Stable Ranks Of Banach Algebras

Throughout this paper, by a Banach algebra we mean a complex Banach algebra which may be not unital. When we speak of ideals of Banach algebras, we will always mean closed two-sided ideals. Given a ring (?), we denote by Lgn((?)) (resp. Rgn((?))) the set of n-tuples of elements of (?) which generate (?) as a left ideal (resp. as a right ideal); the elements of Lgn((?)) (resp. Rgn((?))) are called left (resp. right) unimodular. We denote by GL((?)) the group of all invertible elements of (?). For each stable rank under consideration, when a Banach algebra (?) is not unital, we define the stable rank of (?) to be that of its unitization (?)~.It is an old saying that the problem is the soul of mathematics. One source of algebraic K theory is Serre's Problem. In 1955, J.P.Serre asked that if finitely generated projective modules are free over a polynomial ring, where k is a field.Applying the theorem of Hilbert-Serre, it can be also stated as:Let k be a field, and A=k[t1,…,tn] be the polynomial ring over k. can we always complete an element of Lgm((?)) to an invertible matrix over (?)?Finally, in the January of 1976, Quillen and Suslin, working independently of each other, arrived the first complete solutions of Serre's Problem in full generality. The reader is referred to Lam's excellent monograph《Serre's problem on projective modules》for the background, final solutions and new developments of Serre's Problem.In his investigation of Serre's Problem, H. Bass gave the concept of the so called Bass stable rank. It has a very important application in algebraic K theory. And later, Vaserstein extensively studied the theory of Bass stable rank and got a series of deep and remarkable results. In 1983, in his investigation of K groups of C*-algebras, Rieffel found that there had been few stability results for C*-algebras. One obstruction is a lack of an ap-propriate concept of dimension for C*-algebras. Inspired by the concept of covering dimension for compact Hausdorff spaces, for noncommutative topological spaces-C*-algebras, more generally, Banach algebras, Rieffel introduced a concept of dimension-topological stable rank.The left topological stable rank of a unital Banach algebra (?). denoted by ltsr((?)), is the least positive integer n for which Lgn((?)) is dense in (?)n. When no such integer exists, we set ltsr((?))=∞. The right topological stable rank of (?). denoted by rtsr((?)), is defined analogously. If ltsr((?))= rtsr((?)), we refer to their common value simply as the topological stable rank of (?). written tsr((?)).Along the Vaserstein's discussion of the Bass stable rank, Rieffel studied the con-nections between Banach algebraic-constructions and the topological stable rankIn classical covering dimension theory, it is known that the dimension of an open subset U of a locally compact second countable Hausdorff space X is no greater than the dimension of the whole space X. Comparing with this, we are interested in whether an analog and generalization of the inequality can be obtained in topological stable rank. So it is natural to ask:is it always true that the left topological stable rank of an ideal (?) is no greater than the left topological stable rank of the whole algebra (?)?In the seminal paper《Dimension and stable rank in the K-theory of C*-algebras》, Rieffel asserted it is true under the assumption that (?) has a bounded approximate identity. For nonself-adjoint operator algebras such as nest algebras, the class of ideals with an approximate identity is extremely limited. And assuming that the quotient map splits, Davidson and Ji showed it is also true.In the first part of this thesis, we obtain this conclusion in full generality.Theorem 0.1 Let (?) be a Banach algebra and let (?) be an ideal in (?). Then ltsr((?))≤ltsr((?)), rtsr((?))≤rtsr((?)).While investigating the topological stable rank, Rieffel also defined two other stable ranks:the connected stable rank and the general stable rank. They are collectively referred to as homotopical stable ranks due to their distinctive feature of being homotopy invariants.Comparing with the Bass and the topological stable ranks, the connected and the general stable ranks have been less studied. Recently. Nica systematically studied the connected and the general stable ranks together. The highlight is that a positive answer to Swan's Problem in full generality for homotopical stable ranks had been given by Nica.The first part of this thesis also focuses on the relationships between the stable ranks and Banach-algebraic constructions such as split extensions and pullbacks. And we obtain the following result.Theorem 0.2 Let (?) be a Banach algebra, and let (?) an ideal of (?) such that the quotient mapπof (?) onto (?)/(?) is split. Then1. max{bsr((?)/(?)),bsr((?))}≤bsr((?))≤max{bsr((?)/(?)),gsr((?)/(?)),bsr((?))}.2. max{ltsr((?)/(?)),ltsr((?))}≤ltsr((?))≤max{ltsr((?)/(?)),gsr((?)/(?)),ltsr((?))}; max{rtsr((?)/(?)),rtsr((?))}≤rtsr((?))≤max{rtsr((?)/(?)),gsr((?)/(?)),rtsr((?))}.3. csr((?))= max{csr((?)/(?)),csr((?))}.4. gsr(2t)= max{gsr((?)/(?)),gsr((?))}.We also studied the stable ranks of pullbacks of Banach algebras. For the ho-motopical stable ranks, we demonstrate an example to show that there exist unital Banach algebras (?),(?), (?) andφ:(?)→(?),ψ:(?)→(?) surjective homomorphisms satisfy csr((?))> max{csr((?)), csr((?))} and gsr((?))>max{gsr((?)), gsr((?))}.Comparing with the study of the stable ranks of C*-algebras, there are only a few results of the stable ranks of non-selfadjoint Banach algebras. The deepest result in this area up to this point is that of Treil. He proved that bsr(H∞(D))= 1, where H∞(D) consists of all bounded analytic functions on the open unit disk. And four year later, Suarez concluded that tsr(H∞(D))= 2. But, in fact, very few results have been obtained for the stable ranks of noncommutative and non-selfadjoint Banach algebras. However, the following Rieffel's question prompts the study of the stable ranks of noncommutative and non-selfadjoint Banach algebras.Question 0.3 Does there exist a Banach algebra whose left and right topological stable ranks are different?It is easy to see that if there exists such a Banach algebra, it must be noncommu-tative and non-selfadjoint. In 2007, Davidson et al. obtained a fabulous result, that is, there exist a nest algebra such that ltsr((?))=∞, while rtsr((?))=2. So they an-swered the above question positively. And later, Davidson and Ji computed the right (resp. left) topological and general stable ranks for all nest algebras. And it should be mentioned that, in 1987, Peters had already considered the topological stable rank of semi-crossed product algebras. In fact, a lot of problems in control theory can be trans-ferred into the calculations of the stable ranks of noncommutative and non-selfadjoint Banach algebras.The other part of our research mainly focuses on the stable ranks of noncommuta-tive and non-selfadjoint Banach algebras. Firstly, we study the Bass and the connected stable ranks of nest algebras, and obtain the following result.Theorem 0.4 Suppose that N={Ni, H:i≥0} is a nest of order typeωwith finite rank atoms. Let M be a subnest of finite index in N. Then1. bsr(T(N)) = bsr(T(M)),2. csr(T(N)) = csr(T(M)),3. GL(T(N)) is connected if and only if GL(T(M)) is connected.We also consider the stable ranks of semi-crossed algebras. We estimate the topo-logical stable rank of semi-crossed algebras:and moreover we transform the homotopi-cal stable ranks of the semi-crossed algebra (?)×αZ+ into those of the Banach algebra (?) via showing that Banach algebras (?)×αZ- and (?) are homotopical equivalence. Theorem 0.5 Let (?) be an operator algebra acting on the Hilbert space H, andα: (?)→(?) be an isometric endomorphism, and (?)×αZ+ be the corresponding semi-crossed algebra. Then max{rtsr((?)), 2}≤rtsr((?)×αZ+)≤rtsr((?))+1;Proposition 0.6 Let (?) be an operator algebra acting on the Hilbert space H, andα: (?)→(?) be an isometric endomorphism, and (?)×αZ+ be the corresponding semi-crossed algebra. Then (?)×αZ+ and (?) are homotopical equivalence. In particular, csr((?)×αZ+) = csr((?)), gsr((?)×αZ+) = gsr((?)).Finally, we roughly discuss the Bass stable rank of the noncommutative disc alge-bra and obtain the following small result.Proposition 0.7 If (?)n is the non-commutative disc algebra on n-generators, then bsr((?)n)≥[n/2]+1.