| In 1980s, M. Rieffel provided a notion of "topological stable rank" when he stud-ied the K theory of C*-algebra. He introduced topological stable rank in terms of a non-commutative version of the covering dimension for compact space. At that time, The results on the stability of K group of C*- algebra are almost few, and the reason is that there is no suitable notion of "covering dimension" for C*= algebra. By the in-spiration of related theorems in the covering dimension for compact space, Rieffel give a definition of corresponding dimension-topological stable rank (for short, tsr)for C*-algebra, more generally, Banach algebra, for a non-commutative topological space.For a given Banach algebra 91 with unital, we denote by Rgn((?))(Lgn((?)))the set of n-tuples of elements of (?) which generate (?)as a right(left) ideal, i.e.Then the right(left) topological stable rank of (?), denoted by rtsr ((?))(Itsr((?))), is the least positive integer n for which Rgn(?))(Lgn((?)))is dense in (?)". If such integer does not exist, then we denote by rtsr((?))=oo(Itsr((?))=∞).If the left and right topological stable rank of (?) coincide, we refer to the common value as the topological stable rank of (?), denoted by tsr((?)).Rieffel have proved that the topological stable rank of disc algebra A(D) is 2. And in the paper two open questions was presented:Question 1 If Ais C-algebra with unital, is tsr(A)= Bsr(A)?Question 2 Is there a Banach algebra (?) such that ltsr((?)) ≠rtsr((?))?In 1984, Herman and Vaserstein made an affirm answer to the Question 1 provided by Rieffel:If A is C*-algebra, then tsr (A)= Bsr(A). What is amazing is that Rieffel found out the common properties between Bass stable rank and topological stable rank, when he studied the topological stable rank of C*algebra and the related algebras. This implies that topological stable rank coincides with Bass stable rank.P. W. Jones, D. Marshall and T. Wolff have proved that for disc algebra A(D), its Bass stable rank is 1. However, the topological stable rank of disc algebra A(D) is 2. This shows that there is a Banaeh algebra for which topological stable rank and Bass stable rank may not be equal.The Question 2 give strong impetus to the development of stable rank of non-commutative, non-selfadjoint Banaeh algebra. As far as stable rank of C*-algebra is concerned, the theoretical research on stable rank of non-selfadjoint Banaeh algebra is still not too much, and the research of that focus on the case of commutative algebra. The greater influence is that the calculation of topological stable rank and Bass stable rank of commutative Banaeh algebra H∞(D), which is formed by all bounded analytic function on classic unit disc.If such an algebra is to exist, it must be non-commutative and non-selfadjoint. In 2007, Davidson and several authors got a creative result:a special nest algebra for which the left and right topological stable rank differ.In this paper, we present a Banaeh algebra:a non-commutative version of disc algebra. On one hand, it is a non-commutative analogue of classic disc algebra, on the other, it is a typical closed subalgebra of T(N). Hence, it is a non-commutative and non-selfadjoint Banaeh algebra. Then we get a result in general:定理 1. ltsr(AN)= rtsr (AN)= 2Fixed point theory is a significant tool in existence and uniqueness of the solutions in mathematical model, more extensions of the mapping have been given. Bessem, Calogero and Pasquale proved the existence theorem of fixed point of α-ψ-contraction in complete metric space, they introduced the notion of a-admissible of the mapping in metric space, then presented the results about fixed point in terms of the notion of α-admissible of the mapping. As to existence of the solutions of integral equation, they demonstrated the meaning of α-admissible. Based on the notion of α-admissible, G.Durmaz, G.Mjnak and I.Altun generalized the results in above article, and gave some examples to illustrate these generalization is indeed the extension of the previous results.A generalized metric space has been defined as a space which is the quadrilateral inequality instead of triangle inequality. Although just one condition in metric space has been changed, the property of the space has been changed, too. Many properties in a metric space does not exist in a generalized metric space, which leads to the interest of more researchers. Poom Kumam and Nguyen Van Dung clarified the inherent property in a generalized metric space, different from a metric space, and proved the relationship between two topologies in a generalized metric space and gave some examples. In the paper, fixed point theorem of iterated function systems has been expressed of which aα-ψ-contraction is deduced by α-ψ-contraction in fractal space consisting of the compact sets in metric space.定理 2. Let (X, d) be a complete metric space and f:X'X be an α-ψ-contractive mapping satisfying the following conditions:1. f is a-admissible;2. there exists x0∈X such that a(x0,f(xo))≥1. Then the mapping Ff:A (?)f(A)(A ∈ H{X))is an αα-ψ-contraction(with the same function ψ)from H(X) into itself and Ff has a fixed point, that is, there exists K ∈ H(X) such that Ff(K)= K.And in fractal space consisting of compact sets in a generalized metric space which the limit need not to be unique and the metric need not to be continuous, the fixed point theorem has been proved of Banach contractions. The theory will be applied in topological dynamic systems. |
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