Generalized gyrovector spaces and the positive cone of Contact Information: ∗ Faculty of Engineering a unital C -algebra Niigata University 8050 2no-cho ikarashi Nishi-ku Niigata-shi, Niigata- ken, Japan Toshikazu Abe Phone: +81 (25) 262 7469 Niigata University, Faculty of Engineering Email: [email protected]

Abstract Definition 1. A magma (G, ⊕) is called a gyrogroup if there exists an Note that a real normed space (V, +, ×) is a GGV with ϕ = idV. Results The concept of generalized gyrovecor space (GGV, for short) is a com- element e such that the binary operation ⊕ satisfies the following (G1) On a GGV, we can define the gyromidpoint. It is a algebraic mid- mon generalization of the concept of real normed spaces and of the gyrovector spaces. The addition of a GGV is not necessarily a commutative group but is a to (G5). point on the GGV. On a real normed space, the gyromidpoint is the A Mazur-Ulam Theorem for GGV’s gyrocommutative gyrogroup. A typical example of GGV’s is the positive cone ∀ ∈ ⊕ arithmetric mean. ∗ (G1) a G, e a = a. of a unital C -algebra. (G, ⊕, ⊗) p(a, b) We have a Mazur-Ulam type theorem for GGV’s as follows. This is a (G2) ∀a ∈ G, ∃ ⊖ a s.t. ⊖a ⊕ a = e. Definition 4. Let be a GGV. The gyromidpoint of a, b ∈ (G, ⊕, ⊗) is defined as p(a, b) = 1 ⊗ (a ⊞ b), where ⊞ is the generalization of the celebrated Mazur-Ulam theorem. A proof of this ∀ ∈ ∃ ∈ ⊕ ⊕ ⊕ ⊕ 2 (G3) a, b, c G, ! gyr[a, b]c G s.t. a (b c) = (a b) gyr[a, b]c. gyrogroup coaddition of the gyrogroup (G, ⊕). theorem is a modification of the proof of the Mazur-Ulam theorem due Introduction (G4) For any a, b ∈ G, the map gyr[a, b]: G → G given by c 7→ On a GGV, we can define the gyrometric. It is a ”metric” structure on to Vais¨ al¨ a.¨ gyr[a, b]c, is an automorphism of a magma (G, ⊕), which is called the GGV. (In actuality, it isn’t necessarily a metric.) On a real normed ⊕ ⊗ ⊕ ⊗ The concept of the gyrocommutative gyrogroup is defined by A. A. Theorem 7. Let (G1, 1, 1) and (G2, 2, 2) be GGV’s. Let T : the gyroautomorphism of G generated by a, b ∈ G. space, the gyrometric is the metric induced by its norm. → Ungar. Gyrocommutative gyrogroups are generalzed groups, which is G1 G2 be a surjection. If T is a gyrometric preserving map, then T ∀ ∈ ⊕ ⊕ ⊗ ∥ ⊖ ∥ not necessarily commutative nor associative. A typical example of the (G5) a, b G, gyr[a b, b] = gyr[a, b]. Definition 5. Let (G, , ) be a GGV. Let ϱ(a, b) = ϕ(a b) for preserves the gyromidpoints. That is ∈ ⊕ ⊗ gyrocommutative gyrogroup is the addition of the admissible velocities A gyrogroup is gyrocommutative if the following (G6) is satisfied. all a, b G. We call ϱ the gyrometric on (G, , ). ϱ (a, b) = ϱ (T a,T b)(∀a, b ∈ G ) of the Einstein’s special relativity. It is called the Einstein gyrogroup. ∀a, b ∈ G a ⊕ b = gyr[a, b](b ⊕ a) On a GGV, we can define the gyroline. On a real normed space, the 1 2 1 (G6) , . ⇒ ∀ ∈ Other typical example is the complex open unit disc D. Define the gyroline is the line. p(T a,T b) = T (p(a, b)) ( a, b G1). Note that e is a identity element of the magma (G, ⊕) and ⊖a is the binary operation on D by Definition 6. Let (G, ⊕, ⊗) be a GGV. Define inverse element of a. A (commutative) group is a (gyrocommutative) The following corollary asserts that a surjective gyrometric preserv- a + b L[a, b](t) := a ⊕ t ⊗ (⊖a ⊕ b) a ⊕ b = gyrogroup whose gyroautomorphisms are all trivial. ing map preserves the algebraic structure followed by the left transla- M ⊕ 1 +ab ¯ For study a gyrogroup (G, ), it is useful to consider the second bi- for any a, b ∈ G, t ∈ R. L[a, b](R) is called gyroline in G that passes tions. In particular, if two GGV’s have the same gyrometric structure ⊞ G (G, ⊕) for any a, b ∈ D then (D, ⊕M) is a gyrocommutative gyrogroup. It nary operation on , which is called the coaddition of . If through a and b. then these have the same GGV structure. is motivated by Mobius¨ transformation and is called the Mobius¨ gy- (G, ⊕) is a group, it doesn’t distinguish between ⊕ and ⊞. 1 Note that L[a, b](0) = a, L[a, b](1) = b and L[a, b](2) = p(a, b). Corollary 8. Let (G , ⊕ , ⊗ ) and (G , ⊕ , ⊗ ) be GGV’s. Let ϱ and rogroup. Definition 2. Let (G, ⊕) be a gyrogroup. The gyrogroup coaddition ⊞ 1 1 1 2 2 2 1 ϱ be gyrometrics of G and G , respectively. Suppose that a surjection Some commutative group admit scalar multiplication and an inner is defined by 2 1 2 ∗ T : G →G satisfies product give rise to real inner product spaces. In full analogy, some a ⊞ b = a ⊕ gyr[a, ⊖b]b The positive cone of a unital C -algebra 1 2 gyrocommutative gyrogroup give rise to gyrovector space. The con- for all a, b ∈ G. A ∗ ∥ · ∥ A −1 ϱ (T a,T b) = ϱ (a, b) cept of the gyrovector space is also defined by A. A. Ungar. It is a Let be a unital C -algebra with the norm and + denotes the 2 1 Note that set of all positive invertible elements of A . Put generalization of the real inner product space, which addition is not ∈ · ⊕ · • ⊞ 1 1 for any pair a, b G1. Then T is of the form T ( ) = T (e) 2 T0( ), necessarily a commutative group but a gyrocommutative gyrogroup. the gyrogroup coaddition is commutative if and only if the gy- a ⊕ b = a2ba2 where T0 is an isometrical isomorphism in the sense that the equalities The Einstein gyrogroup and the Mobius¨ gyrogroup give rise to gy- rogroup (G, ⊕) is gyrocommutative; − − a, b ∈ A 1 (A 1, ⊕) rovecor spaces. These are called the Einstein gyrovector space and the for all + . Then + is gyrocommutative gyrogroup • if (G, ⊕) is a group then ⊞ = ⊕. T (a ⊕ b) = T (a) ⊕ T (b); Mobius¨ gyrovector space, respectively. A gyrovector space has rich with the gyroautomorphisms 0 1 0 2 0 ∗ − ∈ A 1 T0(α ⊗1 a) = α ⊗2 T0(a); structures as with an inner product space. In the Mobius¨ gyrovector gyr[a, b]c = XcX , a, b, c + , Generalized gyrovector spaces ϱ (T a,T b) = ϱ (a, b). space, its structures are compatible with hyperbolic geometry of the where X is a unitary element in A given by 2 0 0 1 Poincare´ disk model. 1 1 −1 1 1 ∗ We define the concept of GGV’s. It is a common generalization of the 2 2 2 2 2 ∈ ∈ R It is known that the positive cone of a unital C −algebra has a gyro- X = (a ba ) a b . hold for every a, b G1 and α . concept of the real normed spaces and of the gyrovector spaces. − commutative gyrogroup structure. However, the positive cone doesn’t The identity element of (A 1, ⊕) is the identity of A as the C∗- Definition 3. Let (G, ⊕) be a gyrocommutative gyrogroup with the + give rise to a gyrovector space. In the joint work with O. Hatori, we algebra. The inverse element ⊖a is a−1, the inverse of a as the C∗- map ⊗ : R × G → G. Let ϕ be an injection from G into a real normed An application define the concept of generalized gyrovector spaces (GGV, for short). algebra. space (V, ∥ · ∥). We say that (G, ⊕, ⊗, ϕ) is a generalized gyrovector It is a common generalization of the concept of the real normed space Moreover, put The following theorem was proved by O. Hatori and L. Molnar.´ The space (GGV, for short) if the it obeys the following axioms: and of the gyrovector spaces. The positive cone give rise to a GGV. Its r ⊗ a proof employ a non-commutative Mazur-Ulam theorem. A simple ∥ ∥ ∥ ∥ ∈ GGV structures are compatible with hyperbolic geometry of the posi- (V0) ϕ(gyr[u, v]a) = ϕ(a) for any u, v, a G; ∈ A −1 ∈ R proof of the following theorem is given by as an application of Corol- for every a + , r and tive cone. (V1) 1 ⊗ a = a for every a ∈ G; − lary 8. ϕ = log : A 1 → (A , ∥ · ∥), In this poster, we define the concept of the GGV’s, look the GGV + S −1 −1 ∗ ∗ (V2) (r1 + r2) ⊗ a = (r1 ⊗ a) ⊕ (r2 ⊗ a) for any a ∈ G, r1, r2 ∈ R; Theorem 9. Let A and B be the positive cone of unital C - structure of the positive cone of a unital C −algebra, and present a where A is a real subspace of all self-adjoint elements of A . + + ⊗ ⊗ ⊗ ∈ ∈ R S− algebras A and B, respectively. Let dA (resp. dB) be the Thompson Mazur-Ulam type theorem for GGV’s. (V3) (r1r2) a = r1 (r2 a) for any a G, r1, r2 ; Then (A 1, ⊕, ⊗) is a GGV. A real one-dimensional vector space − − + metric on A 1 (resp. B 1), that is, (V4) (ϕ(|r| ⊗ a))/∥ϕ(r ⊗ a)∥ = ϕ(a)/∥ϕ(a)∥ for any a ∈ G \{e}, r ∈ ∥ A −1 ∥ ⊕′ ⊗′ R × + + ( log( + ) , , ) = ( , +, ) is the usual 1 dimensional real vec- R \{0}, where e is the identity element of (G, ⊕); tor space of the real line. −1 1 −1 −1 −1 Definitions dA (a, b)(resp.dB(a, b)) = ∥ log a 2b a 2∥, a, b ∈ A (resp.B ). (V5) gyr[u, v](r ⊗ a) = r ⊗ gyr[u, v]a for any u, v, a ∈ G, r ∈ R; The gyrometric + + 1 −1 1 (V6) gyr[r1 ⊗ v, r2 ⊗ v] = idG for any v ∈ G, r1, r2 ∈ R; ϱ(a, b) = ∥ϕ(a ⊖ b)∥ = ∥ log a2b a2∥ A −1 → B−1 Gyrocommutative gyrogroups Suppose that T : + + is a surjective isometry; dA (a, b) = − (VV) There exists a real one-dimensional vector space ∥ϕ(G)∥ = d (T a, T b) for any a, b ∈ A 1. Then there exists a Jordan *- The concept of gyrocommutative gyrogroup is a generalization of the ′ is the Thompson metric. The gyromidpoint B + {∥ϕ(a)∥ ∈ R : a ∈ G} with vector addition ⊕ and scalar multi- A → B ∈ B 1 1 1 −1 1 −1 1 isomorphism J : and a central projection p such that T concept of the commutative group which is not necessarily commu- ⊗′ p(a, b) = ⊗ (a ⊞ b) = a2(a2b a2) 2a2 plication which satisfies the following (V7) and (V8); 2 has the form tative nor associative. A magma is a set L with a binary operation ′ (V7) ∥ϕ(r ⊗ a)∥ = |r| ⊗ ∥ϕ(a)∥ for any a ∈ G, r ∈ R; is the geodesic mean. The gyroline is L×L → L. It is known that a magma is a gyrocommutative gyrogroup t t 1 ′ −1 t −1 ∥ ⊕ ∥ ≤ ∥ ∥ ⊕ ∥ ∥ ∈ 1 1 −1 1 −t 1 T (a) = (T (e)2(pJ(a) + (e − p)J(a )) T (e)2) t , a ∈ A . if and only if it is a K-loop. (V8) ϕ(a b) ϕ(a) ϕ(b) for any a, b G. L[a, b](R) = {L[a, b](t) = a2(a2b a2) a2; t ∈ R} +