Bass Stable Rank Research Articles

Simultaneous stabilization in AR(D)

In this note we study the problem of simultaneous stabilization for the algebra $A_\R(\D)$. Invertible pairs $(f_j,g_j)$, $j=1,..., n$, in a commutative unital algebra are called \textit{simultaneously stabilizable} if there exists a pair $(\alpha,\beta)$ of elements such that $\alpha f_j+\beta g_j$ is invertible in this algebra for $j=1,..., n$. For $n=2$, the simultaneous stabilization problem admits a positive solution for any data if and only if the Bass stable rank of the algebra is one. Since $A_\R(\D)$ has stable rank two, we are faced here with a different situation. When $n=2$, necessary and sufficient conditions are given so that we have simultaneous stability in $A_\R(\D)$. For $n\geq 3$ we show that under these conditions simultaneous stabilization is not possible and further connect this result to the question of which pairs $(f,g)$ in $A_\R(\D)^2$ are totally reducible; that is, for which pairs do there exist two units $u$ and $v$ in $A_\R(\D)$ such that $uf+vg=1$.

The Bass and topological stable ranks of and

In this note we prove that the Bass stable rank of is two. This establishes the validity of a conjecture by S. Treil. We accomplish this in two different ways, one by giving a direct proof, and the other, by first showing that the topological stable rank of is two. We apply these results to give new proofs of results by R. Rupp and A. Sasane stating that the Bass stable rank of is two and the topological stable rank of is two, settling a conjecture by the second author. We also present a -free proof of the second author's characterization of the reducible pairs in .

On strong stabilization for linear continuous time time-varying systems

This paper deals with the strong stabilization problem for linear time-varying continuous time systems. The main result is that for such systems internally stabilizable systems may not be strongly stabilizable. This follows from the fact that the algebra of stable systems has infinite (Bass) stable rank.

Stable Ranks of Banach Algebras of Operator-Valued Analytic Functions

Let E be a separable infinite-dimensional Hilbert space, and let H(D; (E)) denote the algebra of all functions f:D (E) that are holomorphic. If is a subalgebra of H(D; (E)) , then using an algebraic result of Corach and Larotonda, we derive that under some conditions, the Bass stable rank of is infinite. In particular, we deduce that the Bass (and hence topological stable ranks) of the Hardy algebra H (D; (E)), the disk algebra A(D; (E)) and the Wiener algebra W+(D; (E)) are all infinite.

Inverse limits of rings and multiplier rings

It is proved that the exchange property, the Bass stable rank and the quasi-Bass property are all preserved under surjective inverse limits. This is then applied to multiplier rings by showing that these in many cases can be obtained as inverse limits.

On the stable rank of simple C*-algebras

In [16] M. A. Rieffel introduced the notion of stable rank for C*-algebras. For A a unital C*-algebra the stable rank, denoted by sr(A), is the least integer n such that the set of n-tuples over A which generate A as a left ideal is dense in An if no such n exists we set sr(A) = oo. For unital C*-algebras the stable rank was shown to coincide with the Bass stable rank, which is defined for rings; see [12]. The stable rank can also be defined for nonunital C*-algebras, but we shall only consider the unital case. Stable rank is related to nonstable K-theory: In [16, Theorem 10.12] and [3, Corollary 7.14] it was shown that if A is a simple unital C*-algebra of stable rank one, then the natural map U(A)/U(A)o -4 Ki(A) is an isomorphism. See also [17], [2] and [18]. It is a natural question which values occur as the stable rank of simple C*algebras. Suppose that A is simple and infinite. Then, by [7, Proposition 1.5], A contains two isometries with orthogonal ranges and so, by [16, Proposition 6.5], sr(A) = oo. In the case of finite, simple C*-algebras the following is known: Whenever A is simple and stably finite and B is a UHF-algebra, the tensor product A0 B has stable rank one; see [19]. This is also the case for every simple approximately homogeneous (AH) algebra with slow dimension growth; see [4] and [8]. So, in short, sr = oo for every infinite simple C*-algebra and sr = 1 for various classes of finite simple C*-algebras. In this paper we answer the question above by showing that every n C N U {oo} is the stable rank of some simple, finite C*-algebra. More precisely, it is shown that for every n = 2, 3, 4,... ., oo there exists a simple, separable and unital AH-algebra A of stable rank n. In particular, the group of invertible elements in A is not dense. And, like the examples in [20], A does not have slow dimension growth and is not approximately divisible; see [5]. I thank Marius Dadarlat for suggesting that I study AH-algebras with connecting homomorphisms as described in the following section, and Juliana Erlijman for drawing my attention to some of the powerful tools of differential topology, and I thank George Elliott for my stay at The Fields Institute, where this paper was written.

The Stable Rank of Topological Algebras and a Problem of R. G. Swan

Various notions of stable ranks are studied for topological algebras. Some partial answers to R. G. Swan's problem (Have two Banach or good Fréchet algebras as in the density theorem inK-theory the same stable rank?) are obtained. For example, a Fréchet dense ∗-subalgebraAof aC*-algebraB, closed underC*-functional calculus of self-adjoint elements, has the same Bass stable rank asB.

Cohomological Stabilization of Maximal Ideal Spaces in Banach Algebras

We prove that for a complex commutative Banach algebra A with finite dimensional character space X, the Bass stable rank condition sr A ≤ n is equivalent to a certain stabilization property for Čech cohomology sequences of pairs ( X, Z), where Z is the hull of an arbitrary closed ideal. In particular, the k-cohomology groups of X, Z and ( X, Z) vanish when k ≥ 2 n + 1. Using this result and Michael′s continuous selection theory we improve previous estimations for sr C ( Y, A), where C ( Y, A) denotes the algebra of continuous functions from a finite dimensional compact space Y into A.

Stable rank and approximation theorems in 𝐻^{∞}

It is conjectured that for H ∞ {H^\infty } the Bass stable rank ( bsr ) (\text {bsr}) is 1 1 and the topological stable rank ( tsr ) (\text {tsr}) is 2 2 . bsr ( H ∞ ) = 1 {\text {bsr}}({H^\infty })= 1 if and only if for every ( f 1 , f 2 ) ∈ H ∞ × H ∞ ({f_1},{f_2})\; \in {H^\infty } \times {H^\infty } which is a corona pair (i.e., there exist g 1 {g_{1}} , g 2 ∈ H ∞ {g_2} \in {H^\infty } such that f 1 g 1 + f 2 g 2 = 1 {f_1}{g_1} + {f_2}{g_2}= 1 ) there exists a g ∈ H ∞ g \in {H^\infty } such that f 1 + f 2 g ∈ ( H ∞ ) − 1 {f_1} + {f_2}g \in {({H^\infty })^{ - 1}} , the invertibles in H ∞ {H^\infty } ; however, it suffices to consider corona pairs ( f 1 , f 2 ) ({f_1},{f_2}) where f 1 {f_1} is a Blaschke product. It is also shown that there exists a g ∈ H ∞ g \in {H^\infty } such that f 1 + f 2 g ∈ exp ⁡ ( H ∞ ) {f_1} + {f_2}g \in \exp ({H^\infty }) if and only if log ⁡ f 1 \log {f_1} can be boundedly, analytically defined on { z ∈ D : | f 2 ( z ) | > δ } \{ {z \in \mathbb {D}:|\,{f_2}(z)| > \delta } \} , for some δ > 0 \delta > 0 . tsr ( H ∞ ) = 2 {\text {tsr}}({H^\infty })= 2 if and only if the corona pairs are uniformly dense in H ∞ × H ∞ {H^\infty } \times {H^\infty } ; however, it suffices to show that the corona pairs are uniformly dense in pairs of Blaschke products. This condition would be satisfied if the interpolating Blaschke products were uniformly dense in the Blaschke products. For b b an inner function, K = H 2 ⊖ b H 2 K= {H^2} \ominus b\,{H^2} is an H ∞ {H^\infty } -module via the compressed Toeplitz operators C f = P K T f | K {C_f}= {P_K}{T_f}{|_K} , for f ∈ H ∞ f \in {H^\infty } , where T f {T_f} is the Toeplitz operator T f g = f g {T_f}g= f\,g , for g ∈ H 2 g \in {H^2} . Some stable rank questions can be recast as lifting questions: for ( f i ) 1 n ⊂ H ∞ ({f_i})_1^n \subset {H^\infty } , there exist ( g i ) 1 n ({g_i})_1^n , ( h i ) 1 n ⊂ H ∞ ({h_i})_1^n \subset {H^\infty } such that ∑ i = 1 n ( f i + b g i ) h i = 1 \sum \nolimits _{i = 1}^n {({f_i} + b\,{g_i}){h_i}= 1} if and only if the compressed Toeplitz operators ( C f i ) 1 n ({C_{{f_i}}})_1^n may be lifted to Toeplitz operators ( T F i ) 1 n ({T_{{F_i}}})_1^n which generate B ( H 2 ) B({H^2}) as an ideal.

Stable Rank of the Disc Algebra

We prove that the Bass stable rank of the disc algebra is one.

Stable rank of the disc algebra

We prove that the Bass stable rank of the disc algebra is one.

Dimension and Stable Rank in the K -Theory of C* -Algebras

In topological K-theory, which can be viewed as the algebraic side of the theory of vector bundles, some of the interesting properties which one investigates are, for example, the conditions under which bundles must possess trivial direct summands, or the extent to which the cancellation property for direct (Whitney) sums of bundles holds. Such properties turn out to be controlled in part by the dimension of the base space, and results describing the nature of this control are among what are frequently called stability results [17]. During the past few years there has been a sudden flowering of a /^-theory for C*algebras. (See surveys [35, 36, 37].) Since C*-algebras are profitably thought of as 'non-commutative locally compact spaces', with the finitely generated projective modules being the appropriate generalization of vector bundles according to Swan's theorem [40], it would be natural to look for stability results for C*-algebras. But until now the theory has been focussed on the /C-groups of C*-algebras, and there has been little discussion of stability properties, presumably in part for lack of an appropriate concept of dimension for C*-algebras. One of the main objectives of this paper is to introduce a concept of dimension for C*-algebras which directly generalizes the classical concept of dimension for compact spaces [16, 20, 22], and to develop some techniques for calculating it, notably for C*algebras which are obtained as crossed-product algebras for an action of the group of integers. In algebraic K-theory, which is the generalization of topological /C-theory to rings, a substantial number of stability properties have been discovered [2, 39, 42-48]. Within that theory the concept which has played a role most analogous to that of dimension is the Bass stable rank. But despite its many successes, the Bass stable rank has often been difficult to calculate even for relatively uncomplicated rings. Another main objective of the present paper is to make accessible for application to C*algebras some of the stability results from algebraic /C-theory. This comes about from the fact that the concept of dimension which we introduce dominates the Bass stable rank. I was led to seek stability results for C*-algebras by my desire to determine whether the cancellation property for projective modules holds for the irrational rotation C*algebras discussed in [30] (which, as Elliott [11] has suggested, can appropriately be called 2-dimensional 'non-commutative tori'). In more concrete terms, the question is whether two projections in such an algebra which have the same trace must be unitarily equivalent. The present paper provides the general theory which is needed for the affirmative answer which I obtained. In a subsequent paper I will derive the