States and orthogonality in C ∗-algebra Proofs and applications

Orthogonality and Gateaux derivative of C ∗-norm

Sushil Singla

Department of Mathematics School of Natural Sciences Shiv Nadar University

June 18, 2021

1 / 24 States and orthogonality in C ∗-algebra Proofs and applications Table of Contents

∗ 1 States and orthogonality in C -algebra

2 Proofs and applications

2 / 24 States and orthogonality in C ∗-algebra Proofs and applications Table of Contents

∗ 1 States and orthogonality in C -algebra

2 Proofs and applications

3 / 24 Theorem (Gelfand-Naimark-Segal) ∗ 2 Let a ∈ A. Then there exists φ ∈ SA such that φ(a a) = kak . 2 ∗ This can be rephrased as : dist(a, {0}) = max{φ(a a): φ ∈ SA}.

A positive functional φ gives a semi inner product on A defined as ∗ ha|biφ = φ(a b). The above theorem can be rephrased as - 2 There exists φ ∈ S(A) such that ha|aiφ = dist(a, {0}) . We generalize this for any subspace B, when a best approximation to a in B exists.

States and orthogonality in C ∗-algebra Proofs and applications Existence of states on a C ∗-algebra

∗ Let A be a C algebra over field C or R. SA will stand for the set of all states on A.

4 / 24 2 ∗ This can be rephrased as : dist(a, {0}) = max{φ(a a): φ ∈ SA}.

A positive functional φ gives a semi inner product on A defined as ∗ ha|biφ = φ(a b). The above theorem can be rephrased as - 2 There exists φ ∈ S(A) such that ha|aiφ = dist(a, {0}) . We generalize this for any subspace B, when a best approximation to a in B exists.

States and orthogonality in C ∗-algebra Proofs and applications Existence of states on a C ∗-algebra

∗ Let A be a C algebra over field C or R. SA will stand for the set of all states on A.

Theorem (Gelfand-Naimark-Segal) ∗ 2 Let a ∈ A. Then there exists φ ∈ SA such that φ(a a) = kak .

4 / 24 A positive functional φ gives a semi inner product on A defined as ∗ ha|biφ = φ(a b). The above theorem can be rephrased as - 2 There exists φ ∈ S(A) such that ha|aiφ = dist(a, {0}) . We generalize this for any subspace B, when a best approximation to a in B exists.

States and orthogonality in C ∗-algebra Proofs and applications Existence of states on a C ∗-algebra

∗ Let A be a C algebra over field C or R. SA will stand for the set of all states on A.

Theorem (Gelfand-Naimark-Segal) ∗ 2 Let a ∈ A. Then there exists φ ∈ SA such that φ(a a) = kak . 2 ∗ This can be rephrased as : dist(a, {0}) = max{φ(a a): φ ∈ SA}.

4 / 24 The above theorem can be rephrased as - 2 There exists φ ∈ S(A) such that ha|aiφ = dist(a, {0}) . We generalize this for any subspace B, when a best approximation to a in B exists.

States and orthogonality in C ∗-algebra Proofs and applications Existence of states on a C ∗-algebra

∗ Let A be a C algebra over field C or R. SA will stand for the set of all states on A.

Theorem (Gelfand-Naimark-Segal) ∗ 2 Let a ∈ A. Then there exists φ ∈ SA such that φ(a a) = kak . 2 ∗ This can be rephrased as : dist(a, {0}) = max{φ(a a): φ ∈ SA}.

A positive functional φ gives a semi inner product on A defined as ∗ ha|biφ = φ(a b).

4 / 24 2 There exists φ ∈ S(A) such that ha|aiφ = dist(a, {0}) . We generalize this for any subspace B, when a best approximation to a in B exists.

States and orthogonality in C ∗-algebra Proofs and applications Existence of states on a C ∗-algebra

∗ Let A be a C algebra over field C or R. SA will stand for the set of all states on A.

Theorem (Gelfand-Naimark-Segal) ∗ 2 Let a ∈ A. Then there exists φ ∈ SA such that φ(a a) = kak . 2 ∗ This can be rephrased as : dist(a, {0}) = max{φ(a a): φ ∈ SA}.

A positive functional φ gives a semi inner product on A defined as ∗ ha|biφ = φ(a b). The above theorem can be rephrased as -

4 / 24 We generalize this for any subspace B, when a best approximation to a in B exists.

States and orthogonality in C ∗-algebra Proofs and applications Existence of states on a C ∗-algebra

∗ Let A be a C algebra over field C or R. SA will stand for the set of all states on A.

Theorem (Gelfand-Naimark-Segal) ∗ 2 Let a ∈ A. Then there exists φ ∈ SA such that φ(a a) = kak . 2 ∗ This can be rephrased as : dist(a, {0}) = max{φ(a a): φ ∈ SA}.

A positive functional φ gives a semi inner product on A defined as ∗ ha|biφ = φ(a b). The above theorem can be rephrased as - 2 There exists φ ∈ S(A) such that ha|aiφ = dist(a, {0}) .

4 / 24 States and orthogonality in C ∗-algebra Proofs and applications Existence of states on a C ∗-algebra

∗ Let A be a C algebra over field C or R. SA will stand for the set of all states on A.

Theorem (Gelfand-Naimark-Segal) ∗ 2 Let a ∈ A. Then there exists φ ∈ SA such that φ(a a) = kak . 2 ∗ This can be rephrased as : dist(a, {0}) = max{φ(a a): φ ∈ SA}.

A positive functional φ gives a semi inner product on A defined as ∗ ha|biφ = φ(a b). The above theorem can be rephrased as - 2 There exists φ ∈ S(A) such that ha|aiφ = dist(a, {0}) . We generalize this for any subspace B, when a best approximation to a in B exists.

4 / 24 States and orthogonality in C ∗-algebra Proofs and applications Existence of states on a C ∗-algebra

∗ Let A be a C algebra over field C or R. SA will stand for the set of all states on A.

Theorem (Gelfand-Naimark-Segal) ∗ 2 Let a ∈ A. Then there exists φ ∈ SA such that φ(a a) = kak . 2 ∗ This can be rephrased as : dist(a, {0}) = max{φ(a a): φ ∈ SA}.

A positive functional φ gives a semi inner product on A defined as ∗ ha|biφ = φ(a b). The above theorem can be rephrased as - 2 There exists φ ∈ S(A) such that ha|aiφ = dist(a, {0}) . We generalize this for any subspace B, when a best approximation to a in B exists.

4 / 24 Definition

An element w0 ∈ W is said to be a best approximation to v in W if and only if dist(v, W ) = kv − w0k . In case w0 = 0, we say v is Birkhoff-James orthogonal to W .

Note that w0 is a best approximation to v in W if and only if v − w0 is Birkhoff-James orthogonal to W . Also we note that in case V is a Hilbert space, this notion of orthogonality matches with usual notion of orthogonality in a Hilbert space. This viewpoint of best approximation as orthogonality enable us to guess results from geometric intuition and then try to prove it algebrically.

States and orthogonality in C ∗-algebra Proofs and applications Birkhoff-James Orthogonality and best approximation

Let W be a subspace of a normed space V .

5 / 24 In case w0 = 0, we say v is Birkhoff-James orthogonal to W .

Note that w0 is a best approximation to v in W if and only if v − w0 is Birkhoff-James orthogonal to W . Also we note that in case V is a Hilbert space, this notion of orthogonality matches with usual notion of orthogonality in a Hilbert space. This viewpoint of best approximation as orthogonality enable us to guess results from geometric intuition and then try to prove it algebrically.

States and orthogonality in C ∗-algebra Proofs and applications Birkhoff-James Orthogonality and best approximation

Let W be a subspace of a normed space V .

Definition

An element w0 ∈ W is said to be a best approximation to v in W if and only if dist(v, W ) = kv − w0k .

5 / 24 Note that w0 is a best approximation to v in W if and only if v − w0 is Birkhoff-James orthogonal to W . Also we note that in case V is a Hilbert space, this notion of orthogonality matches with usual notion of orthogonality in a Hilbert space. This viewpoint of best approximation as orthogonality enable us to guess results from geometric intuition and then try to prove it algebrically.

States and orthogonality in C ∗-algebra Proofs and applications Birkhoff-James Orthogonality and best approximation

Let W be a subspace of a normed space V .

Definition

An element w0 ∈ W is said to be a best approximation to v in W if and only if dist(v, W ) = kv − w0k . In case w0 = 0, we say v is Birkhoff-James orthogonal to W .

5 / 24 Also we note that in case V is a Hilbert space, this notion of orthogonality matches with usual notion of orthogonality in a Hilbert space. This viewpoint of best approximation as orthogonality enable us to guess results from geometric intuition and then try to prove it algebrically.

States and orthogonality in C ∗-algebra Proofs and applications Birkhoff-James Orthogonality and best approximation

Let W be a subspace of a normed space V .

Definition

An element w0 ∈ W is said to be a best approximation to v in W if and only if dist(v, W ) = kv − w0k . In case w0 = 0, we say v is Birkhoff-James orthogonal to W .

Note that w0 is a best approximation to v in W if and only if v − w0 is Birkhoff-James orthogonal to W .

5 / 24 States and orthogonality in C ∗-algebra Proofs and applications Birkhoff-James Orthogonality and best approximation

Let W be a subspace of a normed space V .

Definition

An element w0 ∈ W is said to be a best approximation to v in W if and only if dist(v, W ) = kv − w0k . In case w0 = 0, we say v is Birkhoff-James orthogonal to W .

Note that w0 is a best approximation to v in W if and only if v − w0 is Birkhoff-James orthogonal to W . Also we note that in case V is a Hilbert space, this notion of orthogonality matches with usual notion of orthogonality in a Hilbert space. This viewpoint of best approximation as orthogonality enable us to guess results from geometric intuition and then try to prove it algebrically.

5 / 24 States and orthogonality in C ∗-algebra Proofs and applications Birkhoff-James Orthogonality and best approximation

Let W be a subspace of a normed space V .

Definition

An element w0 ∈ W is said to be a best approximation to v in W if and only if dist(v, W ) = kv − w0k . In case w0 = 0, we say v is Birkhoff-James orthogonal to W .

Note that w0 is a best approximation to v in W if and only if v − w0 is Birkhoff-James orthogonal to W . Also we note that in case V is a Hilbert space, this notion of orthogonality matches with usual notion of orthogonality in a Hilbert space. This viewpoint of best approximation as orthogonality enable us to guess results from geometric intuition and then try to prove it algebrically.

5 / 24 The above theorem says that b0 is a best approximation to a in B if and only if there exists φ ∈ SA such that

ka − b0kφ = ka − b0k and ha − b0|biφ = 0 for all b ∈ B.

The above characterization of orthogonality has following geometric interpretation.

States and orthogonality in C ∗-algebra Proofs and applications Orthogonality characterization in 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.

6 / 24 if and only if there exists φ ∈ SA such that

ka − b0kφ = ka − b0k and ha − b0|biφ = 0 for all b ∈ B.

The above characterization of orthogonality has following geometric interpretation.

States and orthogonality in C ∗-algebra Proofs and applications Orthogonality characterization in 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.

The above theorem says that b0 is a best approximation to a in B

6 / 24 The above characterization of orthogonality has following geometric interpretation.

States and orthogonality in C ∗-algebra Proofs and applications Orthogonality characterization in 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.

The above theorem says that b0 is a best approximation to a in B if and only if there exists φ ∈ SA such that

ka − b0kφ = ka − b0k and ha − b0|biφ = 0 for all b ∈ B.

6 / 24 States and orthogonality in C ∗-algebra Proofs and applications Orthogonality characterization in 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.

The above theorem says that b0 is a best approximation to a in B if and only if there exists φ ∈ SA such that

ka − b0kφ = ka − b0k and ha − b0|biφ = 0 for all b ∈ B.

The above characterization of orthogonality has following geometric interpretation.

6 / 24 States and orthogonality in C ∗-algebra Proofs and applications Orthogonality characterization in 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.

The above theorem says that b0 is a best approximation to a in B if and only if there exists φ ∈ SA such that

ka − b0kφ = ka − b0k and ha − b0|biφ = 0 for all b ∈ B.

The above characterization of orthogonality has following geometric interpretation.

6 / 24 States and orthogonality in C ∗-algebra Proofs and applications Geometric interpretation

a

kakφ ka − bkφ

θa,b 0 φ B b

The above theorem is a generalization of very well known results, which follow as a corollary of the above result.

7 / 24 States and orthogonality in C ∗-algebra Proofs and applications Geometric interpretation

a

kakφ ka − bkφ

θa,b 0 φ B b

The above theorem is a generalization of very well known results, which follow as a corollary of the above result.

7 / 24 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

States and orthogonality in C ∗-algebra Proofs and applications 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 .

8 / 24 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

States and orthogonality in C ∗-algebra Proofs and applications 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:

8 / 24 R 2 (1) is equivalent to (f − g)(f − g) dµ = kf − gk∞. X

States and orthogonality in C ∗-algebra Proofs and applications 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

8 / 24 States and orthogonality in C ∗-algebra Proofs and applications 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 8 / 24 States and orthogonality in C ∗-algebra Proofs and applications 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 8 / 24 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.

Density matrices correspond to states on Mn(F) and trace (A∗AT ) = kA2k is equivalent to A∗AT = kAk2T .

States and orthogonality in C ∗-algebra Proofs and applications

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.

9 / 24 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.

Density matrices correspond to states on Mn(F) and trace (A∗AT ) = kA2k is equivalent to A∗AT = kAk2T .

States and orthogonality in C ∗-algebra Proofs and applications

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).

9 / 24 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.

Density matrices correspond to states on Mn(F) and trace (A∗AT ) = kA2k is equivalent to A∗AT = kAk2T .

States and orthogonality in C ∗-algebra Proofs and applications

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

9 / 24 Density matrices correspond to states on Mn(F) and trace (A∗AT ) = kA2k is equivalent to A∗AT = kAk2T .

States and orthogonality in C ∗-algebra Proofs and applications

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.

9 / 24 States and orthogonality in C ∗-algebra Proofs and applications

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.

Density matrices correspond to states on Mn(F) and trace (A∗AT ) = kA2k is equivalent to A∗AT = kAk2T .

9 / 24 States and orthogonality in C ∗-algebra Proofs and applications

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.

Density matrices correspond to states on Mn(F) and trace (A∗AT ) = kA2k is equivalent to A∗AT = kAk2T .

9 / 24 A related to question will be given a, b ∈ A, if a orthogonal to b, can we find a pur state φ such that φ(a∗a) = kak2 and φ(a∗b) = 0? (answer is still unknown)

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.

States and orthogonality in C ∗-algebra Proofs and applications 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 / 24 A related to question will be given a, b ∈ A, if a orthogonal to b, can we find a pur state φ such that φ(a∗a) = kak2 and φ(a∗b) = 0? (answer is still unknown)

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.

States and orthogonality in C ∗-algebra Proofs and applications 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 / 24 (answer is still unknown)

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.

States and orthogonality in C ∗-algebra Proofs and applications 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.

A related to question will be given a, b ∈ A, if a orthogonal to b, can we find a pur state φ such that φ(a∗a) = kak2 and φ(a∗b) = 0?

10 / 24 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.

States and orthogonality in C ∗-algebra Proofs and applications 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.

A related to question will be given a, b ∈ A, if a orthogonal to b, can we find a pur state φ such that φ(a∗a) = kak2 and φ(a∗b) = 0? (answer is still unknown)

10 / 24 States and orthogonality in C ∗-algebra Proofs and applications 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.

A related to question will be given a, b ∈ A, if a orthogonal to b, can we find a pur state φ such that φ(a∗a) = kak2 and φ(a∗b) = 0? (answer is still unknown)

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.

10 / 24 States and orthogonality in C ∗-algebra Proofs and applications 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.

A related to question will be given a, b ∈ A, if a orthogonal to b, can we find a pur state φ such that φ(a∗a) = kak2 and φ(a∗b) = 0? (answer is still unknown)

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.

10 / 24 States and orthogonality in C ∗-algebra Proofs and applications 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.

A related to question will be given a, b ∈ A, if a orthogonal to b, can we find a pur state φ such that φ(a∗a) = kak2 and φ(a∗b) = 0? (answer is still unknown)

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.

10 / 24 Then there exists φ ∈ SA such that ∗ 2 φ((a − λ01A) (a − λ01A)) = dist(a, C1A) and φ(a − λ01A) = 0. A generalization of this will be : Theorem (Grover P.; Singla S., 2021)

Let a ∈ A. Let B be a subspace of A. Let b0 be a best approximation to a in B. Then

2 ∗ ∗ ∗ ∗ dist(a, B) = max{φ(a a) − φ(b0b0): φ ∈ SA and φ(a b) = φ(b0b) for all b ∈ B}.

States and orthogonality in C ∗-algebra Proofs and applications Application to distance formulas

Theorem (Williams J. P., 1970) For a ∈ A, we have

2 ∗ 2 dist(a, C1A) = max{φ(a a) − |φ(a)| : φ ∈ SA}.

Proof. There exists λ0 ∈ C such that dist(a, C1A) = ka − λ01Ak.

11 / 24 A generalization of this will be : Theorem (Grover P.; Singla S., 2021)

Let a ∈ A. Let B be a subspace of A. Let b0 be a best approximation to a in B. Then

2 ∗ ∗ ∗ ∗ dist(a, B) = max{φ(a a) − φ(b0b0): φ ∈ SA and φ(a b) = φ(b0b) for all b ∈ B}.

States and orthogonality in C ∗-algebra Proofs and applications Application to distance formulas

Theorem (Williams J. P., 1970) For a ∈ A, we have

2 ∗ 2 dist(a, C1A) = max{φ(a a) − |φ(a)| : φ ∈ SA}.

Proof. 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 / 24 A generalization of this will be : Theorem (Grover P.; Singla S., 2021)

Let a ∈ A. Let B be a subspace of A. Let b0 be a best approximation to a in B. Then

2 ∗ ∗ ∗ ∗ dist(a, B) = max{φ(a a) − φ(b0b0): φ ∈ SA and φ(a b) = φ(b0b) for all b ∈ B}.

States and orthogonality in C ∗-algebra Proofs and applications Application to distance formulas

Theorem (Williams J. P., 1970) For a ∈ A, we have

2 ∗ 2 dist(a, C1A) = max{φ(a a) − |φ(a)| : φ ∈ SA}.

Proof. 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 / 24 States and orthogonality in C ∗-algebra Proofs and applications Application to distance formulas

Theorem (Williams J. P., 1970) For a ∈ A, we have

2 ∗ 2 dist(a, C1A) = max{φ(a a) − |φ(a)| : φ ∈ SA}.

Proof. 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. A generalization of this will be : Theorem (Grover P.; Singla S., 2021)

Let a ∈ A. Let B be a subspace of A. Let b0 be a best approximation to a in B. Then

2 ∗ ∗ ∗ ∗ dist(a, B) = max{φ(a a) − φ(b0b0): φ ∈ SA and φ(a b) = φ(b0b) for all b ∈ B}. 11 / 24 Let b0 be a best approximation to a in B. Then there exists φ ∈ SA such that ka − b0kφ = ka − b0k and ha − b0|biφ = 0 for all b ∈ B. Now there exists a cyclic representation (H, π, ξ) such that φ(c) = hπ(c)ξ|ξi for all c ∈ A. So kπ(a − b0)ξk = ka − b0k and hπ(a − b0)ξ|π(b)ξi = 0 for all 1 b ∈ B. Taking η = π(a − b0)ξ, we get the required result. ka−b0k

States and orthogonality in C ∗-algebra Proofs and applications A distance formulas in terms of representation

Theorem (Grover P.; Singla S., 2021) Let a ∈ A. Let B be a subspace of A. Suppose there is a best approximation to a in B. Then  dist(a, B) = max hπ(a)ξ|ηi :(H, π, ξ) is cyclic representation of A, η ∈ H, kηk = 1 and hπ(b)ξ|ηi = 0 for all b ∈ B .

Proof. Clearly RHS ≤ LHS.

12 / 24 Then there exists φ ∈ SA such that ka − b0kφ = ka − b0k and ha − b0|biφ = 0 for all b ∈ B. Now there exists a cyclic representation (H, π, ξ) such that φ(c) = hπ(c)ξ|ξi for all c ∈ A. So kπ(a − b0)ξk = ka − b0k and hπ(a − b0)ξ|π(b)ξi = 0 for all 1 b ∈ B. Taking η = π(a − b0)ξ, we get the required result. ka−b0k

States and orthogonality in C ∗-algebra Proofs and applications A distance formulas in terms of representation

Theorem (Grover P.; Singla S., 2021) Let a ∈ A. Let B be a subspace of A. Suppose there is a best approximation to a in B. Then  dist(a, B) = max hπ(a)ξ|ηi :(H, π, ξ) is cyclic representation of A, η ∈ H, kηk = 1 and hπ(b)ξ|ηi = 0 for all b ∈ B .

Proof. Clearly RHS ≤ LHS. Let b0 be a best approximation to a in B.

12 / 24 Now there exists a cyclic representation (H, π, ξ) such that φ(c) = hπ(c)ξ|ξi for all c ∈ A. So kπ(a − b0)ξk = ka − b0k and hπ(a − b0)ξ|π(b)ξi = 0 for all 1 b ∈ B. Taking η = π(a − b0)ξ, we get the required result. ka−b0k

States and orthogonality in C ∗-algebra Proofs and applications A distance formulas in terms of representation

Theorem (Grover P.; Singla S., 2021) Let a ∈ A. Let B be a subspace of A. Suppose there is a best approximation to a in B. Then  dist(a, B) = max hπ(a)ξ|ηi :(H, π, ξ) is cyclic representation of A, η ∈ H, kηk = 1 and hπ(b)ξ|ηi = 0 for all b ∈ B .

Proof. Clearly RHS ≤ LHS. Let b0 be a best approximation to a in B. Then there exists φ ∈ SA such that ka − b0kφ = ka − b0k and ha − b0|biφ = 0 for all b ∈ B.

12 / 24 So kπ(a − b0)ξk = ka − b0k and hπ(a − b0)ξ|π(b)ξi = 0 for all 1 b ∈ B. Taking η = π(a − b0)ξ, we get the required result. ka−b0k

States and orthogonality in C ∗-algebra Proofs and applications A distance formulas in terms of representation

Theorem (Grover P.; Singla S., 2021) Let a ∈ A. Let B be a subspace of A. Suppose there is a best approximation to a in B. Then  dist(a, B) = max hπ(a)ξ|ηi :(H, π, ξ) is cyclic representation of A, η ∈ H, kηk = 1 and hπ(b)ξ|ηi = 0 for all b ∈ B .

Proof. Clearly RHS ≤ LHS. Let b0 be a best approximation to a in B. Then there exists φ ∈ SA such that ka − b0kφ = ka − b0k and ha − b0|biφ = 0 for all b ∈ B. Now there exists a cyclic representation (H, π, ξ) such that φ(c) = hπ(c)ξ|ξi for all c ∈ A.

12 / 24 1 Taking η = π(a − b0)ξ, we get the required result. ka−b0k

States and orthogonality in C ∗-algebra Proofs and applications A distance formulas in terms of representation

Theorem (Grover P.; Singla S., 2021) Let a ∈ A. Let B be a subspace of A. Suppose there is a best approximation to a in B. Then  dist(a, B) = max hπ(a)ξ|ηi :(H, π, ξ) is cyclic representation of A, η ∈ H, kηk = 1 and hπ(b)ξ|ηi = 0 for all b ∈ B .

Proof. Clearly RHS ≤ LHS. Let b0 be a best approximation to a in B. Then there exists φ ∈ SA such that ka − b0kφ = ka − b0k and ha − b0|biφ = 0 for all b ∈ B. Now there exists a cyclic representation (H, π, ξ) such that φ(c) = hπ(c)ξ|ξi for all c ∈ A. So kπ(a − b0)ξk = ka − b0k and hπ(a − b0)ξ|π(b)ξi = 0 for all b ∈ B.

12 / 24 States and orthogonality in C ∗-algebra Proofs and applications A distance formulas in terms of representation

Theorem (Grover P.; Singla S., 2021) Let a ∈ A. Let B be a subspace of A. Suppose there is a best approximation to a in B. Then  dist(a, B) = max hπ(a)ξ|ηi :(H, π, ξ) is cyclic representation of A, η ∈ H, kηk = 1 and hπ(b)ξ|ηi = 0 for all b ∈ B .

Proof. Clearly RHS ≤ LHS. Let b0 be a best approximation to a in B. Then there exists φ ∈ SA such that ka − b0kφ = ka − b0k and ha − b0|biφ = 0 for all b ∈ B. Now there exists a cyclic representation (H, π, ξ) such that φ(c) = hπ(c)ξ|ξi for all c ∈ A. So kπ(a − b0)ξk = ka − b0k and hπ(a − b0)ξ|π(b)ξi = 0 for all 1 b ∈ B. Taking η = π(a − b0)ξ, we get the required result. ka−b0k

12 / 24 Then

2 ∗ ∗ −1 dist(a, B) = max{trace (A AP − CB(AP) CB(AP)CB(P) : P ≥ 0, trace (P) = 1}.

Theorem ∗ Let a ∈ A. Let B be C -subalgebra of A containing 1A. Then

2 ∗ ∗ dist(a, B) ≥ sup{φ(E(a a) − E(a) E(a)) : φ ∈ SA, E is a conditional expectation from A to B}.

We know equality occurs when B = C1A and when B is central finite dimensional subspace but for general C ∗-subalgebra, we still don’t know.

States and orthogonality in C ∗-algebra Proofs and applications A distance formulas in terms of conditional expectation

Theorem (P. Grover, 2014) ∗ Let B be C -subalgebra of (C) containing idenity matrix. Let CB be orthogonal projection of Mn(C) onto B.

13 / 24 Theorem ∗ Let a ∈ A. Let B be C -subalgebra of A containing 1A. Then

2 ∗ ∗ dist(a, B) ≥ sup{φ(E(a a) − E(a) E(a)) : φ ∈ SA, E is a conditional expectation from A to B}.

We know equality occurs when B = C1A and when B is central finite dimensional subspace but for general C ∗-subalgebra, we still don’t know.

States and orthogonality in C ∗-algebra Proofs and applications A distance formulas in terms of conditional expectation

Theorem (P. Grover, 2014) ∗ Let B be C -subalgebra of (C) containing idenity matrix. Let CB be orthogonal projection of Mn(C) onto B. Then

2 ∗ ∗ −1 dist(a, B) = max{trace (A AP − CB(AP) CB(AP)CB(P) : P ≥ 0, trace (P) = 1}.

13 / 24 Then

2 ∗ ∗ dist(a, B) ≥ sup{φ(E(a a) − E(a) E(a)) : φ ∈ SA, E is a conditional expectation from A to B}.

We know equality occurs when B = C1A and when B is central finite dimensional subspace but for general C ∗-subalgebra, we still don’t know.

States and orthogonality in C ∗-algebra Proofs and applications A distance formulas in terms of conditional expectation

Theorem (P. Grover, 2014) ∗ Let B be C -subalgebra of (C) containing idenity matrix. Let CB be orthogonal projection of Mn(C) onto B. Then

2 ∗ ∗ −1 dist(a, B) = max{trace (A AP − CB(AP) CB(AP)CB(P) : P ≥ 0, trace (P) = 1}.

Theorem ∗ Let a ∈ A. Let B be C -subalgebra of A containing 1A.

13 / 24 We know equality occurs when B = C1A and when B is central finite dimensional subspace but for general C ∗-subalgebra, we still don’t know.

States and orthogonality in C ∗-algebra Proofs and applications A distance formulas in terms of conditional expectation

Theorem (P. Grover, 2014) ∗ Let B be C -subalgebra of (C) containing idenity matrix. Let CB be orthogonal projection of Mn(C) onto B. Then

2 ∗ ∗ −1 dist(a, B) = max{trace (A AP − CB(AP) CB(AP)CB(P) : P ≥ 0, trace (P) = 1}.

Theorem ∗ Let a ∈ A. Let B be C -subalgebra of A containing 1A. Then

2 ∗ ∗ dist(a, B) ≥ sup{φ(E(a a) − E(a) E(a)) : φ ∈ SA, E is a conditional expectation from A to B}.

13 / 24 and when B is central finite dimensional subspace but for general C ∗-subalgebra, we still don’t know.

States and orthogonality in C ∗-algebra Proofs and applications A distance formulas in terms of conditional expectation

Theorem (P. Grover, 2014) ∗ Let B be C -subalgebra of (C) containing idenity matrix. Let CB be orthogonal projection of Mn(C) onto B. Then

2 ∗ ∗ −1 dist(a, B) = max{trace (A AP − CB(AP) CB(AP)CB(P) : P ≥ 0, trace (P) = 1}.

Theorem ∗ Let a ∈ A. Let B be C -subalgebra of A containing 1A. Then

2 ∗ ∗ dist(a, B) ≥ sup{φ(E(a a) − E(a) E(a)) : φ ∈ SA, E is a conditional expectation from A to B}.

We know equality occurs when B = C1A

13 / 24 States and orthogonality in C ∗-algebra Proofs and applications A distance formulas in terms of conditional expectation

Theorem (P. Grover, 2014) ∗ Let B be C -subalgebra of (C) containing idenity matrix. Let CB be orthogonal projection of Mn(C) onto B. Then

2 ∗ ∗ −1 dist(a, B) = max{trace (A AP − CB(AP) CB(AP)CB(P) : P ≥ 0, trace (P) = 1}.

Theorem ∗ Let a ∈ A. Let B be C -subalgebra of A containing 1A. Then

2 ∗ ∗ dist(a, B) ≥ sup{φ(E(a a) − E(a) E(a)) : φ ∈ SA, E is a conditional expectation from A to B}.

We know equality occurs when B = C1A and when B is central finite dimensional subspace but for general C ∗-subalgebra, we still don’t know. 13 / 24 States and orthogonality in C ∗-algebra Proofs and applications A distance formulas in terms of conditional expectation

Theorem (P. Grover, 2014) ∗ Let B be C -subalgebra of (C) containing idenity matrix. Let CB be orthogonal projection of Mn(C) onto B. Then

2 ∗ ∗ −1 dist(a, B) = max{trace (A AP − CB(AP) CB(AP)CB(P) : P ≥ 0, trace (P) = 1}.

Theorem ∗ Let a ∈ A. Let B be C -subalgebra of A containing 1A. Then

2 ∗ ∗ dist(a, B) ≥ sup{φ(E(a a) − E(a) E(a)) : φ ∈ SA, E is a conditional expectation from A to B}.

We know equality occurs when B = C1A and when B is central finite dimensional subspace but for general C ∗-subalgebra, we still don’t know. 13 / 24 States and orthogonality in C ∗-algebra Proofs and applications Table of Contents

∗ 1 States and orthogonality in C -algebra

2 Proofs and applications

14 / 24 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.

States and orthogonality in C ∗-algebra Proofs and applications Birkhoff-James orthogonality as a calculus problem

We consider the function f (λ) = ka + λbk mapping C into R+.

15 / 24 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.

States and orthogonality in C ∗-algebra Proofs and applications 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.

15 / 24 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.

States and orthogonality in C ∗-algebra Proofs and applications 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.

15 / 24 and we will see characterization of orthogonality in terms of Gateaux derivative.

States and orthogonality in C ∗-algebra Proofs and applications 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

15 / 24 States and orthogonality in C ∗-algebra Proofs and applications 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.

15 / 24 States and orthogonality in C ∗-algebra Proofs and applications 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.

15 / 24 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.

States and orthogonality in C ∗-algebra Proofs and applications 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.

16 / 24 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.

States and orthogonality in C ∗-algebra Proofs and applications 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

16 / 24 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.

States and orthogonality in C ∗-algebra Proofs and applications 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.

16 / 24 is called the φ-Gateaux derivative of the norm at the vector x, in the y and φ directions.

States and orthogonality in C ∗-algebra Proofs and applications 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

16 / 24 States and orthogonality in C ∗-algebra Proofs and applications 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.

16 / 24 States and orthogonality in C ∗-algebra Proofs and applications 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.

16 / 24 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.

States and orthogonality in C ∗-algebra Proofs and applications Proof using tools of convex analysis

Lemma (Singla S., 2021)

Let a, b ∈ A. Then

ka + tbk − kak 1 ka∗a + ta∗bk − ka∗ak lim = lim . t→0+ t kak t→0+ t

17 / 24 ∗ 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.

States and orthogonality in C ∗-algebra Proofs and applications Proof using tools of convex analysis

Lemma (Singla S., 2021)

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

17 / 24 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.

States and orthogonality in C ∗-algebra Proofs and applications Proof using tools of convex analysis

Lemma (Singla S., 2021)

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

17 / 24 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.

States and orthogonality in C ∗-algebra Proofs and applications Proof using tools of convex analysis

Lemma (Singla S., 2021)

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.

17 / 24 And φ is required state using fact it attains its norm at a non-zero positive element.

States and orthogonality in C ∗-algebra Proofs and applications Proof using tools of convex analysis

Lemma (Singla S., 2021)

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.

17 / 24 States and orthogonality in C ∗-algebra Proofs and applications Proof using tools of convex analysis

Lemma (Singla S., 2021)

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.

17 / 24 States and orthogonality in C ∗-algebra Proofs and applications Proof using tools of convex analysis

Lemma (Singla S., 2021)

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.

17 / 24 Theorem (Singla S., 2021)

Let a, b ∈ A. Then we have

ka + tbk − kak 1 ∗ ∗ 2 lim = max{Reφ(a b): φ ∈ SA, φ(a a) = kak }. t→0+ t kak

Corollary. For A, B ∈ K (H), we have kA + tBk − kAk 1 lim = max RehAu|Bui. t→0+ t kAk kuk=1,A∗Au=kAk2u

States and orthogonality in C ∗-algebra Proofs and applications Expression for Gateuax derivative

For a normed space V and v, u ∈ V , we have kv + tuk − kvk lim = max{Re f (u): f ∈ V ∗, f (v) = kvk, kf k = 1}. t→0+ t

18 / 24 Corollary. For A, B ∈ K (H), we have kA + tBk − kAk 1 lim = max RehAu|Bui. t→0+ t kAk kuk=1,A∗Au=kAk2u

States and orthogonality in C ∗-algebra Proofs and applications Expression for Gateuax derivative

For a normed space V and v, u ∈ V , we have kv + tuk − kvk lim = max{Re f (u): f ∈ V ∗, f (v) = kvk, kf k = 1}. t→0+ t

Theorem (Singla S., 2021)

Let a, b ∈ A. Then we have

ka + tbk − kak 1 ∗ ∗ 2 lim = max{Reφ(a b): φ ∈ SA, φ(a a) = kak }. t→0+ t kak

18 / 24 States and orthogonality in C ∗-algebra Proofs and applications Expression for Gateuax derivative

For a normed space V and v, u ∈ V , we have kv + tuk − kvk lim = max{Re f (u): f ∈ V ∗, f (v) = kvk, kf k = 1}. t→0+ t

Theorem (Singla S., 2021)

Let a, b ∈ A. Then we have

ka + tbk − kak 1 ∗ ∗ 2 lim = max{Reφ(a b): φ ∈ SA, φ(a a) = kak }. t→0+ t kak

Corollary. For A, B ∈ K (H), we have kA + tBk − kAk 1 lim = max RehAu|Bui. t→0+ t kAk kuk=1,A∗Au=kAk2u

18 / 24 It is a general fact that v is a smooth point of the unit ball of v if kv + tuk − kvk and only if lim exists and in this case, it is equal t→0 t to Re Fv (u). Now using these facts and expression for Gateaux derivative of norm for K (H), we get a characterization of smooth points of K (H).

States and orthogonality in C ∗-algebra Proofs and applications Smooth points

We say that a vector v of norm one is a smooth point of the unit ball of V if there exists a unique functional Fv , called the support functional, such that kFv k = 1 and Fv (v) = 1.

19 / 24 and in this case, it is equal

to Re Fv (u). Now using these facts and expression for Gateaux derivative of norm for K (H), we get a characterization of smooth points of K (H).

States and orthogonality in C ∗-algebra Proofs and applications Smooth points

We say that a vector v of norm one is a smooth point of the unit ball of V if there exists a unique functional Fv , called the support functional, such that kFv k = 1 and Fv (v) = 1. It is a general fact that v is a smooth point of the unit ball of v if kv + tuk − kvk and only if lim exists t→0 t

19 / 24 Now using these facts and expression for Gateaux derivative of norm for K (H), we get a characterization of smooth points of K (H).

States and orthogonality in C ∗-algebra Proofs and applications Smooth points

We say that a vector v of norm one is a smooth point of the unit ball of V if there exists a unique functional Fv , called the support functional, such that kFv k = 1 and Fv (v) = 1. It is a general fact that v is a smooth point of the unit ball of v if kv + tuk − kvk and only if lim exists and in this case, it is equal t→0 t to Re Fv (u).

19 / 24 States and orthogonality in C ∗-algebra Proofs and applications Smooth points

We say that a vector v of norm one is a smooth point of the unit ball of V if there exists a unique functional Fv , called the support functional, such that kFv k = 1 and Fv (v) = 1. It is a general fact that v is a smooth point of the unit ball of v if kv + tuk − kvk and only if lim exists and in this case, it is equal t→0 t to Re Fv (u). Now using these facts and expression for Gateaux derivative of norm for K (H), we get a characterization of smooth points of K (H).

19 / 24 States and orthogonality in C ∗-algebra Proofs and applications Smooth points

We say that a vector v of norm one is a smooth point of the unit ball of V if there exists a unique functional Fv , called the support functional, such that kFv k = 1 and Fv (v) = 1. It is a general fact that v is a smooth point of the unit ball of v if kv + tuk − kvk and only if lim exists and in this case, it is equal t→0 t to Re Fv (u). Now using these facts and expression for Gateaux derivative of norm for K (H), we get a characterization of smooth points of K (H).

19 / 24 Theorem (Holub J. R., 1973) An operator A is a smooth point of the unit ball of B(H) if and only if A attains its norm at a unit vector h such that sup kAxk < kAk. In that case, x⊥h,kxk=1 kA + tBk − kAk lim = RehAh|Bhi. t→0 t

The same result holds true for smooth points of unit ball of B(H). One of the proof can be done by modifying proofs for finding Gateaux derivative in C ∗-algebra when a ∈ I for a two sided ideal I . And we have also been able to find such a formula under the condition dist(a, I) < kak and using tools of M-ideals theory.

States and orthogonality in C ∗-algebra Proofs and applications Smooth points in K (H) and B(H) kA + tBk − kAk 1 Using lim = max RehAu|Bui, t→0+ t kAk kuk=1,A∗Au=kAk2u we get

20 / 24 such that sup kAxk < kAk. In that case, x⊥h,kxk=1 kA + tBk − kAk lim = RehAh|Bhi. t→0 t

The same result holds true for smooth points of unit ball of B(H). One of the proof can be done by modifying proofs for finding Gateaux derivative in C ∗-algebra when a ∈ I for a two sided ideal I . And we have also been able to find such a formula under the condition dist(a, I) < kak and using tools of M-ideals theory.

States and orthogonality in C ∗-algebra Proofs and applications Smooth points in K (H) and B(H) kA + tBk − kAk 1 Using lim = max RehAu|Bui, t→0+ t kAk kuk=1,A∗Au=kAk2u we get Theorem (Holub J. R., 1973) An operator A is a smooth point of the unit ball of B(H) if and only if A attains its norm at a unit vector h

20 / 24 In that case,

kA + tBk − kAk lim = RehAh|Bhi. t→0 t

The same result holds true for smooth points of unit ball of B(H). One of the proof can be done by modifying proofs for finding Gateaux derivative in C ∗-algebra when a ∈ I for a two sided ideal I . And we have also been able to find such a formula under the condition dist(a, I) < kak and using tools of M-ideals theory.

States and orthogonality in C ∗-algebra Proofs and applications Smooth points in K (H) and B(H) kA + tBk − kAk 1 Using lim = max RehAu|Bui, t→0+ t kAk kuk=1,A∗Au=kAk2u we get Theorem (Holub J. R., 1973) An operator A is a smooth point of the unit ball of B(H) if and only if A attains its norm at a unit vector h such that sup kAxk < kAk. x⊥h,kxk=1

20 / 24 The same result holds true for smooth points of unit ball of B(H). One of the proof can be done by modifying proofs for finding Gateaux derivative in C ∗-algebra when a ∈ I for a two sided ideal I . And we have also been able to find such a formula under the condition dist(a, I) < kak and using tools of M-ideals theory.

States and orthogonality in C ∗-algebra Proofs and applications Smooth points in K (H) and B(H) kA + tBk − kAk 1 Using lim = max RehAu|Bui, t→0+ t kAk kuk=1,A∗Au=kAk2u we get Theorem (Holub J. R., 1973) An operator A is a smooth point of the unit ball of B(H) if and only if A attains its norm at a unit vector h such that sup kAxk < kAk. In that case, x⊥h,kxk=1 kA + tBk − kAk lim = RehAh|Bhi. t→0 t

20 / 24 One of the proof can be done by modifying proofs for finding Gateaux derivative in C ∗-algebra when a ∈ I for a two sided ideal I . And we have also been able to find such a formula under the condition dist(a, I) < kak and using tools of M-ideals theory.

States and orthogonality in C ∗-algebra Proofs and applications Smooth points in K (H) and B(H) kA + tBk − kAk 1 Using lim = max RehAu|Bui, t→0+ t kAk kuk=1,A∗Au=kAk2u we get Theorem (Holub J. R., 1973) An operator A is a smooth point of the unit ball of B(H) if and only if A attains its norm at a unit vector h such that sup kAxk < kAk. In that case, x⊥h,kxk=1 kA + tBk − kAk lim = RehAh|Bhi. t→0 t

The same result holds true for smooth points of unit ball of B(H).

20 / 24 And we have also been able to find such a formula under the condition dist(a, I) < kak and using tools of M-ideals theory.

States and orthogonality in C ∗-algebra Proofs and applications Smooth points in K (H) and B(H) kA + tBk − kAk 1 Using lim = max RehAu|Bui, t→0+ t kAk kuk=1,A∗Au=kAk2u we get Theorem (Holub J. R., 1973) An operator A is a smooth point of the unit ball of B(H) if and only if A attains its norm at a unit vector h such that sup kAxk < kAk. In that case, x⊥h,kxk=1 kA + tBk − kAk lim = RehAh|Bhi. t→0 t

The same result holds true for smooth points of unit ball of B(H). One of the proof can be done by modifying proofs for finding Gateaux derivative in C ∗-algebra when a ∈ I for a two sided ideal I .

20 / 24 States and orthogonality in C ∗-algebra Proofs and applications Smooth points in K (H) and B(H) kA + tBk − kAk 1 Using lim = max RehAu|Bui, t→0+ t kAk kuk=1,A∗Au=kAk2u we get Theorem (Holub J. R., 1973) An operator A is a smooth point of the unit ball of B(H) if and only if A attains its norm at a unit vector h such that sup kAxk < kAk. In that case, x⊥h,kxk=1 kA + tBk − kAk lim = RehAh|Bhi. t→0 t

The same result holds true for smooth points of unit ball of B(H). One of the proof can be done by modifying proofs for finding Gateaux derivative in C ∗-algebra when a ∈ I for a two sided ideal I . And we have also been able to find such a formula under the condition dist(a, I) < kak and using tools of M-ideals theory. 20 / 24 States and orthogonality in C ∗-algebra Proofs and applications Smooth points in K (H) and B(H) kA + tBk − kAk 1 Using lim = max RehAu|Bui, t→0+ t kAk kuk=1,A∗Au=kAk2u we get Theorem (Holub J. R., 1973) An operator A is a smooth point of the unit ball of B(H) if and only if A attains its norm at a unit vector h such that sup kAxk < kAk. In that case, x⊥h,kxk=1 kA + tBk − kAk lim = RehAh|Bhi. t→0 t

The same result holds true for smooth points of unit ball of B(H). One of the proof can be done by modifying proofs for finding Gateaux derivative in C ∗-algebra when a ∈ I for a two sided ideal I . And we have also been able to find such a formula under the condition dist(a, I) < kak and using tools of M-ideals theory. 20 / 24 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.

States and orthogonality in C ∗-algebra Proofs and applications 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.

21 / 24 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.

States and orthogonality in C ∗-algebra Proofs and applications 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,

21 / 24 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.

States and orthogonality in C ∗-algebra Proofs and applications 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.

21 / 24 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.

States and orthogonality in C ∗-algebra Proofs and applications 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.

21 / 24 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.

States and orthogonality in C ∗-algebra Proofs and applications 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.

21 / 24 States and orthogonality in C ∗-algebra Proofs and applications 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.

21 / 24 States and orthogonality in C ∗-algebra Proofs and applications 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.

21 / 24 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.

States and orthogonality in C ∗-algebra Proofs and applications 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.

22 / 24 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.

States and orthogonality in C ∗-algebra Proofs and applications 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.

22 / 24 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.

States and orthogonality in C ∗-algebra Proofs and applications 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

22 / 24 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.

States and orthogonality in C ∗-algebra Proofs and applications 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

22 / 24 8 Define φ ∈ A∗ as φ(c) = hξ|π(c)ξi.

States and orthogonality in C ∗-algebra Proofs and applications 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.

22 / 24 States and orthogonality in C ∗-algebra Proofs and applications 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.

22 / 24 States and orthogonality in C ∗-algebra Proofs and applications 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.

22 / 24 States and orthogonality in C ∗-algebra Proofs and applications References

5 Bhatia R.; Semrlˇ P. : Orthogonality of matrices and some distance problems. Linear Algebra Appl. 287 (1999), 77–85. 6 Holub J. R. : On the metric geometry of ideals of operators on Hilbert space. Math. Ann. 201 (1973), 157–163. 7 Rieffel M. A. : Leibniz seminorms and best approximation from C ∗-subalgebras. Sci. China Math. 54 (2011), 2259–2274. 8 Singer I. : Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces. Springer-Verlag, Berlin, 1970. 9 Williams J. P. : Finite operators. Proc. Amer. Math. Soc. 26 (1970), 129-136.

23 / 24 States and orthogonality in C ∗-algebra Proofs and applications References

1 James R. C. : Orthogonality and linear functionals in normed linear spaces. Trans. Amer. Math. Soc. 61 (1947), 265–292. 2 Grover P. : Orthogonality to matrix subspaces, and a distance formula. Linear Algebra Appl. 445 (2014), 280–288. 3 Grover P. ; Singla S. : Best Approximations, distance formulas and orthogonality in C ∗-algebras. J. Ramanujan Math. Soc. 36 (2021), 85–91. 4 Grover P. ; Singla S. : Birkhoff-James orthogonality and applications : A survey. Operator Theory, Functional Analysis and Applications, Birkh¨auser,Springer, vol. 282, 2021. 5 Singla S. : Gateaux derivative of C ∗ norm. communicated.

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