From (4.2 - 4.3 4.2 - 4.3 corresponding proofs in Theorems 3.2 and 3.3, respectively, together with the permanencorrespondingf fixed proofsindex, we have i(Theorems{1}^, P)=1 (4.4) and i(T, P_{r^_{2}}, P)=1.
Since A ( B ¯ r 1 ) ⊂ P, we have, from the permanence property of fixed point index and Lemma 2.5, that deg ( I − A, B r 1, 0 ) = i ( A, B r 1 ∩ P, P ) = 0, (3.12).
Thus, for any (yinpartial P_{r^_{0}}), in view of (4.1) and the corresponding proof in Theorem 3.1 together with the permanence property of fixed point index, we have i(T, P_{r^_{0}}, P =0.
Then we have from the permanence property and the homotopy invariance property of fixed point index that deg ( I − A ˜, B r 3, 0 ) = i ( A ˜, B r 3 ∩ P, P ) = i ( 0, B r 3 ∩ P, P ) = 1.
Moreover, according to (2.16) and the permanence property of the fixed point index, we have i(A,Omega_{i},E =i(A,Omega_{i}cap W_{i},W_{i})=i(A, Omega_{i},W_{i})=0, quad i=3,4.
From (4.8) and (4.9), the corresponding proofs in Theorems 3.2 and 3.3, respectively, together with the permanence property of fixed point index, we have i(T, P_{tilde{r}_{1}^, P =0 (4.10) and i(T, P_{tilde{r}_{3}^, P =0.
We then give some permanence properties which are similar to those of exact groups.
We will study some permanence properties of C∗-unique groups in details.
In [11], one can see a detailed study of sufficient conditions for this property as well as their permanence properties.
Perhaps most remarkable, however, are the permanence properties of Nc-SD Lagrangians; their calculus is relatively manageable, and their applications are quite broad.
More precisely, we will give some interesting equivalent formulations as well as some permanence properties for both property (T) and strong property (T).
