Maps which preserve equality of distance. Must they be linear?

Transcription

1 Maps which preserve equality of distance. Must they be linear? Colloquium Fullerton College Bernard Russo University of California, Irvine November 16, 2017 Bernard Russo (UCI) Maps which preserve equality of distance. Must they be linear? 1 / 10

2 Part 1: Tingley 1987 (motivation) Isometries of the unit sphere Part 2: Mankiewicz 1972 (used by Tingley and Tanaka) On extension of isometries in normed linear spaces Part 3: Tanaka 2016, 2017 (convex subsets of unit sphere) The solution of Tingley s problem for the operator norm unit sphere of complex n n matrices Part 4: So 1990 (faces in the unit ball) Facial structures of Schatten p-norms Part 5: Hatori-Molnar 2014 (isometries of unitaries) Isometries of the unitary groups in C*-algebras Part 6: Tanaka 2017 (main result) Spherical isometries of finite dimensional C*-algebras Part 7: Polo-Peralta 2017 (survey) On the extension of isometries between the unit spheres of von Neumann algebras Bernard Russo (UCI) Maps which preserve equality of distance. Must they be linear? 2 / 10

3 Part 1 Tingley 1987 (motivation) Notation If X is a Banach space with norm, its unit ball and unit sphere are B = B(X ) = {x X : x 1} S = S(X ) = {x X : x = 1} Mazur-Ulam 1932 If f ; X X is a surjective isometry (not assumed linear), then f is linear (or affine) Mankiewicz 1972 If f : B B is a surjective isometry, then f extends to a linear (or affine) surjective isometry from X to X. Bernard Russo (UCI) Maps which preserve equality of distance. Must they be linear? 3 / 10

4 Example 1 An isometry that is not linear or affine (Hint: it is not onto): X = R, X = R 2, (x, y) = max{ x, y }, f (x) = (x, x) if x 0, f (x) = (x, x) if x < 0. Question (Tingley s problem) If f : S S is a surjective isometry, is f the restriction to S of a linear (or affine) transformation? Not known at this time even for dimension 2. Theorem (Tingley 1987) Suppose that S and S are the unit spheres of finite dimensional Banach spaces X and X. If f : S S is a surjective isometry, then f ( x) = f (x) for all x S Example 2 X = R, x = x, S = { 1, 1} There are four functions f : S S, two of which are surjective. To make progress, give X some more structure. Bernard Russo (UCI) Maps which preserve equality of distance. Must they be linear? 4 / 10

5 Part 2 Mankiewicz 1972 (used by Tingley and Tanaka) Theorem 1 Let V be an open non-empty connected subset of X and let W be an open subset of Y. Then every isometry T of V onto W can be uniquely extended to an affine isometry of X onto Y. (see the example below) Theorem 2 Let X and Y be normed linear spaces, and let V be a convex body in X and W a convex body in Y. Then every isometry T of V onto W can be uniquely extended to an affine isometry from X onto Y. A convex body is a closed convex set with non-empty interior. Example surjective isometry on open set, not linear In R 2 with the max norm, let V = {(x, y) R 2 : (x, y) < 1} {(x, y) R 2 : (x, y) (4, 0) < 1) and define f : V V by f (x, y) = (x, y) if (x, y) {(x, y) R 2 : (x, y) < 1} and f (x, y) = (x, y) if (x, y) {(x, y) R 2 : (x, y) (4, 0) < 1). Bernard Russo (UCI) Maps which preserve equality of distance. Must they be linear? 5 / 10

6 Part 3 Tanaka 2016, 2017 (convex subsets of unit sphere) Lemma 1 Let X be a Banach space. Suppose that C is a maximal convex subset of the unit sphere S(X ) of X. Then C is a norm exposed face of B(X ). Definitions Convex set, extreme point, face, exposed point, exposed face Lemma 2 Let X and Y be Banach spaces, and let T : S(X ) S(Y ) be a surjective isometry. Then C is a maximal convex subset of S(X ) if and only if T (C) is that of S(Y ). So by Lemma 1, faces are mapped into faces. Bernard Russo (UCI) Maps which preserve equality of distance. Must they be linear? 6 / 10

7 Part 4 So 1990 (faces in the unit ball) Theorem F is a proper closed face of B(M n (C)) if and only if there exists 1 r n and unitary matrices U, V M n (C) such that [ ] Ir 0 F = {U V : A B(M 0 A n r (C)} In other words, Every closed face of B(M n (C)) is associated with a unique partial isometry v M n (C) such that F = v + (1 vv )B(M n (C))(1 v v) = {a M n (C) : av = vv }. This is true for weakly closed faces of the unit ball of a von Neumann algebra. Much more is true (JBW*-triples) Bernard Russo (UCI) Maps which preserve equality of distance. Must they be linear? 7 / 10

8 Part 5 Hatori-Molnar 2014 (isometries of unitaries) Theorem Let U(n) be the set of unitary n n matrices. The map φ : U(n) U(n) is a surjective isometry if and only if there is a unitary w U(n) such that either φ(a) = φ(1)waw for all a U(n) or φ(a) = φ(1)wa t w for all a U(n). More generally, Theorem Let M j br a von Neumann algebra, and U j its unitary group, j = 1, 2. The map φ : U 1 U 2 is a surjective isometry if and only if there is a central projection p M 2 and a Jordan *-isomorphism J : M 1 M 2 such that φ is of the form φ(a) = φ(1)(pj(a) + (1 p)j(a) ), a U 1 Bernard Russo (UCI) Maps which preserve equality of distance. Must they be linear? 8 / 10

9 Part 6 Tanaka 2017 (main result) Lemma Let A 1 and A 2 be finite dimensional C*-algebras. Suppose that T : S(A 1 ) S(A 2 ) is a surjective isometry. Then T is locally affine, that is, if a, b, ta + (1 t)b S(A 1 ) for some t (0, 1), then sa + (1 s)b S(A 1 ) for all s [0, 1], and T (sa + (1 s)b) = st (a) + (1 s)t (b), s [0, 1]. Theorem Let A 1 and A 2 be finite dimensional C*-algebras. Suppose that T : S(A 1 ) S(A 2 ) is a surjective isometry. Then there is a central projection p A 2 and a Jordan *-homomorphism J : A 1 A 2 such that T (a) = T (1)(pJ(a) + (1 p)j(a) ) for each a A 1. Bernard Russo (UCI) Maps which preserve equality of distance. Must they be linear? 9 / 10

10 Part 7 Polo-Peralta 2017 (survey) Theorem Let f : S(M) S(N) be a surjective isometry between the unit spheres of two von Neumann algebras. Then there is a surjective real-linear isometry T : M N whose restriction to S(M) is f. Theorem (La piéce de résistance) Let f : S(A) S(B) be a surjective isometry between the unit spheres of two atomic JBW*-triples. Then there is a surjective real-linear isometry T : A B whose restriction to S(A) is f. Bernard Russo (UCI) Maps which preserve equality of distance. Must they be linear? 10 / 10