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 8 2 ⊕ 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) 8a 2 G, 9 ⊖ a s.t. ⊖a ⊕ a = e: Definition 4. Let be a GGV. The gyromidpoint of a; b 2 (G; ⊕; ⊗) is defined as p(a; b) = 1 ⊗ (a ⊞ b), where ⊞ is the generalization of the celebrated Mazur-Ulam theorem. A proof of this 8 2 9 2 ⊕ ⊕ ⊕ ⊕ 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 2 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 2 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 8 2 ⊕ ⊕ ⊗ k ⊖ k 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 %(a; b) = ϕ(a b) for preserves the gyromidpoints. That is 2 ⊕ ⊗ 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 % the gyrometric on (G; ; ). % (a; b) = % (T a;T b)(8a; b 2 G ) of the Einstein’s special relativity. It is called the Einstein gyrogroup. 8a; b 2 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) , . ) 8 2 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 2 G, t 2 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 2 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 % and rogroup. Definition 2. Let (G; ⊕) be a gyrogroup. The gyrogroup coaddition ⊞ 1 1 1 2 2 2 1 % 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 2 G. A ∗ k · k A −1 % (T a;T b) = % (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 2 · ⊕ · • ⊞ 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 2 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 ∗ − 2 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 % (T a;T b) = % (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 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; k · k). 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 k k k k 2 GGV structures are compatible with hyperbolic geometry of the posi- (V0) ϕ(gyr[u; v]a) = ϕ(a) for any u; v; a G; 2 A −1 2 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 2 G; − lary 8. ϕ = log : A 1 ! (A ; k · k); 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 2 G; r1; r2 2 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 . + + ⊗ ⊗ ⊗ 2 2 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(jrj ⊗ a))=kϕ(r ⊗ a)k = ϕ(a)=kϕ(a)k for any a 2 G n feg; r 2 k A −1 k ⊕0 ⊗0 R × + + ( log( + ) ; ; ) = ( ; +; ) is the usual 1 dimensional real vec- R n f0g, where e is the identity element of (G; ⊕); tor space of the real line.