Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra
Best approximations and orthogonality in Hilbert C ∗-modules
Sushil Singla
Department of Mathematics School of Natural Sciences Shiv Nadar University
May 11, 2021
1 / 23 Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Table of Contents
1 Birkhoff-James orthogonality in Hilbert C ∗-modules
2 Proofs of characterization of orthogonality in C ∗-algebra
2 / 23 Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Table of Contents
1 Birkhoff-James orthogonality in Hilbert C ∗-modules
2 Proofs of characterization of orthogonality in C ∗-algebra
3 / 23 If we can define some kind of orthogonality in a normed space, then we might be able to guess some results from geometric intuition and then try to prove it. Let us see a characterization of a vector being orthogonal to a subspace in an inner product space.
Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Idea of Birkhoff-James Orthogonality
Question: Is there a way to define orthogonality of a vector to a subspace/subset in a normed space?
4 / 23 Let us see a characterization of a vector being orthogonal to a subspace in an inner product space.
Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Idea of Birkhoff-James Orthogonality
Question: Is there a way to define orthogonality of a vector to a subspace/subset in a normed space? If we can define some kind of orthogonality in a normed space, then we might be able to guess some results from geometric intuition and then try to prove it.
4 / 23 Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Idea of Birkhoff-James Orthogonality
Question: Is there a way to define orthogonality of a vector to a subspace/subset in a normed space? If we can define some kind of orthogonality in a normed space, then we might be able to guess some results from geometric intuition and then try to prove it. Let us see a characterization of a vector being orthogonal to a subspace in an inner product space.
4 / 23 Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Idea of Birkhoff-James Orthogonality
Question: Is there a way to define orthogonality of a vector to a subspace/subset in a normed space? If we can define some kind of orthogonality in a normed space, then we might be able to guess some results from geometric intuition and then try to prove it. Let us see a characterization of a vector being orthogonal to a subspace in an inner product space.
4 / 23 James, in 1945, introduced a notion of orthogonality in any normed space using this characterization.
Definition (Birkhoff-James Orthogonality) Let (V , k·k) be a normed space. Then a vector v is said to be Birkhoff-James orthogonal to a subspace W of V if kvk ≤ kv − wk for all w ∈ W .
Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Birkhoff-James Orthogonality
Theorem: In an inner product space V , v is orthogonal to a subspace W of V if and only if kvk ≤ kv − wk for all w ∈ W .
5 / 23 Definition (Birkhoff-James Orthogonality) Let (V , k·k) be a normed space. Then a vector v is said to be Birkhoff-James orthogonal to a subspace W of V if kvk ≤ kv − wk for all w ∈ W .
Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Birkhoff-James Orthogonality
Theorem: In an inner product space V , v is orthogonal to a subspace W of V if and only if kvk ≤ kv − wk for all w ∈ W . James, in 1945, introduced a notion of orthogonality in any normed space using this characterization.
5 / 23 Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Birkhoff-James Orthogonality
Theorem: In an inner product space V , v is orthogonal to a subspace W of V if and only if kvk ≤ kv − wk for all w ∈ W . James, in 1945, introduced a notion of orthogonality in any normed space using this characterization.
Definition (Birkhoff-James Orthogonality) Let (V , k·k) be a normed space. Then a vector v is said to be Birkhoff-James orthogonal to a subspace W of V if kvk ≤ kv − wk for all w ∈ W .
5 / 23 Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Birkhoff-James Orthogonality
Theorem: In an inner product space V , v is orthogonal to a subspace W of V if and only if kvk ≤ kv − wk for all w ∈ W . James, in 1945, introduced a notion of orthogonality in any normed space using this characterization.
Definition (Birkhoff-James Orthogonality) Let (V , k·k) be a normed space. Then a vector v is said to be Birkhoff-James orthogonal to a subspace W of V if kvk ≤ kv − wk for all w ∈ W .
5 / 23 Now, kvk ≤ kv − wk for all w ∈ W is same as dist(v, W ) = kvk. Thus, a vector v is Birkhoff-James orthogonal to W if and only if dist(v, W ) is attained at 0. This is abbreviated as v has a best approximation to W at 0.
Definition
An element w0 ∈ W is said to be best approximation to v in W if and only if dist(v, W ) = kv − w0k . Equivalently, v − w0 is Birkhoff-James orthogonal to W .
So results about Birkhoff-James orthogonality give results for best approximations of a point to a subspace and vice versa.
Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Birkhoff-James Orthogonality and best approximation
Let W be a subspace of normed space V .
6 / 23 Thus, a vector v is Birkhoff-James orthogonal to W if and only if dist(v, W ) is attained at 0. This is abbreviated as v has a best approximation to W at 0.
Definition
An element w0 ∈ W is said to be best approximation to v in W if and only if dist(v, W ) = kv − w0k . Equivalently, v − w0 is Birkhoff-James orthogonal to W .
So results about Birkhoff-James orthogonality give results for best approximations of a point to a subspace and vice versa.
Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Birkhoff-James Orthogonality and best approximation
Let W be a subspace of normed space V . Now, kvk ≤ kv − wk for all w ∈ W is same as dist(v, W ) = kvk.
6 / 23 This is abbreviated as v has a best approximation to W at 0.
Definition
An element w0 ∈ W is said to be best approximation to v in W if and only if dist(v, W ) = kv − w0k . Equivalently, v − w0 is Birkhoff-James orthogonal to W .
So results about Birkhoff-James orthogonality give results for best approximations of a point to a subspace and vice versa.
Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Birkhoff-James Orthogonality and best approximation
Let W be a subspace of normed space V . Now, kvk ≤ kv − wk for all w ∈ W is same as dist(v, W ) = kvk. Thus, a vector v is Birkhoff-James orthogonal to W if and only if dist(v, W ) is attained at 0.
6 / 23 Definition
An element w0 ∈ W is said to be best approximation to v in W if and only if dist(v, W ) = kv − w0k . Equivalently, v − w0 is Birkhoff-James orthogonal to W .
So results about Birkhoff-James orthogonality give results for best approximations of a point to a subspace and vice versa.
Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Birkhoff-James Orthogonality and best approximation
Let W be a subspace of normed space V . Now, kvk ≤ kv − wk for all w ∈ W is same as dist(v, W ) = kvk. Thus, a vector v is Birkhoff-James orthogonal to W if and only if dist(v, W ) is attained at 0. This is abbreviated as v has a best approximation to W at 0.
6 / 23 So results about Birkhoff-James orthogonality give results for best approximations of a point to a subspace and vice versa.
Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Birkhoff-James Orthogonality and best approximation
Let W be a subspace of normed space V . Now, kvk ≤ kv − wk for all w ∈ W is same as dist(v, W ) = kvk. Thus, a vector v is Birkhoff-James orthogonal to W if and only if dist(v, W ) is attained at 0. This is abbreviated as v has a best approximation to W at 0.
Definition
An element w0 ∈ W is said to be best approximation to v in W if and only if dist(v, W ) = kv − w0k . Equivalently, v − w0 is Birkhoff-James orthogonal to W .
6 / 23 Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Birkhoff-James Orthogonality and best approximation
Let W be a subspace of normed space V . Now, kvk ≤ kv − wk for all w ∈ W is same as dist(v, W ) = kvk. Thus, a vector v is Birkhoff-James orthogonal to W if and only if dist(v, W ) is attained at 0. This is abbreviated as v has a best approximation to W at 0.
Definition
An element w0 ∈ W is said to be best approximation to v in W if and only if dist(v, W ) = kv − w0k . Equivalently, v − w0 is Birkhoff-James orthogonal to W .
So results about Birkhoff-James orthogonality give results for best approximations of a point to a subspace and vice versa.
6 / 23 Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Birkhoff-James Orthogonality and best approximation
Let W be a subspace of normed space V . Now, kvk ≤ kv − wk for all w ∈ W is same as dist(v, W ) = kvk. Thus, a vector v is Birkhoff-James orthogonal to W if and only if dist(v, W ) is attained at 0. This is abbreviated as v has a best approximation to W at 0.
Definition
An element w0 ∈ W is said to be best approximation to v in W if and only if dist(v, W ) = kv − w0k . Equivalently, v − w0 is Birkhoff-James orthogonal to W .
So results about Birkhoff-James orthogonality give results for best approximations of a point to a subspace and vice versa.
6 / 23 Theorem (Singer I., 1970) Let f ∈ C(X ) and W is a subspace of C(X ). Let g ∈ W , then the following are equivalent:
1 g is a best approximation to f in W .
2 There exists a regular Borel probability measure µ on X such that a) the support of µ is contained in the set {x ∈ X : |(f − g)(x)| = kf − gk∞} and b) R (f − g)h dµ = 0 for all h ∈ W . X
R 2 (1) is equivalent to (f − g)(f − g) dµ = kf − gk∞. X
Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Characterization of best approximation in C(X )
Notation: F will stand for C or R. Let (C(X ), k · k∞) be the space of F-valued continuous functions on a compact Hausdorff space X .
7 / 23 Let g ∈ W , then the following are equivalent:
1 g is a best approximation to f in W .
2 There exists a regular Borel probability measure µ on X such that a) the support of µ is contained in the set {x ∈ X : |(f − g)(x)| = kf − gk∞} and b) R (f − g)h dµ = 0 for all h ∈ W . X
R 2 (1) is equivalent to (f − g)(f − g) dµ = kf − gk∞. X
Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Characterization of best approximation in C(X )
Notation: F will stand for C or R. Let (C(X ), k · k∞) be the space of F-valued continuous functions on a compact Hausdorff space X . Theorem (Singer I., 1970) Let f ∈ C(X ) and W is a subspace of C(X ).
7 / 23 1 g is a best approximation to f in W .
2 There exists a regular Borel probability measure µ on X such that a) the support of µ is contained in the set {x ∈ X : |(f − g)(x)| = kf − gk∞} and b) R (f − g)h dµ = 0 for all h ∈ W . X
R 2 (1) is equivalent to (f − g)(f − g) dµ = kf − gk∞. X
Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Characterization of best approximation in C(X )
Notation: F will stand for C or R. Let (C(X ), k · k∞) be the space of F-valued continuous functions on a compact Hausdorff space X . Theorem (Singer I., 1970) Let f ∈ C(X ) and W is a subspace of C(X ). Let g ∈ W , then the following are equivalent:
7 / 23 2 There exists a regular Borel probability measure µ on X such that a) the support of µ is contained in the set {x ∈ X : |(f − g)(x)| = kf − gk∞} and b) R (f − g)h dµ = 0 for all h ∈ W . X
R 2 (1) is equivalent to (f − g)(f − g) dµ = kf − gk∞. X
Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Characterization of best approximation in C(X )
Notation: F will stand for C or R. Let (C(X ), k · k∞) be the space of F-valued continuous functions on a compact Hausdorff space X . Theorem (Singer I., 1970) Let f ∈ C(X ) and W is a subspace of C(X ). Let g ∈ W , then the following are equivalent:
1 g is a best approximation to f in W .
7 / 23 b) R (f − g)h dµ = 0 for all h ∈ W . X
R 2 (1) is equivalent to (f − g)(f − g) dµ = kf − gk∞. X
Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Characterization of best approximation in C(X )
Notation: F will stand for C or R. Let (C(X ), k · k∞) be the space of F-valued continuous functions on a compact Hausdorff space X . Theorem (Singer I., 1970) Let f ∈ C(X ) and W is a subspace of C(X ). Let g ∈ W , then the following are equivalent:
1 g is a best approximation to f in W .
2 There exists a regular Borel probability measure µ on X such that a) the support of µ is contained in the set {x ∈ X : |(f − g)(x)| = kf − gk∞} and
7 / 23 R 2 (1) is equivalent to (f − g)(f − g) dµ = kf − gk∞. X
Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Characterization of best approximation in C(X )
Notation: F will stand for C or R. Let (C(X ), k · k∞) be the space of F-valued continuous functions on a compact Hausdorff space X . Theorem (Singer I., 1970) Let f ∈ C(X ) and W is a subspace of C(X ). Let g ∈ W , then the following are equivalent:
1 g is a best approximation to f in W .
2 There exists a regular Borel probability measure µ on X such that a) the support of µ is contained in the set {x ∈ X : |(f − g)(x)| = kf − gk∞} and b) R (f − g)h dµ = 0 for all h ∈ W . X
7 / 23 Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Characterization of best approximation in C(X )
Notation: F will stand for C or R. Let (C(X ), k · k∞) be the space of F-valued continuous functions on a compact Hausdorff space X . Theorem (Singer I., 1970) Let f ∈ C(X ) and W is a subspace of C(X ). Let g ∈ W , then the following are equivalent:
1 g is a best approximation to f in W .
2 There exists a regular Borel probability measure µ on X such that a) the support of µ is contained in the set {x ∈ X : |(f − g)(x)| = kf − gk∞} and b) R (f − g)h dµ = 0 for all h ∈ W . X
R 2 (1) is equivalent to (f − g)(f − g) dµ = kf − gk∞. X 7 / 23 Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Characterization of best approximation in C(X )
Notation: F will stand for C or R. Let (C(X ), k · k∞) be the space of F-valued continuous functions on a compact Hausdorff space X . Theorem (Singer I., 1970) Let f ∈ C(X ) and W is a subspace of C(X ). Let g ∈ W , then the following are equivalent:
1 g is a best approximation to f in W .
2 There exists a regular Borel probability measure µ on X such that a) the support of µ is contained in the set {x ∈ X : |(f − g)(x)| = kf − gk∞} and b) R (f − g)h dµ = 0 for all h ∈ W . X
R 2 (1) is equivalent to (f − g)(f − g) dµ = kf − gk∞. X 7 / 23 Theorem (Grover P., 2014)
Let A ∈ Mn(F) and W be a subspace of Mn(F). Then A is Birkhoff-James orthogonal to W if and only if there exists a ∗ 2 density matrix T ∈ Mn(F) such that A AT = kAk T and trace (B∗AT ) = 0 for all B ∈ W.
∗ C(X ) and Mn(F) are particular examples of C -algebra and probability measure and density matrices correspond to states on a C ∗-algebra.
Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra
Characterization of Birkhoff-James orthogonality in Mn(F)
Let Mn(F) be the space of n × n matrices with entries in F.A density matrix A ∈ Mn(F) is a non-negative matrix with trace (A) = 1.
8 / 23 Then A is Birkhoff-James orthogonal to W if and only if there exists a ∗ 2 density matrix T ∈ Mn(F) such that A AT = kAk T and trace (B∗AT ) = 0 for all B ∈ W.
∗ C(X ) and Mn(F) are particular examples of C -algebra and probability measure and density matrices correspond to states on a C ∗-algebra.
Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra
Characterization of Birkhoff-James orthogonality in Mn(F)
Let Mn(F) be the space of n × n matrices with entries in F.A density matrix A ∈ Mn(F) is a non-negative matrix with trace (A) = 1.
Theorem (Grover P., 2014)
Let A ∈ Mn(F) and W be a subspace of Mn(F).
8 / 23 if and only if there exists a ∗ 2 density matrix T ∈ Mn(F) such that A AT = kAk T and trace (B∗AT ) = 0 for all B ∈ W.
∗ C(X ) and Mn(F) are particular examples of C -algebra and probability measure and density matrices correspond to states on a C ∗-algebra.
Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra
Characterization of Birkhoff-James orthogonality in Mn(F)
Let Mn(F) be the space of n × n matrices with entries in F.A density matrix A ∈ Mn(F) is a non-negative matrix with trace (A) = 1.
Theorem (Grover P., 2014)
Let A ∈ Mn(F) and W be a subspace of Mn(F). Then A is Birkhoff-James orthogonal to W
8 / 23 ∗ C(X ) and Mn(F) are particular examples of C -algebra and probability measure and density matrices correspond to states on a C ∗-algebra.
Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra
Characterization of Birkhoff-James orthogonality in Mn(F)
Let Mn(F) be the space of n × n matrices with entries in F.A density matrix A ∈ Mn(F) is a non-negative matrix with trace (A) = 1.
Theorem (Grover P., 2014)
Let A ∈ Mn(F) and W be a subspace of Mn(F). Then A is Birkhoff-James orthogonal to W if and only if there exists a ∗ 2 density matrix T ∈ Mn(F) such that A AT = kAk T and trace (B∗AT ) = 0 for all B ∈ W.
8 / 23 Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra
Characterization of Birkhoff-James orthogonality in Mn(F)
Let Mn(F) be the space of n × n matrices with entries in F.A density matrix A ∈ Mn(F) is a non-negative matrix with trace (A) = 1.
Theorem (Grover P., 2014)
Let A ∈ Mn(F) and W be a subspace of Mn(F). Then A is Birkhoff-James orthogonal to W if and only if there exists a ∗ 2 density matrix T ∈ Mn(F) such that A AT = kAk T and trace (B∗AT ) = 0 for all B ∈ W.
∗ C(X ) and Mn(F) are particular examples of C -algebra and probability measure and density matrices correspond to states on a C ∗-algebra.
8 / 23 Theorem (Grover P.; Singla S., 2021) Let a ∈ A. Let B be a subspace of A. Then a is Birkhoff-James orthogonal to B if and only if there exists φ ∈ SA such that φ(a∗a) = kak2 and φ(a∗b) = 0 for all b ∈ B.
This is generalization of characterization of best approximation in C(X ) and characterization of Birkhoff-James orthogonality in Mn(F).
Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Orthogonality characterization in C ∗-algebra
∗ A will stand for unital C algebra with unity 1A. SA will stand for the set of all states on A.
9 / 23 This is generalization of characterization of best approximation in C(X ) and characterization of Birkhoff-James orthogonality in Mn(F).
Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Orthogonality characterization in C ∗-algebra
∗ A will stand for unital C algebra with unity 1A. SA will stand for the set of all states on A.
Theorem (Grover P.; Singla S., 2021) Let a ∈ A. Let B be a subspace of A. Then a is Birkhoff-James orthogonal to B if and only if there exists φ ∈ SA such that φ(a∗a) = kak2 and φ(a∗b) = 0 for all b ∈ B.
9 / 23 Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Orthogonality characterization in C ∗-algebra
∗ A will stand for unital C algebra with unity 1A. SA will stand for the set of all states on A.
Theorem (Grover P.; Singla S., 2021) Let a ∈ A. Let B be a subspace of A. Then a is Birkhoff-James orthogonal to B if and only if there exists φ ∈ SA such that φ(a∗a) = kak2 and φ(a∗b) = 0 for all b ∈ B.
This is generalization of characterization of best approximation in C(X ) and characterization of Birkhoff-James orthogonality in Mn(F).
9 / 23 Theorem (Arambaˇsi´cL.; Raji´cR., 2012) Let a ∈ A and B be one dimensional subspace. Then a is Birkhoff-James orthogonal to B if and only if there exists φ ∈ SA such that φ(a∗a) = kak2 and φ(a∗b) = 0 for all b ∈ B.
Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Corollaries
Theorem (Bhatia R.; Semrlˇ P., 1999) A matrix A is orthogonal to B if and only if there exist unit vector x such that kAxk = kAk and hAx|Bxi = 0.
10 / 23 Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Corollaries
Theorem (Bhatia R.; Semrlˇ P., 1999) A matrix A is orthogonal to B if and only if there exist unit vector x such that kAxk = kAk and hAx|Bxi = 0.
Theorem (Arambaˇsi´cL.; Raji´cR., 2012) Let a ∈ A and B be one dimensional subspace. Then a is Birkhoff-James orthogonal to B if and only if there exists φ ∈ SA such that φ(a∗a) = kak2 and φ(a∗b) = 0 for all b ∈ B.
10 / 23 Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Corollaries
Theorem (Bhatia R.; Semrlˇ P., 1999) A matrix A is orthogonal to B if and only if there exist unit vector x such that kAxk = kAk and hAx|Bxi = 0.
Theorem (Arambaˇsi´cL.; Raji´cR., 2012) Let a ∈ A and B be one dimensional subspace. Then a is Birkhoff-James orthogonal to B if and only if there exists φ ∈ SA such that φ(a∗a) = kak2 and φ(a∗b) = 0 for all b ∈ B.
10 / 23 Theorem (Williams J. P., 1970) For a ∈ A, we have
2 ∗ 2 dist(a, C1A) = max{φ(a a) − |φ(a)| : φ ∈ SA}. (1)
Proof of 1: There exists λ0 ∈ C such that dist(a, C1A) = ka − λ01Ak. Then there exists φ ∈ SA such that ∗ 2 φ((a − λ01A) (a − λ01A)) = dist(a, C1A) and φ(a − λ01A) = 0.
Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Corollaries
Theorem (Rieffel M. A., 2011) Let a ∈ A be a Hermitian element and B be a C ∗-subalgebra of a A. If a is Birkhoff-James orthogonal to B, then there exists 2 2 ∗ φ ∈ SA such that φ(a ) = kak and φ(ab + b a) = 0 for all b ∈ B.
11 / 23 Then there exists φ ∈ SA such that ∗ 2 φ((a − λ01A) (a − λ01A)) = dist(a, C1A) and φ(a − λ01A) = 0.
Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Corollaries
Theorem (Rieffel M. A., 2011) Let a ∈ A be a Hermitian element and B be a C ∗-subalgebra of a A. If a is Birkhoff-James orthogonal to B, then there exists 2 2 ∗ φ ∈ SA such that φ(a ) = kak and φ(ab + b a) = 0 for all b ∈ B.
Theorem (Williams J. P., 1970) For a ∈ A, we have
2 ∗ 2 dist(a, C1A) = max{φ(a a) − |φ(a)| : φ ∈ SA}. (1)
Proof of 1: There exists λ0 ∈ C such that dist(a, C1A) = ka − λ01Ak.
11 / 23 Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Corollaries
Theorem (Rieffel M. A., 2011) Let a ∈ A be a Hermitian element and B be a C ∗-subalgebra of a A. If a is Birkhoff-James orthogonal to B, then there exists 2 2 ∗ φ ∈ SA such that φ(a ) = kak and φ(ab + b a) = 0 for all b ∈ B.
Theorem (Williams J. P., 1970) For a ∈ A, we have
2 ∗ 2 dist(a, C1A) = max{φ(a a) − |φ(a)| : φ ∈ SA}. (1)
Proof of 1: There exists λ0 ∈ C such that dist(a, C1A) = ka − λ01Ak. Then there exists φ ∈ SA such that ∗ 2 φ((a − λ01A) (a − λ01A)) = dist(a, C1A) and φ(a − λ01A) = 0.
11 / 23 Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Corollaries
Theorem (Rieffel M. A., 2011) Let a ∈ A be a Hermitian element and B be a C ∗-subalgebra of a A. If a is Birkhoff-James orthogonal to B, then there exists 2 2 ∗ φ ∈ SA such that φ(a ) = kak and φ(ab + b a) = 0 for all b ∈ B.
Theorem (Williams J. P., 1970) For a ∈ A, we have
2 ∗ 2 dist(a, C1A) = max{φ(a a) − |φ(a)| : φ ∈ SA}. (1)
Proof of 1: There exists λ0 ∈ C such that dist(a, C1A) = ka − λ01Ak. Then there exists φ ∈ SA such that ∗ 2 φ((a − λ01A) (a − λ01A)) = dist(a, C1A) and φ(a − λ01A) = 0.
11 / 23 with a function h·, ·i : E × E → A, known as A-valued semi-inner product, with the following properties - for ξ, η, ζ ∈ E, a ∈ A, λ ∈ C, 1 hξ, η + ζi = hξ, ηi + hξ, ζi and hξ, ληi = λ hξ, ηi, 2 hξ, ηai = hξ, ηi a, ∗ 3 hξ, ηi = hη, ξi , 4 hξ, ξi is a positive element of A.
Theorem E can be isometrically embedded into B(H, K) for some Hilbert spaces H and K. Further, we can find a faithful representation π : A → B(H) and isometric embedding L : E → B(H, K) satisfies hL(e1)h1|L(e2)h2i = hh1|π(he1, e2))h2i for all e1, e2 ∈ E and h1, h2 ∈ H.
Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Hilbert C ∗-modules
Definition A Hilbert C ∗-module E over A is a right A-module
12 / 23 with the following properties - for ξ, η, ζ ∈ E, a ∈ A, λ ∈ C, 1 hξ, η + ζi = hξ, ηi + hξ, ζi and hξ, ληi = λ hξ, ηi, 2 hξ, ηai = hξ, ηi a, ∗ 3 hξ, ηi = hη, ξi , 4 hξ, ξi is a positive element of A.
Theorem E can be isometrically embedded into B(H, K) for some Hilbert spaces H and K. Further, we can find a faithful representation π : A → B(H) and isometric embedding L : E → B(H, K) satisfies hL(e1)h1|L(e2)h2i = hh1|π(he1, e2))h2i for all e1, e2 ∈ E and h1, h2 ∈ H.
Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Hilbert C ∗-modules
Definition A Hilbert C ∗-module E over A is a right A-module with a function h·, ·i : E × E → A, known as A-valued semi-inner product,
12 / 23 2 hξ, ηai = hξ, ηi a, ∗ 3 hξ, ηi = hη, ξi , 4 hξ, ξi is a positive element of A.
Theorem E can be isometrically embedded into B(H, K) for some Hilbert spaces H and K. Further, we can find a faithful representation π : A → B(H) and isometric embedding L : E → B(H, K) satisfies hL(e1)h1|L(e2)h2i = hh1|π(he1, e2))h2i for all e1, e2 ∈ E and h1, h2 ∈ H.
Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Hilbert C ∗-modules
Definition A Hilbert C ∗-module E over A is a right A-module with a function h·, ·i : E × E → A, known as A-valued semi-inner product, with the following properties - for ξ, η, ζ ∈ E, a ∈ A, λ ∈ C, 1 hξ, η + ζi = hξ, ηi + hξ, ζi and hξ, ληi = λ hξ, ηi,
12 / 23 ∗ 3 hξ, ηi = hη, ξi , 4 hξ, ξi is a positive element of A.
Theorem E can be isometrically embedded into B(H, K) for some Hilbert spaces H and K. Further, we can find a faithful representation π : A → B(H) and isometric embedding L : E → B(H, K) satisfies hL(e1)h1|L(e2)h2i = hh1|π(he1, e2))h2i for all e1, e2 ∈ E and h1, h2 ∈ H.
Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Hilbert C ∗-modules
Definition A Hilbert C ∗-module E over A is a right A-module with a function h·, ·i : E × E → A, known as A-valued semi-inner product, with the following properties - for ξ, η, ζ ∈ E, a ∈ A, λ ∈ C, 1 hξ, η + ζi = hξ, ηi + hξ, ζi and hξ, ληi = λ hξ, ηi, 2 hξ, ηai = hξ, ηi a,
12 / 23 4 hξ, ξi is a positive element of A.
Theorem E can be isometrically embedded into B(H, K) for some Hilbert spaces H and K. Further, we can find a faithful representation π : A → B(H) and isometric embedding L : E → B(H, K) satisfies hL(e1)h1|L(e2)h2i = hh1|π(he1, e2))h2i for all e1, e2 ∈ E and h1, h2 ∈ H.
Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Hilbert C ∗-modules
Definition A Hilbert C ∗-module E over A is a right A-module with a function h·, ·i : E × E → A, known as A-valued semi-inner product, with the following properties - for ξ, η, ζ ∈ E, a ∈ A, λ ∈ C, 1 hξ, η + ζi = hξ, ηi + hξ, ζi and hξ, ληi = λ hξ, ηi, 2 hξ, ηai = hξ, ηi a, ∗ 3 hξ, ηi = hη, ξi ,
12 / 23 Theorem E can be isometrically embedded into B(H, K) for some Hilbert spaces H and K. Further, we can find a faithful representation π : A → B(H) and isometric embedding L : E → B(H, K) satisfies hL(e1)h1|L(e2)h2i = hh1|π(he1, e2))h2i for all e1, e2 ∈ E and h1, h2 ∈ H.
Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Hilbert C ∗-modules
Definition A Hilbert C ∗-module E over A is a right A-module with a function h·, ·i : E × E → A, known as A-valued semi-inner product, with the following properties - for ξ, η, ζ ∈ E, a ∈ A, λ ∈ C, 1 hξ, η + ζi = hξ, ηi + hξ, ζi and hξ, ληi = λ hξ, ηi, 2 hξ, ηai = hξ, ηi a, ∗ 3 hξ, ηi = hη, ξi , 4 hξ, ξi is a positive element of A.
12 / 23 Further, we can find a faithful representation π : A → B(H) and isometric embedding L : E → B(H, K) satisfies hL(e1)h1|L(e2)h2i = hh1|π(he1, e2))h2i for all e1, e2 ∈ E and h1, h2 ∈ H.
Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Hilbert C ∗-modules
Definition A Hilbert C ∗-module E over A is a right A-module with a function h·, ·i : E × E → A, known as A-valued semi-inner product, with the following properties - for ξ, η, ζ ∈ E, a ∈ A, λ ∈ C, 1 hξ, η + ζi = hξ, ηi + hξ, ζi and hξ, ληi = λ hξ, ηi, 2 hξ, ηai = hξ, ηi a, ∗ 3 hξ, ηi = hη, ξi , 4 hξ, ξi is a positive element of A.
Theorem E can be isometrically embedded into B(H, K) for some Hilbert spaces H and K.
12 / 23 Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Hilbert C ∗-modules
Definition A Hilbert C ∗-module E over A is a right A-module with a function h·, ·i : E × E → A, known as A-valued semi-inner product, with the following properties - for ξ, η, ζ ∈ E, a ∈ A, λ ∈ C, 1 hξ, η + ζi = hξ, ηi + hξ, ζi and hξ, ληi = λ hξ, ηi, 2 hξ, ηai = hξ, ηi a, ∗ 3 hξ, ηi = hη, ξi , 4 hξ, ξi is a positive element of A.
Theorem E can be isometrically embedded into B(H, K) for some Hilbert spaces H and K. Further, we can find a faithful representation π : A → B(H) and isometric embedding L : E → B(H, K) satisfies hL(e1)h1|L(e2)h2i = hh1|π(he1, e2))h2i for all e1, e2 ∈ E and h1, h2 ∈ H. 12 / 23 Proof: We only need to prove the theorem for the special case E = B(H, K). For any operator t ∈ B(H, K) we denote by t˜, the 0 0 operator on H ⊕ K given by t˜ = . t 0 Let e be orthogonal to B. We have e is Birkhoff-James orthogonal to B if and only ife ˜ is Birkhoff-James orthogonal to B˜ = {b˜ : b ∈ B}. Now using characterization of C ∗-algebra, we ˜ get that there exists φ ∈ SB(H⊕K) such that φ˜(˜e∗e˜) = ke˜k2 and φ˜(˜e∗b˜) = 0 for all b˜ ∈ B˜. Now φ defined as φ(e) = φ˜(˜e) is the required state.
Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Orthogonality in Hilbert C ∗-modules
Theorem (Grover P.; Singla S., 2021) Let e ∈ E. Let B be a subspace of E. Then e is Birkhoff-James orthogonal to B in the Banach space E if and only if there exists 2 φ ∈ SA such that φ(he, ei) = kek and φ(he, bi) = 0 for all b ∈ B.
13 / 23 For any operator t ∈ B(H, K) we denote by t˜, the 0 0 operator on H ⊕ K given by t˜ = . t 0 Let e be orthogonal to B. We have e is Birkhoff-James orthogonal to B if and only ife ˜ is Birkhoff-James orthogonal to B˜ = {b˜ : b ∈ B}. Now using characterization of C ∗-algebra, we ˜ get that there exists φ ∈ SB(H⊕K) such that φ˜(˜e∗e˜) = ke˜k2 and φ˜(˜e∗b˜) = 0 for all b˜ ∈ B˜. Now φ defined as φ(e) = φ˜(˜e) is the required state.
Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Orthogonality in Hilbert C ∗-modules
Theorem (Grover P.; Singla S., 2021) Let e ∈ E. Let B be a subspace of E. Then e is Birkhoff-James orthogonal to B in the Banach space E if and only if there exists 2 φ ∈ SA such that φ(he, ei) = kek and φ(he, bi) = 0 for all b ∈ B.
Proof: We only need to prove the theorem for the special case E = B(H, K).
13 / 23 Let e be orthogonal to B. We have e is Birkhoff-James orthogonal to B if and only ife ˜ is Birkhoff-James orthogonal to B˜ = {b˜ : b ∈ B}. Now using characterization of C ∗-algebra, we ˜ get that there exists φ ∈ SB(H⊕K) such that φ˜(˜e∗e˜) = ke˜k2 and φ˜(˜e∗b˜) = 0 for all b˜ ∈ B˜. Now φ defined as φ(e) = φ˜(˜e) is the required state.
Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Orthogonality in Hilbert C ∗-modules
Theorem (Grover P.; Singla S., 2021) Let e ∈ E. Let B be a subspace of E. Then e is Birkhoff-James orthogonal to B in the Banach space E if and only if there exists 2 φ ∈ SA such that φ(he, ei) = kek and φ(he, bi) = 0 for all b ∈ B.
Proof: We only need to prove the theorem for the special case E = B(H, K). For any operator t ∈ B(H, K) we denote by t˜, the 0 0 operator on H ⊕ K given by t˜ = . t 0
13 / 23 We have e is Birkhoff-James orthogonal to B if and only ife ˜ is Birkhoff-James orthogonal to B˜ = {b˜ : b ∈ B}. Now using characterization of C ∗-algebra, we ˜ get that there exists φ ∈ SB(H⊕K) such that φ˜(˜e∗e˜) = ke˜k2 and φ˜(˜e∗b˜) = 0 for all b˜ ∈ B˜. Now φ defined as φ(e) = φ˜(˜e) is the required state.
Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Orthogonality in Hilbert C ∗-modules
Theorem (Grover P.; Singla S., 2021) Let e ∈ E. Let B be a subspace of E. Then e is Birkhoff-James orthogonal to B in the Banach space E if and only if there exists 2 φ ∈ SA such that φ(he, ei) = kek and φ(he, bi) = 0 for all b ∈ B.
Proof: We only need to prove the theorem for the special case E = B(H, K). For any operator t ∈ B(H, K) we denote by t˜, the 0 0 operator on H ⊕ K given by t˜ = . t 0 Let e be orthogonal to B.
13 / 23 Now using characterization of C ∗-algebra, we ˜ get that there exists φ ∈ SB(H⊕K) such that φ˜(˜e∗e˜) = ke˜k2 and φ˜(˜e∗b˜) = 0 for all b˜ ∈ B˜. Now φ defined as φ(e) = φ˜(˜e) is the required state.
Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Orthogonality in Hilbert C ∗-modules
Theorem (Grover P.; Singla S., 2021) Let e ∈ E. Let B be a subspace of E. Then e is Birkhoff-James orthogonal to B in the Banach space E if and only if there exists 2 φ ∈ SA such that φ(he, ei) = kek and φ(he, bi) = 0 for all b ∈ B.
Proof: We only need to prove the theorem for the special case E = B(H, K). For any operator t ∈ B(H, K) we denote by t˜, the 0 0 operator on H ⊕ K given by t˜ = . t 0 Let e be orthogonal to B. We have e is Birkhoff-James orthogonal to B if and only ife ˜ is Birkhoff-James orthogonal to B˜ = {b˜ : b ∈ B}.
13 / 23 we ˜ get that there exists φ ∈ SB(H⊕K) such that φ˜(˜e∗e˜) = ke˜k2 and φ˜(˜e∗b˜) = 0 for all b˜ ∈ B˜. Now φ defined as φ(e) = φ˜(˜e) is the required state.
Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Orthogonality in Hilbert C ∗-modules
Theorem (Grover P.; Singla S., 2021) Let e ∈ E. Let B be a subspace of E. Then e is Birkhoff-James orthogonal to B in the Banach space E if and only if there exists 2 φ ∈ SA such that φ(he, ei) = kek and φ(he, bi) = 0 for all b ∈ B.
Proof: We only need to prove the theorem for the special case E = B(H, K). For any operator t ∈ B(H, K) we denote by t˜, the 0 0 operator on H ⊕ K given by t˜ = . t 0 Let e be orthogonal to B. We have e is Birkhoff-James orthogonal to B if and only ife ˜ is Birkhoff-James orthogonal to B˜ = {b˜ : b ∈ B}. Now using characterization of C ∗-algebra,
13 / 23 Now φ defined as φ(e) = φ˜(˜e) is the required state.
Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Orthogonality in Hilbert C ∗-modules
Theorem (Grover P.; Singla S., 2021) Let e ∈ E. Let B be a subspace of E. Then e is Birkhoff-James orthogonal to B in the Banach space E if and only if there exists 2 φ ∈ SA such that φ(he, ei) = kek and φ(he, bi) = 0 for all b ∈ B.
Proof: We only need to prove the theorem for the special case E = B(H, K). For any operator t ∈ B(H, K) we denote by t˜, the 0 0 operator on H ⊕ K given by t˜ = . t 0 Let e be orthogonal to B. We have e is Birkhoff-James orthogonal to B if and only ife ˜ is Birkhoff-James orthogonal to B˜ = {b˜ : b ∈ B}. Now using characterization of C ∗-algebra, we ˜ get that there exists φ ∈ SB(H⊕K) such that φ˜(˜e∗e˜) = ke˜k2 and φ˜(˜e∗b˜) = 0 for all b˜ ∈ B˜.
13 / 23 Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Orthogonality in Hilbert C ∗-modules
Theorem (Grover P.; Singla S., 2021) Let e ∈ E. Let B be a subspace of E. Then e is Birkhoff-James orthogonal to B in the Banach space E if and only if there exists 2 φ ∈ SA such that φ(he, ei) = kek and φ(he, bi) = 0 for all b ∈ B.
Proof: We only need to prove the theorem for the special case E = B(H, K). For any operator t ∈ B(H, K) we denote by t˜, the 0 0 operator on H ⊕ K given by t˜ = . t 0 Let e be orthogonal to B. We have e is Birkhoff-James orthogonal to B if and only ife ˜ is Birkhoff-James orthogonal to B˜ = {b˜ : b ∈ B}. Now using characterization of C ∗-algebra, we ˜ get that there exists φ ∈ SB(H⊕K) such that φ˜(˜e∗e˜) = ke˜k2 and φ˜(˜e∗b˜) = 0 for all b˜ ∈ B˜. Now φ defined as φ(e) = φ˜(˜e) is the required state.
13 / 23 Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Orthogonality in Hilbert C ∗-modules
Theorem (Grover P.; Singla S., 2021) Let e ∈ E. Let B be a subspace of E. Then e is Birkhoff-James orthogonal to B in the Banach space E if and only if there exists 2 φ ∈ SA such that φ(he, ei) = kek and φ(he, bi) = 0 for all b ∈ B.
Proof: We only need to prove the theorem for the special case E = B(H, K). For any operator t ∈ B(H, K) we denote by t˜, the 0 0 operator on H ⊕ K given by t˜ = . t 0 Let e be orthogonal to B. We have e is Birkhoff-James orthogonal to B if and only ife ˜ is Birkhoff-James orthogonal to B˜ = {b˜ : b ∈ B}. Now using characterization of C ∗-algebra, we ˜ get that there exists φ ∈ SB(H⊕K) such that φ˜(˜e∗e˜) = ke˜k2 and φ˜(˜e∗b˜) = 0 for all b˜ ∈ B˜. Now φ defined as φ(e) = φ˜(˜e) is the required state.
13 / 23 The main theorem can be restated as: If a ∈ A is Birkhoff-James orthogonal to a subspace B of A, then can we find φ ∈ SA such that a is orthogonal to B in (A, h|iφ). a
kakφ ka − bkφ
θa,b 0 φ B b
Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Geometric interpretation
A positive functional φ gives a semi inner product on A defined as ∗ ha|biφ = φ(a b).
14 / 23 If a ∈ A is Birkhoff-James orthogonal to a subspace B of A, then can we find φ ∈ SA such that a is orthogonal to B in (A, h|iφ). a
kakφ ka − bkφ
θa,b 0 φ B b
Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Geometric interpretation
A positive functional φ gives a semi inner product on A defined as ∗ ha|biφ = φ(a b). The main theorem can be restated as:
14 / 23 a
kakφ ka − bkφ
θa,b 0 φ B b
Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Geometric interpretation
A positive functional φ gives a semi inner product on A defined as ∗ ha|biφ = φ(a b). The main theorem can be restated as: If a ∈ A is Birkhoff-James orthogonal to a subspace B of A, then can we find φ ∈ SA such that a is orthogonal to B in (A, h|iφ).
14 / 23 Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Geometric interpretation
A positive functional φ gives a semi inner product on A defined as ∗ ha|biφ = φ(a b). The main theorem can be restated as: If a ∈ A is Birkhoff-James orthogonal to a subspace B of A, then can we find φ ∈ SA such that a is orthogonal to B in (A, h|iφ). a
kakφ ka − bkφ
θa,b 0 φ B b
14 / 23 Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Table of Contents
1 Birkhoff-James orthogonality in Hilbert C ∗-modules
2 Proofs of characterization of orthogonality in C ∗-algebra
15 / 23 Definition: A representation π : A → B(H) is called cyclic if there exists a unit vector ξ ∈ H such that {π(a)ξ : a ∈ A} = H. A tuple (H, π, ξ) will stand for a cyclic representation of A.
Theorem For a functional ψ ∈ A∗, there exists a cyclic representation (H, π, ξ) and a vector η ∈ H such that ψ(a) = hη|π(a)ξi for all a ∈ A.
Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Functional and Representations
Definition: A representation of A is a ∗-homomorphism π : A → B(H), for some Hilbert space H.
16 / 23 A tuple (H, π, ξ) will stand for a cyclic representation of A.
Theorem For a functional ψ ∈ A∗, there exists a cyclic representation (H, π, ξ) and a vector η ∈ H such that ψ(a) = hη|π(a)ξi for all a ∈ A.
Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Functional and Representations
Definition: A representation of A is a ∗-homomorphism π : A → B(H), for some Hilbert space H. Definition: A representation π : A → B(H) is called cyclic if there exists a unit vector ξ ∈ H such that {π(a)ξ : a ∈ A} = H.
16 / 23 Theorem For a functional ψ ∈ A∗, there exists a cyclic representation (H, π, ξ) and a vector η ∈ H such that ψ(a) = hη|π(a)ξi for all a ∈ A.
Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Functional and Representations
Definition: A representation of A is a ∗-homomorphism π : A → B(H), for some Hilbert space H. Definition: A representation π : A → B(H) is called cyclic if there exists a unit vector ξ ∈ H such that {π(a)ξ : a ∈ A} = H. A tuple (H, π, ξ) will stand for a cyclic representation of A.
16 / 23 Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Functional and Representations
Definition: A representation of A is a ∗-homomorphism π : A → B(H), for some Hilbert space H. Definition: A representation π : A → B(H) is called cyclic if there exists a unit vector ξ ∈ H such that {π(a)ξ : a ∈ A} = H. A tuple (H, π, ξ) will stand for a cyclic representation of A.
Theorem For a functional ψ ∈ A∗, there exists a cyclic representation (H, π, ξ) and a vector η ∈ H such that ψ(a) = hη|π(a)ξi for all a ∈ A.
16 / 23 Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Functional and Representations
Definition: A representation of A is a ∗-homomorphism π : A → B(H), for some Hilbert space H. Definition: A representation π : A → B(H) is called cyclic if there exists a unit vector ξ ∈ H such that {π(a)ξ : a ∈ A} = H. A tuple (H, π, ξ) will stand for a cyclic representation of A.
Theorem For a functional ψ ∈ A∗, there exists a cyclic representation (H, π, ξ) and a vector η ∈ H such that ψ(a) = hη|π(a)ξi for all a ∈ A.
16 / 23 Proof. 1 Reverse direction is easy. For all b ∈ B, kak2 = φ(a∗a) ≤ φ(a∗a)+φ(b∗b) = φ((a−b)∗(a−b)) ≤ ka−bk2.
2 Let a be Birkhoff-James orthogonal to B i.e. dist(a, B) = kak. 3 By the Hahn-Banach theorem, there exists ψ ∈ A∗ such that kψk = 1, ψ(a) = kak and ψ(b) = 0 for all b ∈ B. 4 Hence there exists a cyclic representation (H, π, ξ) of A and a unit vector η ∈ H such that ψ(c) = hη|π(c)ξi for all c ∈ A.
Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Proof using representations of C ∗-algebra
Theorem (Grover P.; Singla S., 2021) Let a ∈ A. Let B be a subspace of A. Then a is Birkhoff-James orthogonal to B if and only if there exists φ ∈ SA such that φ(a∗a) = kak2 and φ(a∗b) = 0 for all b ∈ B.
17 / 23 kak2 = φ(a∗a) ≤ φ(a∗a)+φ(b∗b) = φ((a−b)∗(a−b)) ≤ ka−bk2.
2 Let a be Birkhoff-James orthogonal to B i.e. dist(a, B) = kak. 3 By the Hahn-Banach theorem, there exists ψ ∈ A∗ such that kψk = 1, ψ(a) = kak and ψ(b) = 0 for all b ∈ B. 4 Hence there exists a cyclic representation (H, π, ξ) of A and a unit vector η ∈ H such that ψ(c) = hη|π(c)ξi for all c ∈ A.
Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Proof using representations of C ∗-algebra
Theorem (Grover P.; Singla S., 2021) Let a ∈ A. Let B be a subspace of A. Then a is Birkhoff-James orthogonal to B if and only if there exists φ ∈ SA such that φ(a∗a) = kak2 and φ(a∗b) = 0 for all b ∈ B.
Proof. 1 Reverse direction is easy. For all b ∈ B,
17 / 23 2 Let a be Birkhoff-James orthogonal to B i.e. dist(a, B) = kak. 3 By the Hahn-Banach theorem, there exists ψ ∈ A∗ such that kψk = 1, ψ(a) = kak and ψ(b) = 0 for all b ∈ B. 4 Hence there exists a cyclic representation (H, π, ξ) of A and a unit vector η ∈ H such that ψ(c) = hη|π(c)ξi for all c ∈ A.
Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Proof using representations of C ∗-algebra
Theorem (Grover P.; Singla S., 2021) Let a ∈ A. Let B be a subspace of A. Then a is Birkhoff-James orthogonal to B if and only if there exists φ ∈ SA such that φ(a∗a) = kak2 and φ(a∗b) = 0 for all b ∈ B.
Proof. 1 Reverse direction is easy. For all b ∈ B, kak2 = φ(a∗a) ≤ φ(a∗a)+φ(b∗b) = φ((a−b)∗(a−b)) ≤ ka−bk2.
17 / 23 3 By the Hahn-Banach theorem, there exists ψ ∈ A∗ such that kψk = 1, ψ(a) = kak and ψ(b) = 0 for all b ∈ B. 4 Hence there exists a cyclic representation (H, π, ξ) of A and a unit vector η ∈ H such that ψ(c) = hη|π(c)ξi for all c ∈ A.
Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Proof using representations of C ∗-algebra
Theorem (Grover P.; Singla S., 2021) Let a ∈ A. Let B be a subspace of A. Then a is Birkhoff-James orthogonal to B if and only if there exists φ ∈ SA such that φ(a∗a) = kak2 and φ(a∗b) = 0 for all b ∈ B.
Proof. 1 Reverse direction is easy. For all b ∈ B, kak2 = φ(a∗a) ≤ φ(a∗a)+φ(b∗b) = φ((a−b)∗(a−b)) ≤ ka−bk2.
2 Let a be Birkhoff-James orthogonal to B i.e. dist(a, B) = kak.
17 / 23 4 Hence there exists a cyclic representation (H, π, ξ) of A and a unit vector η ∈ H such that ψ(c) = hη|π(c)ξi for all c ∈ A.
Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Proof using representations of C ∗-algebra
Theorem (Grover P.; Singla S., 2021) Let a ∈ A. Let B be a subspace of A. Then a is Birkhoff-James orthogonal to B if and only if there exists φ ∈ SA such that φ(a∗a) = kak2 and φ(a∗b) = 0 for all b ∈ B.
Proof. 1 Reverse direction is easy. For all b ∈ B, kak2 = φ(a∗a) ≤ φ(a∗a)+φ(b∗b) = φ((a−b)∗(a−b)) ≤ ka−bk2.
2 Let a be Birkhoff-James orthogonal to B i.e. dist(a, B) = kak. 3 By the Hahn-Banach theorem, there exists ψ ∈ A∗ such that kψk = 1, ψ(a) = kak and ψ(b) = 0 for all b ∈ B.
17 / 23 Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Proof using representations of C ∗-algebra
Theorem (Grover P.; Singla S., 2021) Let a ∈ A. Let B be a subspace of A. Then a is Birkhoff-James orthogonal to B if and only if there exists φ ∈ SA such that φ(a∗a) = kak2 and φ(a∗b) = 0 for all b ∈ B.
Proof. 1 Reverse direction is easy. For all b ∈ B, kak2 = φ(a∗a) ≤ φ(a∗a)+φ(b∗b) = φ((a−b)∗(a−b)) ≤ ka−bk2.
2 Let a be Birkhoff-James orthogonal to B i.e. dist(a, B) = kak. 3 By the Hahn-Banach theorem, there exists ψ ∈ A∗ such that kψk = 1, ψ(a) = kak and ψ(b) = 0 for all b ∈ B. 4 Hence there exists a cyclic representation (H, π, ξ) of A and a unit vector η ∈ H such that ψ(c) = hη|π(c)ξi for all c ∈ A.
17 / 23 Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Proof using representations of C ∗-algebra
Theorem (Grover P.; Singla S., 2021) Let a ∈ A. Let B be a subspace of A. Then a is Birkhoff-James orthogonal to B if and only if there exists φ ∈ SA such that φ(a∗a) = kak2 and φ(a∗b) = 0 for all b ∈ B.
Proof. 1 Reverse direction is easy. For all b ∈ B, kak2 = φ(a∗a) ≤ φ(a∗a)+φ(b∗b) = φ((a−b)∗(a−b)) ≤ ka−bk2.
2 Let a be Birkhoff-James orthogonal to B i.e. dist(a, B) = kak. 3 By the Hahn-Banach theorem, there exists ψ ∈ A∗ such that kψk = 1, ψ(a) = kak and ψ(b) = 0 for all b ∈ B. 4 Hence there exists a cyclic representation (H, π, ξ) of A and a unit vector η ∈ H such that ψ(c) = hη|π(c)ξi for all c ∈ A.
17 / 23 5 Now ψ(a) = hη|π(a)ξi = kak. So by using the condition for equality in Cauchy-Schwarz inequality, we obtain kakη = π(a)ξ. 1 6 This gives ψ(c) = hπ(a)ξ|π(c)ξi for all c ∈ A. kak
7 Therefore, hπ(a)ξ|π(a)ξi = kak2 and hπ(a)ξ|π(b)ξi = 0 for all b ∈ B.
8 Define φ ∈ A∗ as φ(c) = hξ|π(c)ξi.
Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Proof Continued
Theorem (Grover P.; Singla S., 2021) Let a ∈ A. Let B be a subspace of A. Then a is Birkhoff-James orthogonal to B if and only if there exists φ ∈ SA such that φ(a∗a) = kak2 and φ(a∗b) = 0 for all b ∈ B.
18 / 23 1 6 This gives ψ(c) = hπ(a)ξ|π(c)ξi for all c ∈ A. kak
7 Therefore, hπ(a)ξ|π(a)ξi = kak2 and hπ(a)ξ|π(b)ξi = 0 for all b ∈ B.
8 Define φ ∈ A∗ as φ(c) = hξ|π(c)ξi.
Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Proof Continued
Theorem (Grover P.; Singla S., 2021) Let a ∈ A. Let B be a subspace of A. Then a is Birkhoff-James orthogonal to B if and only if there exists φ ∈ SA such that φ(a∗a) = kak2 and φ(a∗b) = 0 for all b ∈ B.
5 Now ψ(a) = hη|π(a)ξi = kak. So by using the condition for equality in Cauchy-Schwarz inequality, we obtain kakη = π(a)ξ.
18 / 23 7 Therefore, hπ(a)ξ|π(a)ξi = kak2 and hπ(a)ξ|π(b)ξi = 0 for all b ∈ B.
8 Define φ ∈ A∗ as φ(c) = hξ|π(c)ξi.
Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Proof Continued
Theorem (Grover P.; Singla S., 2021) Let a ∈ A. Let B be a subspace of A. Then a is Birkhoff-James orthogonal to B if and only if there exists φ ∈ SA such that φ(a∗a) = kak2 and φ(a∗b) = 0 for all b ∈ B.
5 Now ψ(a) = hη|π(a)ξi = kak. So by using the condition for equality in Cauchy-Schwarz inequality, we obtain kakη = π(a)ξ. 1 6 This gives ψ(c) = hπ(a)ξ|π(c)ξi for all c ∈ A. kak
18 / 23 7 Therefore, hπ(a)ξ|π(a)ξi = kak2 and hπ(a)ξ|π(b)ξi = 0 for all b ∈ B.
8 Define φ ∈ A∗ as φ(c) = hξ|π(c)ξi.
Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Proof Continued
Theorem (Grover P.; Singla S., 2021) Let a ∈ A. Let B be a subspace of A. Then a is Birkhoff-James orthogonal to B if and only if there exists φ ∈ SA such that φ(a∗a) = kak2 and φ(a∗b) = 0 for all b ∈ B.
5 Now ψ(a) = hη|π(a)ξi = kak. So by using the condition for equality in Cauchy-Schwarz inequality, we obtain kakη = π(a)ξ. 1 6 This gives ψ(c) = hπ(a)ξ|π(c)ξi for all c ∈ A. kak
18 / 23 8 Define φ ∈ A∗ as φ(c) = hξ|π(c)ξi.
Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Proof Continued
Theorem (Grover P.; Singla S., 2021) Let a ∈ A. Let B be a subspace of A. Then a is Birkhoff-James orthogonal to B if and only if there exists φ ∈ SA such that φ(a∗a) = kak2 and φ(a∗b) = 0 for all b ∈ B.
5 Now ψ(a) = hη|π(a)ξi = kak. So by using the condition for equality in Cauchy-Schwarz inequality, we obtain kakη = π(a)ξ. 1 6 This gives ψ(c) = hπ(a)ξ|π(c)ξi for all c ∈ A. kak
7 Therefore, hπ(a)ξ|π(a)ξi = kak2 and hπ(a)ξ|π(b)ξi = 0 for all b ∈ B.
18 / 23 Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Proof Continued
Theorem (Grover P.; Singla S., 2021) Let a ∈ A. Let B be a subspace of A. Then a is Birkhoff-James orthogonal to B if and only if there exists φ ∈ SA such that φ(a∗a) = kak2 and φ(a∗b) = 0 for all b ∈ B.
5 Now ψ(a) = hη|π(a)ξi = kak. So by using the condition for equality in Cauchy-Schwarz inequality, we obtain kakη = π(a)ξ. 1 6 This gives ψ(c) = hπ(a)ξ|π(c)ξi for all c ∈ A. kak
7 Therefore, hπ(a)ξ|π(a)ξi = kak2 and hπ(a)ξ|π(b)ξi = 0 for all b ∈ B.
8 Define φ ∈ A∗ as φ(c) = hξ|π(c)ξi.
18 / 23 Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Proof Continued
Theorem (Grover P.; Singla S., 2021) Let a ∈ A. Let B be a subspace of A. Then a is Birkhoff-James orthogonal to B if and only if there exists φ ∈ SA such that φ(a∗a) = kak2 and φ(a∗b) = 0 for all b ∈ B.
5 Now ψ(a) = hη|π(a)ξi = kak. So by using the condition for equality in Cauchy-Schwarz inequality, we obtain kakη = π(a)ξ. 1 6 This gives ψ(c) = hπ(a)ξ|π(c)ξi for all c ∈ A. kak
7 Therefore, hπ(a)ξ|π(a)ξi = kak2 and hπ(a)ξ|π(b)ξi = 0 for all b ∈ B.
8 Define φ ∈ A∗ as φ(c) = hξ|π(c)ξi.
18 / 23 To say that a is orthogonal to b is to say that f attains its minimum at the point 0. This is clearly a calculus problem, except that the function k · k is not differentiable. So we can’t use first derivative test. But k · k is also convex, this gives motivation to define Gateaux derivative and we will see characterization of orthogonality in terms of Gateaux derivative.
Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Birkhoff-James orthogonality as a calculus problem
We consider the function f (λ) = ka + λbk mapping C into R+.
19 / 23 This is clearly a calculus problem, except that the function k · k is not differentiable. So we can’t use first derivative test. But k · k is also convex, this gives motivation to define Gateaux derivative and we will see characterization of orthogonality in terms of Gateaux derivative.
Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Birkhoff-James orthogonality as a calculus problem
We consider the function f (λ) = ka + λbk mapping C into R+. To say that a is orthogonal to b is to say that f attains its minimum at the point 0.
19 / 23 So we can’t use first derivative test. But k · k is also convex, this gives motivation to define Gateaux derivative and we will see characterization of orthogonality in terms of Gateaux derivative.
Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Birkhoff-James orthogonality as a calculus problem
We consider the function f (λ) = ka + λbk mapping C into R+. To say that a is orthogonal to b is to say that f attains its minimum at the point 0. This is clearly a calculus problem, except that the function k · k is not differentiable.
19 / 23 and we will see characterization of orthogonality in terms of Gateaux derivative.
Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Birkhoff-James orthogonality as a calculus problem
We consider the function f (λ) = ka + λbk mapping C into R+. To say that a is orthogonal to b is to say that f attains its minimum at the point 0. This is clearly a calculus problem, except that the function k · k is not differentiable. So we can’t use first derivative test. But k · k is also convex, this gives motivation to define Gateaux derivative
19 / 23 Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Birkhoff-James orthogonality as a calculus problem
We consider the function f (λ) = ka + λbk mapping C into R+. To say that a is orthogonal to b is to say that f attains its minimum at the point 0. This is clearly a calculus problem, except that the function k · k is not differentiable. So we can’t use first derivative test. But k · k is also convex, this gives motivation to define Gateaux derivative and we will see characterization of orthogonality in terms of Gateaux derivative.
19 / 23 Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Birkhoff-James orthogonality as a calculus problem
We consider the function f (λ) = ka + λbk mapping C into R+. To say that a is orthogonal to b is to say that f attains its minimum at the point 0. This is clearly a calculus problem, except that the function k · k is not differentiable. So we can’t use first derivative test. But k · k is also convex, this gives motivation to define Gateaux derivative and we will see characterization of orthogonality in terms of Gateaux derivative.
19 / 23 Hence the kx + tyk − kxk limit D0,x (y) = lim always exists. t→0+ t 2 We have kx + tyk ≥ kxk for all t ∈ R if and only if the inequality D0,x (y) ≥ 0 holds.
3 And x is orthogonal to y if and only if infφ Dφ,x (y) ≥ 0 where kx + teiφyk − kxk Dφ,x (y) = lim is called the φ-Gateaux t→0+ t derivative of the norm at the vector x, in the y and φ directions.
Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Gateaux derivative and orthogonality
Theorem Let X be a Banach space, x, y ∈ X , and φ ∈ [0, 2π). 1 The function α : R → R, α(t) = kx + tyk is convex.
20 / 23 2 We have kx + tyk ≥ kxk for all t ∈ R if and only if the inequality D0,x (y) ≥ 0 holds.
3 And x is orthogonal to y if and only if infφ Dφ,x (y) ≥ 0 where kx + teiφyk − kxk Dφ,x (y) = lim is called the φ-Gateaux t→0+ t derivative of the norm at the vector x, in the y and φ directions.
Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Gateaux derivative and orthogonality
Theorem Let X be a Banach space, x, y ∈ X , and φ ∈ [0, 2π). 1 The function α : R → R, α(t) = kx + tyk is convex.Hence the kx + tyk − kxk limit D0,x (y) = lim always exists. t→0+ t
20 / 23 3 And x is orthogonal to y if and only if infφ Dφ,x (y) ≥ 0 where kx + teiφyk − kxk Dφ,x (y) = lim is called the φ-Gateaux t→0+ t derivative of the norm at the vector x, in the y and φ directions.
Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Gateaux derivative and orthogonality
Theorem Let X be a Banach space, x, y ∈ X , and φ ∈ [0, 2π). 1 The function α : R → R, α(t) = kx + tyk is convex.Hence the kx + tyk − kxk limit D0,x (y) = lim always exists. t→0+ t 2 We have kx + tyk ≥ kxk for all t ∈ R if and only if the inequality D0,x (y) ≥ 0 holds.
20 / 23 is called the φ-Gateaux derivative of the norm at the vector x, in the y and φ directions.
Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Gateaux derivative and orthogonality
Theorem Let X be a Banach space, x, y ∈ X , and φ ∈ [0, 2π). 1 The function α : R → R, α(t) = kx + tyk is convex.Hence the kx + tyk − kxk limit D0,x (y) = lim always exists. t→0+ t 2 We have kx + tyk ≥ kxk for all t ∈ R if and only if the inequality D0,x (y) ≥ 0 holds.
3 And x is orthogonal to y if and only if infφ Dφ,x (y) ≥ 0 where kx + teiφyk − kxk Dφ,x (y) = lim t→0+ t
20 / 23 Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Gateaux derivative and orthogonality
Theorem Let X be a Banach space, x, y ∈ X , and φ ∈ [0, 2π). 1 The function α : R → R, α(t) = kx + tyk is convex.Hence the kx + tyk − kxk limit D0,x (y) = lim always exists. t→0+ t 2 We have kx + tyk ≥ kxk for all t ∈ R if and only if the inequality D0,x (y) ≥ 0 holds.
3 And x is orthogonal to y if and only if infφ Dφ,x (y) ≥ 0 where kx + teiφyk − kxk Dφ,x (y) = lim is called the φ-Gateaux t→0+ t derivative of the norm at the vector x, in the y and φ directions.
20 / 23 Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Gateaux derivative and orthogonality
Theorem Let X be a Banach space, x, y ∈ X , and φ ∈ [0, 2π). 1 The function α : R → R, α(t) = kx + tyk is convex.Hence the kx + tyk − kxk limit D0,x (y) = lim always exists. t→0+ t 2 We have kx + tyk ≥ kxk for all t ∈ R if and only if the inequality D0,x (y) ≥ 0 holds.
3 And x is orthogonal to y if and only if infφ Dφ,x (y) ≥ 0 where kx + teiφyk − kxk Dφ,x (y) = lim is called the φ-Gateaux t→0+ t derivative of the norm at the vector x, in the y and φ directions.
20 / 23 1 ∗ Thus we get D (b) = D ∗ (a b). φ,a kak φ,a a ∗ This gives infφ Dφ,a(b) ≥ 0 if and only if infφ Dφ,a∗a(a b) ≥ 0 i.e. a is orthogonal to b if and only if a∗a is orthogonal to a∗b. Then by the Hahn-Banach Theorem, there exists φ ∈ A∗ such that kφk = 1, φ(a∗a) = kak2 and φ(a∗b) = 0 for all b ∈ B. And φ is required state using fact it attains its norm at a non-zero positive element.
Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Proof using tools of convex analysis
Lemma Let a, b ∈ A. Then
ka + tbk − kak 1 ka∗a + ta∗bk − ka∗ak lim = lim . t→0+ t kak t→0+ t
21 / 23 ∗ This gives infφ Dφ,a(b) ≥ 0 if and only if infφ Dφ,a∗a(a b) ≥ 0 i.e. a is orthogonal to b if and only if a∗a is orthogonal to a∗b. Then by the Hahn-Banach Theorem, there exists φ ∈ A∗ such that kφk = 1, φ(a∗a) = kak2 and φ(a∗b) = 0 for all b ∈ B. And φ is required state using fact it attains its norm at a non-zero positive element.
Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Proof using tools of convex analysis
Lemma Let a, b ∈ A. Then
ka + tbk − kak 1 ka∗a + ta∗bk − ka∗ak lim = lim . t→0+ t kak t→0+ t
1 ∗ Thus we get D (b) = D ∗ (a b). φ,a kak φ,a a
21 / 23 i.e. a is orthogonal to b if and only if a∗a is orthogonal to a∗b. Then by the Hahn-Banach Theorem, there exists φ ∈ A∗ such that kφk = 1, φ(a∗a) = kak2 and φ(a∗b) = 0 for all b ∈ B. And φ is required state using fact it attains its norm at a non-zero positive element.
Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Proof using tools of convex analysis
Lemma Let a, b ∈ A. Then
ka + tbk − kak 1 ka∗a + ta∗bk − ka∗ak lim = lim . t→0+ t kak t→0+ t
1 ∗ Thus we get D (b) = D ∗ (a b). φ,a kak φ,a a ∗ This gives infφ Dφ,a(b) ≥ 0 if and only if infφ Dφ,a∗a(a b) ≥ 0
21 / 23 Then by the Hahn-Banach Theorem, there exists φ ∈ A∗ such that kφk = 1, φ(a∗a) = kak2 and φ(a∗b) = 0 for all b ∈ B. And φ is required state using fact it attains its norm at a non-zero positive element.
Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Proof using tools of convex analysis
Lemma Let a, b ∈ A. Then
ka + tbk − kak 1 ka∗a + ta∗bk − ka∗ak lim = lim . t→0+ t kak t→0+ t
1 ∗ Thus we get D (b) = D ∗ (a b). φ,a kak φ,a a ∗ This gives infφ Dφ,a(b) ≥ 0 if and only if infφ Dφ,a∗a(a b) ≥ 0 i.e. a is orthogonal to b if and only if a∗a is orthogonal to a∗b.
21 / 23 And φ is required state using fact it attains its norm at a non-zero positive element.
Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Proof using tools of convex analysis
Lemma Let a, b ∈ A. Then
ka + tbk − kak 1 ka∗a + ta∗bk − ka∗ak lim = lim . t→0+ t kak t→0+ t
1 ∗ Thus we get D (b) = D ∗ (a b). φ,a kak φ,a a ∗ This gives infφ Dφ,a(b) ≥ 0 if and only if infφ Dφ,a∗a(a b) ≥ 0 i.e. a is orthogonal to b if and only if a∗a is orthogonal to a∗b. Then by the Hahn-Banach Theorem, there exists φ ∈ A∗ such that kφk = 1, φ(a∗a) = kak2 and φ(a∗b) = 0 for all b ∈ B.
21 / 23 Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Proof using tools of convex analysis
Lemma Let a, b ∈ A. Then
ka + tbk − kak 1 ka∗a + ta∗bk − ka∗ak lim = lim . t→0+ t kak t→0+ t
1 ∗ Thus we get D (b) = D ∗ (a b). φ,a kak φ,a a ∗ This gives infφ Dφ,a(b) ≥ 0 if and only if infφ Dφ,a∗a(a b) ≥ 0 i.e. a is orthogonal to b if and only if a∗a is orthogonal to a∗b. Then by the Hahn-Banach Theorem, there exists φ ∈ A∗ such that kφk = 1, φ(a∗a) = kak2 and φ(a∗b) = 0 for all b ∈ B. And φ is required state using fact it attains its norm at a non-zero positive element.
21 / 23 Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra Proof using tools of convex analysis
Lemma Let a, b ∈ A. Then
ka + tbk − kak 1 ka∗a + ta∗bk − ka∗ak lim = lim . t→0+ t kak t→0+ t
1 ∗ Thus we get D (b) = D ∗ (a b). φ,a kak φ,a a ∗ This gives infφ Dφ,a(b) ≥ 0 if and only if infφ Dφ,a∗a(a b) ≥ 0 i.e. a is orthogonal to b if and only if a∗a is orthogonal to a∗b. Then by the Hahn-Banach Theorem, there exists φ ∈ A∗ such that kφk = 1, φ(a∗a) = kak2 and φ(a∗b) = 0 for all b ∈ B. And φ is required state using fact it attains its norm at a non-zero positive element.
21 / 23 Birkhoff-James orthogonality in Hilbert C ∗-modules Proofs of characterization of orthogonality in C ∗-algebra References
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5 Bhatia R.; Semrlˇ P. : Orthogonality of matrices and some distance problems. Linear Algebra Appl. 287 (1999), 77–85. 6 Williams J. P. : Finite operators. Proc. Amer. Math. Soc. 26 (1970), 129-136. 7 Rieffel M. A. : Leibniz seminorms and best approximation from C ∗-subalgebras. Sci. China Math. 54 (2011), 2259–2274. 8 Arambaˇsi´cL. ; Raji´cR. : The Birkhoff-James orthogonality in Hilbert C ∗-modules. Linear Algebra Appl. 437 (2012), 1913–1929. 9 Grover P. ; Singla S. : Birkhoff-James orthogonality and applications : A survey. Operator Theory, Functional Analysis and Applications, Birkh¨auser,Springer, vol. 282, 2021.
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