We describe the structure of all continuous algebraic endomorphisms of the open unit ball B of ℝ3 equipped with the Einstein velocity addition. We show that any nonzero such transformation originates from an orthogonal linear transformation on ℝ3.

1.
T.
Abe
, “
Gyrometric preserving maps on Einstein gyrogroups, Möbius gyrogroups and proper velocity gyrogroups
,”
Nonlinear Funct. Anal. Appl.
19
,
1
17
(
2014
).
2.
G.
Cassinelli
,
E.
De Vito
,
P. J.
Lahti
, and
A.
Levrero
, in
The Theory of Symmetry Actions in Quantum Mechanics
,
Lecture Notes in Physics
Vol.
654
(
Springer
,
2004
).
3.
A.
Einstein
,
Einsteins Miraculous Years: Five Papers that Changed the Face of Physics
(
Princeton University
,
Princeton, NJ
,
1998
).
4.
S.
Kim
, “
Distances of qubit density matrices on Bloch sphere
,”
J. Math. Phys.
52
,
102303
(
2011
).
5.
L.
Molnár
, in
Selected Preserver Problems on Algebraic Structures of Linear Operators and on Function Spaces
,
Lecture Notes in Mathematics
Vol.
1895
(
Springer
,
2007
), p.
236
.
6.
L.
Molnár
and
P.
Szokol
, “
Transformations on positive definite matrices preserving generalized distance measures
,”
Linear Algebra Appl.
466
,
141
159
(
2015
).
7.
L.
Molnár
and
D.
Virosztek
, “
Continuous Jordan triple endomorphisms of ℙ2
,”
J. Math. Anal. Appl.
(submitted).
8.
A. A.
Ungar
,
Analytic Hyperbolic Geometry and Albert Einsteins Special Theory of Relativity
(
World Scientific
,
Singapore
,
2008
).
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