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220 W. WernerDefinition 2.10. Suppose X is a (closed) subspace of L.H /. EquipM n .X/ D ˚.x ij / j x ij 2 X for i;j D 1;:::;n with the norm it carries as a subspace of M n .L.H // D L.H ˚˚H/, and, for abounded operator T W X ! X, denote by T .n/ D id Mn .C/ ˝ X W M n .X/ ! M n .X/the operator .x ij / 7 ! .T x ij /. Then T is said to be completely bounded, iffkT k cb WD sup kT .n/ k < 1:n2NSimilarly, T is called a complete isometry iff each of the maps T .n/ is an isometry.We have the followingTheorem 2.11 ([3], [9]). Let E be a TRO. Then M n .E/ carries a distinguished TROstructure,and the group of TRO-automorphisms of E coincides with the group ofcomplete isometries.Theorem 2.12. Let U be the open unit ball of a TRO E, equipped with the canonicalHilbert C*-structure, and suppose M n .E/ carries the standard TRO-structure foreach n 2 N. Then a diffeomorphism ˆW U ! U is a Hilbert C*-automorphism iffˆ D T ıM a , where a 2 U , and T is the restriction of a linear and completely isometricmapping of E to U .Proof. That each map of the form T ı M a is in Aut U follows from the constructionof the Hilbert C*-structure on U as well as from Theorem 2.11. Since the derivativeof any element ˆ 2 Aut U is complex linear by definition, Aut U Hol U , and henceˆ D T ı M a for some a 2 U and an isometry T . Because the linear map T fixes theorigin, dT must be a TRO-automorphism, and the result follows, by another applicationof Theorem 2.11.2.13. Let M again be a Hilbert C*-manifold. Then the tangent space at each pointm of M carries an essentially unique norm, naturally connected to the TRO-structureby kxk m Dkfx;x;xg m k 1=3 . Since TROs embed completely isometrically into somespace of bounded Hilbert space operators, this norm naturally extends to the spacesM n .T m M/. We will call a Banach manifold M an operator Finsler manifold in caseeach tangent space carries the structure of an operator space depending continuously onbase points. In this sense, any Hilbert C*-manifold carries an operator Finsler structurein a natural way. If, as before, U is the open unit ball of a fixed TRO .E; f; ; g; kk/,furnished with its invariant Hilbert C*-structure then, using that complete isometriesand automorphisms of a TRO are the same mappings, we haveTheorem. If the open unit ball of a TRO is equipped with the natural invariant operatorFinsler structure its automorphism groups coincides with the automorphism group ofthe underlying homogeneous Hilbert C*-manifold.