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 2 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 2 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 2 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 2 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 2 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 2 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 2 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 2 W . 5 / 23 Now, kvk ≤ kv − wk for all w 2 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 2 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 2 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 2 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 2 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 2 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 2 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 2 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 2 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 2 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 2 W is same as dist(v; W ) = kvk.