α is positive.

Ying-Fen Lin Positive extensions

Motivation Generalisations to ℓ2(X) Generalisations to measure spaces

Schur multipliers and positive extensions

Definition A function ψ : κ → B(K) is called an (operator-valued) Schur multiplier if Sψ(tx,y) := (tx,yψ(x, y))x,y∈X ∈ B(H ⊗ K) for every (tx,y)x,y∈X ∈ S(κ). A Schur multiplier φ : X × X → B(K) is called positive if Sφ is positive, i.e., for every positive T ∈ B(H), the operator Sφ(T) ∈ B(H ⊗ K) is positive. Let ψ : κ → B(K) be a Schur multiplier. We say that ψ is partially positive if for α ⊆ X with α × α ⊆ κ, the Schur multiplier ψ|α×α is positive.

Ying-Fen Lin Positive extensions

Motivation Generalisations to ℓ2(X) Generalisations to measure spaces

Positive extensions on ℓ2(X)

Theorem (Levene, L., Todorov) Let κ ⊆ X × X be a graph. The following conditions are equivalent:

1 every partially positive Schur multiplier ψ : κ → B(K) has a

positive extension;

2 κ is chordal; 3 every positive operator in S(κ) is a weak* limit of sums of

rank one positive operators in S(κ).

Ying-Fen Lin Positive extensions

Motivation Generalisations to ℓ2(X) Generalisations to measure spaces

Let (X, µ) be an arbitrary σ-finite measure space. If k ∈ L2(X × X), the Hilbert-Schmidt operator Tk on L2(X, µ) with integral kernel k is defined by Tkf (y) =

• X

k(y, x)f (x)dµ(x) for f ∈ L2(X, µ), y ∈ X. For any measurable subset κ ⊆ X × X, let S(κ) := {Tk : k ∈ L2(κ)}

w∗

.

Ying-Fen Lin Positive extensions

Motivation Generalisations to ℓ2(X) Generalisations to measure spaces

Let (X, µ) be an arbitrary σ-finite measure space. If k ∈ L2(X × X), the Hilbert-Schmidt operator Tk on L2(X, µ) with integral kernel k is defined by Tkf (y) =

• X

k(y, x)f (x)dµ(x) for f ∈ L2(X, µ), y ∈ X. For any measurable subset κ ⊆ X × X, let S(κ) := {Tk : k ∈ L2(κ)}

w∗

.

Ying-Fen Lin Positive extensions

Motivation Generalisations to ℓ2(X) Generalisations to measure spaces

Let (X, µ) be an arbitrary σ-finite measure space. If k ∈ L2(X × X), the Hilbert-Schmidt operator Tk on L2(X, µ) with integral kernel k is defined by Tkf (y) =

• X

k(y, x)f (x)dµ(x) for f ∈ L2(X, µ), y ∈ X. For any measurable subset κ ⊆ X × X, let S(κ) := {Tk : k ∈ L2(κ)}

w∗

.

Ying-Fen Lin Positive extensions

Motivation Generalisations to ℓ2(X) Generalisations to measure spaces

ω-topology (Erdos, Katavolos and Shulman, 1998)

Let (X, µ) be a σ-finite measure space. A subset E ⊆ X × X is called marginally null if E ⊆ (M × X) ∪ (X × M), where M ⊆ X is null. Two subsets E, F ⊆ X × X are called marginally equivalent, denoted by E ∼ = F, if E △ F is marginally null. A set κ ⊆ X × X is called a rectangle if κ = α × β, where α, β are measurable subsets of X; it is called a square if κ = α × α. A set κ ⊆ X × X is called ω-open if κ ∼ =

i αi × βi, where

αi, βi ⊆ X are measurable.

Ying-Fen Lin Positive extensions

Motivation Generalisations to ℓ2(X) Generalisations to measure spaces

ω-topology (Erdos, Katavolos and Shulman, 1998)

Let (X, µ) be a σ-finite measure space. A subset E ⊆ X × X is called marginally null if E ⊆ (M × X) ∪ (X × M), where M ⊆ X is null. Two subsets E, F ⊆ X × X are called marginally equivalent, denoted by E ∼ = F, if E △ F is marginally null. A set κ ⊆ X × X is called a rectangle if κ = α × β, where α, β are measurable subsets of X; it is called a square if κ = α × α. A set κ ⊆ X × X is called ω-open if κ ∼ =

i αi × βi, where

αi, βi ⊆ X are measurable.

Ying-Fen Lin Positive extensions

Motivation Generalisations to ℓ2(X) Generalisations to measure spaces

ω-topology (Erdos, Katavolos and Shulman, 1998)

Let (X, µ) be a σ-finite measure space. A subset E ⊆ X × X is called marginally null if E ⊆ (M × X) ∪ (X × M), where M ⊆ X is null. Two subsets E, F ⊆ X × X are called marginally equivalent, denoted by E ∼ = F, if E △ F is marginally null. A set κ ⊆ X × X is called a rectangle if κ = α × β, where α, β are measurable subsets of X; it is called a square if κ = α × α. A set κ ⊆ X × X is called ω-open if κ ∼ =

i αi × βi, where

αi, βi ⊆ X are measurable.

Ying-Fen Lin Positive extensions

Motivation Generalisations to ℓ2(X) Generalisations to measure spaces

ω-topology (Erdos, Katavolos and Shulman, 1998)

Let (X, µ) be a σ-finite measure space. A subset E ⊆ X × X is called marginally null if E ⊆ (M × X) ∪ (X × M), where M ⊆ X is null. Two subsets E, F ⊆ X × X are called marginally equivalent, denoted by E ∼ = F, if E △ F is marginally null. A set κ ⊆ X × X is called a rectangle if κ = α × β, where α, β are measurable subsets of X; it is called a square if κ = α × α. A set κ ⊆ X × X is called ω-open if κ ∼ =

i αi × βi, where

αi, βi ⊆ X are measurable.

Ying-Fen Lin Positive extensions

Motivation Generalisations to ℓ2(X) Generalisations to measure spaces

For any measurable ω-closed subset κ ⊆ X × X, let S(κ) := {Tk : k ∈ L2(κ)}

w∗

. Proposition Let κ ⊆ X × X be ω-closed. The following are equivalent:

1 S(κ) is an operator system; 2 κ is symmetric, i.e., κ ∼

= ˆ κ and △ := {(x, x) : x ∈ X} ⊆ω κ, where ˆ κ := {(x, y) ∈ X × X : (y, x) ∈ κ}.

Ying-Fen Lin Positive extensions

Motivation Generalisations to ℓ2(X) Generalisations to measure spaces

For any measurable ω-closed subset κ ⊆ X × X, let S(κ) := {Tk : k ∈ L2(κ)}

w∗

. Proposition Let κ ⊆ X × X be ω-closed. The following are equivalent:

1 S(κ) is an operator system; 2 κ is symmetric, i.e., κ ∼

= ˆ κ and △ := {(x, x) : x ∈ X} ⊆ω κ, where ˆ κ := {(x, y) ∈ X × X : (y, x) ∈ κ}.

Ying-Fen Lin Positive extensions

Motivation Generalisations to ℓ2(X) Generalisations to measure spaces

Positivity domains

Definition An ω-closed set κ ⊆ X × X is called a positivity domain if κ ∼ = ˆ κ, △ ⊆ω κ and κ ∼ = clω(intω(κ)). Here intω(κ) =

ω{α × β : α × β ⊆ κ} and clω(κ) = intω(κc)c.

Proposition: For an ω-closed set κ, we have that κ ∼ clω(intω(κ)) if and only if S(κ) is the weak* closed span of the rank one

• perators it contains.

Note: There exists a positivity domain κ such that S(κ) has no positive rank one operators.

Ying-Fen Lin Positive extensions

Motivation Generalisations to ℓ2(X) Generalisations to measure spaces

Positivity domains

Definition An ω-closed set κ ⊆ X × X is called a positivity domain if κ ∼ = ˆ κ, △ ⊆ω κ and κ ∼ = clω(intω(κ)). Here intω(κ) =

ω{α × β : α × β ⊆ κ} and clω(κ) = intω(κc)c.

Proposition: For an ω-closed set κ, we have that κ ∼ clω(intω(κ)) if and only if S(κ) is the weak* closed span of the rank one

• perators it contains.

Note: There exists a positivity domain κ such that S(κ) has no positive rank one operators.

Ying-Fen Lin Positive extensions

Motivation Generalisations to ℓ2(X) Generalisations to measure spaces

Positivity domains

Definition An ω-closed set κ ⊆ X × X is called a positivity domain if κ ∼ = ˆ κ, △ ⊆ω κ and κ ∼ = clω(intω(κ)). Here intω(κ) =

ω{α × β : α × β ⊆ κ} and clω(κ) = intω(κc)c.

Proposition: For an ω-closed set κ, we have that κ ∼ clω(intω(κ)) if and only if S(κ) is the weak* closed span of the rank one

• perators it contains.

Note: There exists a positivity domain κ such that S(κ) has no positive rank one operators.

Ying-Fen Lin Positive extensions

Motivation Generalisations to ℓ2(X) Generalisations to measure spaces

For a positivity domain κ ⊆ X × X, let [S+

1 (κ)] =

• n
• i=1

Ri : Ri ∈ S(κ)+ is of rank one, 1 ≤ i ≤ n

• .

Theorem The following are equivalent, for a positivity domain κ: there exists a family (αi)i∈N of mutually disjoint measurable subsets of X s.t. αi × αi ⊆ω κ for each i and ∞

i=1 αi = X;

I ∈ [S+

1 (κ)] w∗

; κ is generated by squares, i.e. κ ∼ = clω(sqintω(κ)), where sqintω(κ) =

• ω

{Q : Q is a square with Q ⊆ κ}.

Ying-Fen Lin Positive extensions

Motivation Generalisations to ℓ2(X) Generalisations to measure spaces

For a positivity domain κ ⊆ X × X, let [S+

1 (κ)] =

• n
• i=1

Ri : Ri ∈ S(κ)+ is of rank one, 1 ≤ i ≤ n

• .

Theorem The following are equivalent, for a positivity domain κ: there exists a family (αi)i∈N of mutually disjoint measurable subsets of X s.t. αi × αi ⊆ω κ for each i and ∞

i=1 αi = X;

I ∈ [S+

1 (κ)] w∗

; κ is generated by squares, i.e. κ ∼ = clω(sqintω(κ)), where sqintω(κ) =

• ω

{Q : Q is a square with Q ⊆ κ}.

Ying-Fen Lin Positive extensions

Motivation Generalisations to ℓ2(X) Generalisations to measure spaces

For a positivity domain κ ⊆ X × X, let [S+

1 (κ)] =

• n
• i=1

Ri : Ri ∈ S(κ)+ is of rank one, 1 ≤ i ≤ n

• .

Theorem The following are equivalent, for a positivity domain κ: there exists a family (αi)i∈N of mutually disjoint measurable subsets of X s.t. αi × αi ⊆ω κ for each i and ∞

i=1 αi = X;

I ∈ [S+

1 (κ)] w∗

; κ is generated by squares, i.e. κ ∼ = clω(sqintω(κ)), where sqintω(κ) =

• ω

{Q : Q is a square with Q ⊆ κ}.

Ying-Fen Lin Positive extensions

Motivation Generalisations to ℓ2(X) Generalisations to measure spaces

For a positivity domain κ ⊆ X × X, let [S+

1 (κ)] =

• n
• i=1

Ri : Ri ∈ S(κ)+ is of rank one, 1 ≤ i ≤ n

• .

Theorem The following are equivalent, for a positivity domain κ: there exists a family (αi)i∈N of mutually disjoint measurable subsets of X s.t. αi × αi ⊆ω κ for each i and ∞

i=1 αi = X;

I ∈ [S+

1 (κ)] w∗

; κ is generated by squares, i.e. κ ∼ = clω(sqintω(κ)), where sqintω(κ) =

• ω

{Q : Q is a square with Q ⊆ κ}.

Ying-Fen Lin Positive extensions

Motivation Generalisations to ℓ2(X) Generalisations to measure spaces

Schur multipliers

Let κ ⊆ X × X be a positivity domain. A measurable function ϕ : κ → C is called Schur multiplier if ∃C > 0 such that Tϕk ≤ CTk for every k ∈ L2(κ). Proposition Let κ ⊆ X × X be a positivity domain and let ϕ : κ → C be a measurable function. The following are equivalent: ϕ is a Schur multiplier; ∃ a Schur multiplier ψ : X × X → C such that ψ|κ ∼ ϕ; ∃ a completely bounded weak*-conti. map Φ : S(κ) → S(κ) such that Φ(Tk) = Tϕk for k ∈ L2(κ).

Ying-Fen Lin Positive extensions

Motivation Generalisations to ℓ2(X) Generalisations to measure spaces

Schur multipliers

Let κ ⊆ X × X be a positivity domain. A measurable function ϕ : κ → C is called Schur multiplier if ∃C > 0 such that Tϕk ≤ CTk for every k ∈ L2(κ). Proposition Let κ ⊆ X × X be a positivity domain and let ϕ : κ → C be a measurable function. The following are equivalent: ϕ is a Schur multiplier; ∃ a Schur multiplier ψ : X × X → C such that ψ|κ ∼ ϕ; ∃ a completely bounded weak*-conti. map Φ : S(κ) → S(κ) such that Φ(Tk) = Tϕk for k ∈ L2(κ).

Ying-Fen Lin Positive extensions

Motivation Generalisations to ℓ2(X) Generalisations to measure spaces

A characterisation of partially positive Schur multipliers

Let κ ⊆ X × X be a positivity domain. A Schur multiplier ϕ : κ → C is called partially positive if ϕ|α×α is a positive Schur multiplier whenever α ⊆ X is a measurable set with α × α ⊆ κ. Proposition Let κ be a positivity domain. A Schur multiplier ϕ : κ → C is partially positive if and only if Sϕ(S+

1 (κ)) ⊆ B(L2(X))+.

Ying-Fen Lin Positive extensions

Motivation Generalisations to ℓ2(X) Generalisations to measure spaces

A characterisation of partially positive Schur multipliers

Let κ ⊆ X × X be a positivity domain. A Schur multiplier ϕ : κ → C is called partially positive if ϕ|α×α is a positive Schur multiplier whenever α ⊆ X is a measurable set with α × α ⊆ κ. Proposition Let κ be a positivity domain. A Schur multiplier ϕ : κ → C is partially positive if and only if Sϕ(S+

1 (κ)) ⊆ B(L2(X))+.

Ying-Fen Lin Positive extensions

Motivation Generalisations to ℓ2(X) Generalisations to measure spaces

A characterisation of partially positive Schur multipliers

Let κ ⊆ X × X be a positivity domain. A Schur multiplier ϕ : κ → C is called partially positive if ϕ|α×α is a positive Schur multiplier whenever α ⊆ X is a measurable set with α × α ⊆ κ. Proposition Let κ be a positivity domain. A Schur multiplier ϕ : κ → C is partially positive if and only if Sϕ(S+

1 (κ)) ⊆ B(L2(X))+.

Ying-Fen Lin Positive extensions

Motivation Generalisations to ℓ2(X) Generalisations to measure spaces

Proof: (⇐) Suppose Sϕ(S+

1 (κ)) ⊆ B(L2(X))+ and α × α ⊆ κ for a

measurable set α ⊆ X. If T is positive rank one supported by α × α, then Sϕ(T) ≥ 0. Since Sϕ is weak*-conti. and B(P(α)L2(X))+ = span{T : T positive rank one supported by α × α}

w∗

⇒ ϕ|α×α is positive Schur multiplier. (⇒) Suppose ϕ is partially positive and T ∈ S(κ) is a positive rank one operator, say, T = η ⊗ η∗ for some η ∈ L2(X). If supp η = α, then α × α ⊆ω κ (Erdos, Katavolos and Shulman). Then Sϕ(T) ≥ 0 by assumption.

Ying-Fen Lin Positive extensions

Motivation Generalisations to ℓ2(X) Generalisations to measure spaces

Proof: (⇐) Suppose Sϕ(S+

1 (κ)) ⊆ B(L2(X))+ and α × α ⊆ κ for a

measurable set α ⊆ X. If T is positive rank one supported by α × α, then Sϕ(T) ≥ 0. Since Sϕ is weak*-conti. and B(P(α)L2(X))+ = span{T : T positive rank one supported by α × α}

w∗

⇒ ϕ|α×α is positive Schur multiplier. (⇒) Suppose ϕ is partially positive and T ∈ S(κ) is a positive rank one operator, say, T = η ⊗ η∗ for some η ∈ L2(X). If supp η = α, then α × α ⊆ω κ (Erdos, Katavolos and Shulman). Then Sϕ(T) ≥ 0 by assumption.

Ying-Fen Lin Positive extensions

Motivation Generalisations to ℓ2(X) Generalisations to measure spaces

Proof: (⇐) Suppose Sϕ(S+

1 (κ)) ⊆ B(L2(X))+ and α × α ⊆ κ for a

measurable set α ⊆ X. If T is positive rank one supported by α × α, then Sϕ(T) ≥ 0. Since Sϕ is weak*-conti. and B(P(α)L2(X))+ = span{T : T positive rank one supported by α × α}

w∗

⇒ ϕ|α×α is positive Schur multiplier. (⇒) Suppose ϕ is partially positive and T ∈ S(κ) is a positive rank one operator, say, T = η ⊗ η∗ for some η ∈ L2(X). If supp η = α, then α × α ⊆ω κ (Erdos, Katavolos and Shulman). Then Sϕ(T) ≥ 0 by assumption.

Ying-Fen Lin Positive extensions

Motivation Generalisations to ℓ2(X) Generalisations to measure spaces

Proof: (⇐) Suppose Sϕ(S+

1 (κ)) ⊆ B(L2(X))+ and α × α ⊆ κ for a

measurable set α ⊆ X. If T is positive rank one supported by α × α, then Sϕ(T) ≥ 0. Since Sϕ is weak*-conti. and B(P(α)L2(X))+ = span{T : T positive rank one supported by α × α}

w∗

⇒ ϕ|α×α is positive Schur multiplier. (⇒) Suppose ϕ is partially positive and T ∈ S(κ) is a positive rank one operator, say, T = η ⊗ η∗ for some η ∈ L2(X). If supp η = α, then α × α ⊆ω κ (Erdos, Katavolos and Shulman). Then Sϕ(T) ≥ 0 by assumption.

Ying-Fen Lin Positive extensions

Motivation Generalisations to ℓ2(X) Generalisations to measure spaces

Proof: (⇐) Suppose Sϕ(S+

1 (κ)) ⊆ B(L2(X))+ and α × α ⊆ κ for a

measurable set α ⊆ X. If T is positive rank one supported by α × α, then Sϕ(T) ≥ 0. Since Sϕ is weak*-conti. and B(P(α)L2(X))+ = span{T : T positive rank one supported by α × α}

w∗

⇒ ϕ|α×α is positive Schur multiplier. (⇒) Suppose ϕ is partially positive and T ∈ S(κ) is a positive rank one operator, say, T = η ⊗ η∗ for some η ∈ L2(X). If supp η = α, then α × α ⊆ω κ (Erdos, Katavolos and Shulman). Then Sϕ(T) ≥ 0 by assumption.

Ying-Fen Lin Positive extensions

Motivation Generalisations to ℓ2(X) Generalisations to measure spaces

Proof: (⇐) Suppose Sϕ(S+

1 (κ)) ⊆ B(L2(X))+ and α × α ⊆ κ for a

measurable set α ⊆ X. If T is positive rank one supported by α × α, then Sϕ(T) ≥ 0. Since Sϕ is weak*-conti. and B(P(α)L2(X))+ = span{T : T positive rank one supported by α × α}

w∗

⇒ ϕ|α×α is positive Schur multiplier. (⇒) Suppose ϕ is partially positive and T ∈ S(κ) is a positive rank one operator, say, T = η ⊗ η∗ for some η ∈ L2(X). If supp η = α, then α × α ⊆ω κ (Erdos, Katavolos and Shulman). Then Sϕ(T) ≥ 0 by assumption.

Ying-Fen Lin Positive extensions

Motivation Generalisations to ℓ2(X) Generalisations to measure spaces

Definition Let κ ⊆ X × X be a positivity domain and ϕ : κ → C be a Schur

• multiplier. We say that a measurable function ψ : X × X → C is a

positive extension of ϕ if ψ is a positive Schur multiplier and ψ|κ ∼ ϕ. Theorem Let κ be a positivity domain. The following are equivalent for a partially positive Schur multiplier ϕ : κ → C:

1 ϕ has a positive extension; 2 the map Sϕ : S(κ) → S(κ) is positive; 3 the map Sϕ : S(κ) → S(κ) is completely positive. Ying-Fen Lin Positive extensions

Motivation Generalisations to ℓ2(X) Generalisations to measure spaces

Definition Let κ ⊆ X × X be a positivity domain and ϕ : κ → C be a Schur

• multiplier. We say that a measurable function ψ : X × X → C is a

positive extension of ϕ if ψ is a positive Schur multiplier and ψ|κ ∼ ϕ. Theorem Let κ be a positivity domain. The following are equivalent for a partially positive Schur multiplier ϕ : κ → C:

1 ϕ has a positive extension; 2 the map Sϕ : S(κ) → S(κ) is positive; 3 the map Sϕ : S(κ) → S(κ) is completely positive. Ying-Fen Lin Positive extensions

Motivation Generalisations to ℓ2(X) Generalisations to measure spaces

Definition Let κ ⊆ X × X be a positivity domain and ϕ : κ → C be a Schur

• multiplier. We say that a measurable function ψ : X × X → C is a

positive extension of ϕ if ψ is a positive Schur multiplier and ψ|κ ∼ ϕ. Theorem Let κ be a positivity domain. The following are equivalent for a partially positive Schur multiplier ϕ : κ → C:

1 ϕ has a positive extension; 2 the map Sϕ : S(κ) → S(κ) is positive; 3 the map Sϕ : S(κ) → S(κ) is completely positive. Ying-Fen Lin Positive extensions

Motivation Generalisations to ℓ2(X) Generalisations to measure spaces

Definition Let κ ⊆ X × X be a positivity domain and ϕ : κ → C be a Schur

• multiplier. We say that a measurable function ψ : X × X → C is a

positive extension of ϕ if ψ is a positive Schur multiplier and ψ|κ ∼ ϕ. Theorem Let κ be a positivity domain. The following are equivalent for a partially positive Schur multiplier ϕ : κ → C:

1 ϕ has a positive extension; 2 the map Sϕ : S(κ) → S(κ) is positive; 3 the map Sϕ : S(κ) → S(κ) is completely positive. Ying-Fen Lin Positive extensions

Motivation Generalisations to ℓ2(X) Generalisations to measure spaces

Definition Let κ ⊆ X × X be a positivity domain and ϕ : κ → C be a Schur

• multiplier. We say that a measurable function ψ : X × X → C is a

positive extension of ϕ if ψ is a positive Schur multiplier and ψ|κ ∼ ϕ. Theorem Let κ be a positivity domain. The following are equivalent for a partially positive Schur multiplier ϕ : κ → C:

1 ϕ has a positive extension; 2 the map Sϕ : S(κ) → S(κ) is positive; 3 the map Sϕ : S(κ) → S(κ) is completely positive. Ying-Fen Lin Positive extensions

Motivation Generalisations to ℓ2(X) Generalisations to measure spaces

Positive extensions of Schur multipliers

Theorem Let κ be an positivity domain. The following are equivalent:

1 every partially positive Schur multiplier ϕ : κ → C has a

positive extension;

2 S(κ)+ = [S+

1 (κ)] w∗

;

3 S0(κ)+ = [S+

1 (κ)] ·,

where S0(κ) = {Tk : k ∈ L2(κ)}

·.

Ying-Fen Lin Positive extensions

Motivation Generalisations to ℓ2(X) Generalisations to measure spaces

Positive extensions of Schur multipliers

Theorem Let κ be an positivity domain. The following are equivalent:

1 every partially positive Schur multiplier ϕ : κ → C has a

positive extension;

2 S(κ)+ = [S+

1 (κ)] w∗

;

3 S0(κ)+ = [S+

1 (κ)] ·,

where S0(κ) = {Tk : k ∈ L2(κ)}

·.

Ying-Fen Lin Positive extensions

Motivation Generalisations to ℓ2(X) Generalisations to measure spaces

Positive extensions of Schur multipliers

Theorem Let κ be an positivity domain. The following are equivalent:

1 every partially positive Schur multiplier ϕ : κ → C has a

positive extension;

2 S(κ)+ = [S+

1 (κ)] w∗

;

3 S0(κ)+ = [S+

1 (κ)] ·,

where S0(κ) = {Tk : k ∈ L2(κ)}

·.

Ying-Fen Lin Positive extensions

Motivation Generalisations to ℓ2(X) Generalisations to measure spaces

THANK YOU!!

Ying-Fen Lin Positive extensions