Motivation
Generalisations to `2 (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 `2 (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
of 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 `2 (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
of 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 `2 (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
of 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 `2 (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 `2 (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 `2 (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) 6∈ J}.
Note that SJ is an operator system if and only if J is symmetric.
Ying-Fen Lin
Positive extensions
Motivation
Generalisations to `2 (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) 6∈ J}.
Note that SJ is an operator system if and only if J is symmetric.
Ying-Fen Lin
Positive extensions
Motivation
Generalisations to `2 (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) 6∈ J}.
Note that SJ is an operator system if and only if J is symmetric.
Ying-Fen Lin
Positive extensions
Motivation
Generalisations to `2 (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) 6∈ J}.
Note that SJ is an operator system if and only if J is symmetric.
Ying-Fen Lin
Positive extensions
Motivation
Generalisations to `2 (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) 6∈ J}.
Note that SJ is an operator system if and only if J is symmetric.
Ying-Fen Lin
Positive extensions
Motivation
Generalisations to `2 (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 )) = (aij tij ) is positive,
P
ΨT : SJ → C defined by ΨT ((aij )) = ij aij tij is positive.
3
Ying-Fen Lin
Positive extensions
Motivation
Generalisations to `2 (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 )) = (aij tij ) is positive,
P
ΨT : SJ → C defined by ΨT ((aij )) = ij aij tij is positive.
3
Ying-Fen Lin
Positive extensions
Motivation
Generalisations to `2 (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 )) = (aij tij ) is positive,
P
ΨT : SJ → C defined by ΨT ((aij )) = ij aij tij is positive.
3
Ying-Fen Lin
Positive extensions
Motivation
Generalisations to `2 (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 `2 (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 `2 (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 `2 (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 `2 (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 `2 (X )
Generalisations to measure spaces
On the setting of `2 (X )
Let X be a set and H = `2 (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
2
κ is symmetric,
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 `2 (X )
Generalisations to measure spaces
On the setting of `2 (X )
Let X be a set and H = `2 (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
2
κ is symmetric,
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 `2 (X )
Generalisations to measure spaces
On the setting of `2 (X )
Let X be a set and H = `2 (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
2
κ is symmetric,
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 `2 (X )
Generalisations to measure spaces
On the setting of `2 (X )
Let X be a set and H = `2 (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
2
κ is symmetric,
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 `2 (X )
Generalisations to measure spaces
On the setting of `2 (X )
Let X be a set and H = `2 (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
2
κ is symmetric,
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 `2 (X )
Generalisations to measure spaces
On the setting of `2 (X )
Let X be a set and H = `2 (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
2
κ is symmetric,
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 `2 (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,y ψ(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 `2 (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,y ψ(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 `2 (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,y ψ(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 `2 (X )
Generalisations to measure spaces
Positive extensions on `2 (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 `2 (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
Z
Tk f (y ) =
k(y , x)f (x)dµ(x) for f ∈ L2 (X , µ), y ∈ X .
X
For any measurable subset κ ⊆ X × X , let
S(κ) := {Tk : k ∈ L2 (κ)}
Ying-Fen Lin
w∗
Positive extensions
.
Motivation
Generalisations to `2 (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
Z
Tk f (y ) =
k(y , x)f (x)dµ(x) for f ∈ L2 (X , µ), y ∈ X .
X
For any measurable subset κ ⊆ X × X , let
S(κ) := {Tk : k ∈ L2 (κ)}
Ying-Fen Lin
w∗
Positive extensions
.
Motivation
Generalisations to `2 (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
Z
Tk f (y ) =
k(y , x)f (x)dµ(x) for f ∈ L2 (X , µ), y ∈ X .
X
For any measurable subset κ ⊆ X × X , let
S(κ) := {Tk : k ∈ L2 (κ)}
Ying-Fen Lin
w∗
Positive extensions
.
Motivation
Generalisations to `2 (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 M 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.
S
A set κ ⊆ X × X is called ω-open if κ ∼
αi × βi , where
=
i
αi , βi ⊆ X are measurable.
Ying-Fen Lin
Positive extensions
Motivation
Generalisations to `2 (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 M 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.
S
A set κ ⊆ X × X is called ω-open if κ ∼
αi × βi , where
=
i
αi , βi ⊆ X are measurable.
Ying-Fen Lin
Positive extensions
Motivation
Generalisations to `2 (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 M 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.
S
A set κ ⊆ X × X is called ω-open if κ ∼
αi × βi , where
=
i
αi , βi ⊆ X are measurable.
Ying-Fen Lin
Positive extensions
Motivation
Generalisations to `2 (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 M 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.
S
A set κ ⊆ X × X is called ω-open if κ ∼
αi × βi , where
=
i
αi , βi ⊆ X are measurable.
Ying-Fen Lin
Positive extensions
Motivation
Generalisations to `2 (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
2
S(κ) is an operator system;
κ is symmetric, i.e., κ ∼
= κ̂ and 4 := {(x, x) : x ∈ X } ⊆ω κ,
where κ̂ := {(x, y ) ∈ X × X : (y , x) ∈ κ}.
Ying-Fen Lin
Positive extensions
Motivation
Generalisations to `2 (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
2
S(κ) is an operator system;
κ is symmetric, i.e., κ ∼
= κ̂ and 4 := {(x, x) : x ∈ X } ⊆ω κ,
where κ̂ := {(x, y ) ∈ X × X : (y , x) ∈ κ}.
Ying-Fen Lin
Positive extensions
Motivation
Generalisations to `2 (X )
Generalisations to measure spaces
Positivity domains
Definition
An ω-closed set κ ⊆ X × X is called a positivity domain if κ ∼
= κ̂,
∼
4 ⊆ω κ and κ = clω (intω (κ)).
S
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
operators 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 `2 (X )
Generalisations to measure spaces
Positivity domains
Definition
An ω-closed set κ ⊆ X × X is called a positivity domain if κ ∼
= κ̂,
∼
4 ⊆ω κ and κ = clω (intω (κ)).
S
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
operators 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 `2 (X )
Generalisations to measure spaces
Positivity domains
Definition
An ω-closed set κ ⊆ X × X is called a positivity domain if κ ∼
= κ̂,
∼
4 ⊆ω κ and κ = clω (intω (κ)).
S
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
operators 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 `2 (X )
Generalisations to measure spaces
For a positivity domain κ ⊆ X × X , let
[S1+ (κ)] =
n
nX
o
Ri : Ri ∈ S(κ)+ is of rank one, 1 ≤ i ≤ n .
i=1
Theorem
The following are equivalent, for a positivity domain κ:
there exists a family (αi )i∈N of mutually disjoint
S measurable
subsets of X s.t. αi × αi ⊆ω κ for each i and ∞
i=1 αi = X ;
w∗
I ∈ [S1+ (κ)]
;
κ is generated by squares, i.e. κ ∼
= clω (sqintω (κ)), where
[
sqintω (κ) = {Q : Q is a square with Q ⊆ κ}.
ω
Ying-Fen Lin
Positive extensions
Motivation
Generalisations to `2 (X )
Generalisations to measure spaces
For a positivity domain κ ⊆ X × X , let
[S1+ (κ)] =
n
nX
o
Ri : Ri ∈ S(κ)+ is of rank one, 1 ≤ i ≤ n .
i=1
Theorem
The following are equivalent, for a positivity domain κ:
there exists a family (αi )i∈N of mutually disjoint
S measurable
subsets of X s.t. αi × αi ⊆ω κ for each i and ∞
i=1 αi = X ;
w∗
I ∈ [S1+ (κ)]
;
κ is generated by squares, i.e. κ ∼
= clω (sqintω (κ)), where
[
sqintω (κ) = {Q : Q is a square with Q ⊆ κ}.
ω
Ying-Fen Lin
Positive extensions
Motivation
Generalisations to `2 (X )
Generalisations to measure spaces
For a positivity domain κ ⊆ X × X , let
[S1+ (κ)] =
n
nX
o
Ri : Ri ∈ S(κ)+ is of rank one, 1 ≤ i ≤ n .
i=1
Theorem
The following are equivalent, for a positivity domain κ:
there exists a family (αi )i∈N of mutually disjoint
S measurable
subsets of X s.t. αi × αi ⊆ω κ for each i and ∞
i=1 αi = X ;
w∗
I ∈ [S1+ (κ)]
;
κ is generated by squares, i.e. κ ∼
= clω (sqintω (κ)), where
[
sqintω (κ) = {Q : Q is a square with Q ⊆ κ}.
ω
Ying-Fen Lin
Positive extensions
Motivation
Generalisations to `2 (X )
Generalisations to measure spaces
For a positivity domain κ ⊆ X × X , let
[S1+ (κ)] =
n
nX
o
Ri : Ri ∈ S(κ)+ is of rank one, 1 ≤ i ≤ n .
i=1
Theorem
The following are equivalent, for a positivity domain κ:
there exists a family (αi )i∈N of mutually disjoint
S measurable
subsets of X s.t. αi × αi ⊆ω κ for each i and ∞
i=1 αi = X ;
w∗
I ∈ [S1+ (κ)]
;
κ is generated by squares, i.e. κ ∼
= clω (sqintω (κ)), where
[
sqintω (κ) = {Q : Q is a square with Q ⊆ κ}.
ω
Ying-Fen Lin
Positive extensions
Motivation
Generalisations to `2 (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
kTϕk k ≤ C kTk k 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 `2 (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
kTϕk k ≤ C kTk k 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 `2 (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 (S1+ (κ)) ⊆ B(L2 (X ))+ .
Ying-Fen Lin
Positive extensions
Motivation
Generalisations to `2 (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 (S1+ (κ)) ⊆ B(L2 (X ))+ .
Ying-Fen Lin
Positive extensions
Motivation
Generalisations to `2 (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 (S1+ (κ)) ⊆ B(L2 (X ))+ .
Ying-Fen Lin
Positive extensions
Motivation
Generalisations to `2 (X )
Generalisations to measure spaces
Proof:
(⇐) Suppose Sϕ (S1+ (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}
⇒ 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
w∗
Motivation
Generalisations to `2 (X )
Generalisations to measure spaces
Proof:
(⇐) Suppose Sϕ (S1+ (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}
⇒ 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
w∗
Motivation
Generalisations to `2 (X )
Generalisations to measure spaces
Proof:
(⇐) Suppose Sϕ (S1+ (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}
⇒ 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
w∗
Motivation
Generalisations to `2 (X )
Generalisations to measure spaces
Proof:
(⇐) Suppose Sϕ (S1+ (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}
⇒ 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
w∗
Motivation
Generalisations to `2 (X )
Generalisations to measure spaces
Proof:
(⇐) Suppose Sϕ (S1+ (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}
⇒ 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
w∗
Motivation
Generalisations to `2 (X )
Generalisations to measure spaces
Proof:
(⇐) Suppose Sϕ (S1+ (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}
⇒ 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
w∗
Motivation
Generalisations to `2 (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 `2 (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 `2 (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 `2 (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 `2 (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 `2 (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;
w∗
2
S(κ)+ = [S1+ (κ)]
3
S0 (κ)+ = [S1+ (κ)]
;
k·k
,
k·k
where S0 (κ) = {Tk : k ∈ L2 (κ)}
Ying-Fen Lin
.
Positive extensions
Motivation
Generalisations to `2 (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;
w∗
2
S(κ)+ = [S1+ (κ)]
3
S0 (κ)+ = [S1+ (κ)]
;
k·k
,
k·k
where S0 (κ) = {Tk : k ∈ L2 (κ)}
Ying-Fen Lin
.
Positive extensions
Motivation
Generalisations to `2 (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;
w∗
2
S(κ)+ = [S1+ (κ)]
3
S0 (κ)+ = [S1+ (κ)]
;
k·k
,
k·k
where S0 (κ) = {Tk : k ∈ L2 (κ)}
Ying-Fen Lin
.
Positive extensions
Motivation
Generalisations to `2 (X )
Generalisations to measure spaces
THANK YOU!!
Ying-Fen Lin
Positive extensions