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 2 A. Then there exists φ 2 SA such that φ(a a) = kak . 2 ∗ This can be rephrased as : dist(a; f0g) = maxfφ(a a): φ 2 SAg. A positive functional φ gives a semi inner product on A defined as ∗ hajbiφ = φ(a b). The above theorem can be rephrased as - 2 There exists φ 2 S(A) such that hajaiφ = dist(a; f0g) . 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; f0g) = maxfφ(a a): φ 2 SAg. A positive functional φ gives a semi inner product on A defined as ∗ hajbiφ = φ(a b). The above theorem can be rephrased as - 2 There exists φ 2 S(A) such that hajaiφ = dist(a; f0g) . 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 2 A. Then there exists φ 2 SA such that φ(a a) = kak . 4 / 24 A positive functional φ gives a semi inner product on A defined as ∗ hajbiφ = φ(a b). The above theorem can be rephrased as - 2 There exists φ 2 S(A) such that hajaiφ = dist(a; f0g) . 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 2 A. Then there exists φ 2 SA such that φ(a a) = kak . 2 ∗ This can be rephrased as : dist(a; f0g) = maxfφ(a a): φ 2 SAg. 4 / 24 The above theorem can be rephrased as - 2 There exists φ 2 S(A) such that hajaiφ = dist(a; f0g) . 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 2 A. Then there exists φ 2 SA such that φ(a a) = kak . 2 ∗ This can be rephrased as : dist(a; f0g) = maxfφ(a a): φ 2 SAg. A positive functional φ gives a semi inner product on A defined as ∗ hajbiφ = φ(a b). 4 / 24 2 There exists φ 2 S(A) such that hajaiφ = dist(a; f0g) . 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 2 A. Then there exists φ 2 SA such that φ(a a) = kak . 2 ∗ This can be rephrased as : dist(a; f0g) = maxfφ(a a): φ 2 SAg. A positive functional φ gives a semi inner product on A defined as ∗ hajbiφ = 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 2 A. Then there exists φ 2 SA such that φ(a a) = kak . 2 ∗ This can be rephrased as : dist(a; f0g) = maxfφ(a a): φ 2 SAg. A positive functional φ gives a semi inner product on A defined as ∗ hajbiφ = φ(a b). The above theorem can be rephrased as - 2 There exists φ 2 S(A) such that hajaiφ = dist(a; f0g) . 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 2 A. Then there exists φ 2 SA such that φ(a a) = kak . 2 ∗ This can be rephrased as : dist(a; f0g) = maxfφ(a a): φ 2 SAg. A positive functional φ gives a semi inner product on A defined as ∗ hajbiφ = φ(a b). The above theorem can be rephrased as - 2 There exists φ 2 S(A) such that hajaiφ = dist(a; f0g) . 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 2 A. Then there exists φ 2 SA such that φ(a a) = kak . 2 ∗ This can be rephrased as : dist(a; f0g) = maxfφ(a a): φ 2 SAg. A positive functional φ gives a semi inner product on A defined as ∗ hajbiφ = φ(a b). The above theorem can be rephrased as - 2 There exists φ 2 S(A) such that hajaiφ = dist(a; f0g) . We generalize this for any subspace B, when a best approximation to a in B exists. 4 / 24 Definition An element w0 2 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 2 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 2 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 2 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 .