Topological Stable Rank Research Articles

Topological stable rank of H ∞(Ω) for circular domains Ω

Let Ω be a circular domain, that is, an open disk with finitely many closed disjoint disks removed. Denote by H ∞(Ω) the Banach algebra of all bounded holomorphic functions on Ω, with pointwise operations and the supremum norm. We show that the topological stable rank of H ∞(Ω) is equal to 2. The proof is based on Suárez’s theorem that the topological stable rank of H ∞(\( \mathbb{D} \)) is equal to 2, where \( \mathbb{D} \) is the unit disk. We also show that for circular domains symmetric to the real axis, the Bass and topological stable ranks of the real-symmetric algebra H ∞ℝ (Ω) are 2.

Matricial topological ranks for two algebras of bounded holomorphic functions

Let N and D be two matrices over the algebra H ∞ of bounded analytic functions in the disk, or its real counterpart . Suppose that N and D have the same number n of columns. In a generalization of the notion of topological stable rank 2, it is shown that N and D can be approximated (in the operator norm) by two matrices Ñ and , so that the Aryabhatta–Bezout equation admits a solution. This has particular interesting consequences in systems theory. Moreover, in case that N is a square matrix, X can be chosen to be invertible in the case of the algebra H ∞, but not always in the case of .

Spectral characteristics and stable ranks for the Sarason algebra $H^\inf + C$

We prove a Corona type theorem with bounds for the Sarason algebra $H^\infty+C$ and determine its spectral characteristics. We also determine the Bass, the dense, and the topological stable ranks of $H^\infty+C$.

On the stable rank and reducibility in algebras of real symmetric functions

Let Aℝ(𝔻) denote the set of functions belonging to the disc algebra having real Fourier coefficients. We show that Aℝ(𝔻) has Bass and topological stable ranks equal to 2, which settles the conjecture made by Brett Wick in [18]. We also give a necessary and sufficient condition for reducibility in some real algebras of functions on symmetric domains with holes, which is a generalization of the main theorem in [18]. A sufficient topological condition on the symmetric open set 𝔻 is given for the corresponding real algebra Aℝ(𝔻) to have Bass stable rank equal to 1 (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Projective modules over noncommutative tori are multi-window Gabor frames for modulation spaces

In the present investigation we link noncommutative geometry over noncommutative tori with Gabor analysis, where the first has its roots in operator algebras and the second in time-frequency analysis. We are therefore in the position to invoke modern methods of operator algebras, e.g. topological stable rank of Banach algebras, to display the deeper properties of Gabor frames. Furthermore, we are able to extend results due to Connes and Rieffel on projective modules over noncommutative tori to Banach algebras, which arise in a natural manner in Gabor analysis. The main goal of this investigation is twofold: (i) an interpretation of projective modules over noncommutative tori in terms of Gabor analysis and (ii) to show that the Morita–Rieffel equivalence between noncommutative tori is the natural framework for the duality theory of Gabor frames. More concretely, we interpret generators of projective modules over noncommutative tori as the Gabor atoms of multi-window Gabor frames for modulation spaces. Moreover, we show that this implies the existence of good multi-window Gabor frames for modulation spaces with Gabor atoms in e.g. Feichtinger's algebra or in Schwartz space.

Ideal Structure and Stable Rank of $${\mathbb {C} e+ \ell^2(I)}$$ with the Hadamard Product

Let I be any index set. We consider the Banach algebra \({\mathbb {C} e+ \ell^2(I)}\) with the Hadamard product, and prove that its Bass and topological stable ranks are both equal to 1. We also characterize divisors, maximal ideals, closed ideals and closed principal ideals. For \({I=\mathbb {N}}\) we also characterize all prime z-ideals in this Banach algebra.

Bass and Topological Stable Ranks of Complex and Real Algebras of Measures, Functions and Sequences

We compute the Bass stable rank and the topological stable rank of several convolution Banach algebras of complex measures on (-∞,∞) or on [0,∞) consisting of a discrete measure and/or of an absolutely continuous measure. We also compute the stable ranks of the convolution algebras , , l1(S) and , where S is an arbitrary subgroup of , of the almost periodic algebra AP and of , etc. We answer affirmatively the question posed by Mortini (Studia Mathematica 103(3):275–281, 1992). For the above algebras, the polydisc algebra , the algebra of continuous functions, and others, we also study their subsets (real Banach algebras) of real-valued measures, real-valued sequences or real-symmetric functions, and of corresponding exponentially stable algebras (for example, the Callier–Desoer algebra of causal exponentially decaying measures and L1 functions), and we compute their stable ranks. Finally, we show that in some of these real algebras a variant of the parity interlacing property is equivalent to reducibility of a unimodular (or coprime) pair. Also corona theorems are presented and the existence of coprime fractions is studied; in particular, we list which of these algebras are Bezout domains.

Cancellation for inclusions of C*-algebras of finite depth

Let 1 ∈ A ⊂ B be a pair of C* - algebras with common unit. We prove that if E: B → A is a conditional expectation with index-finite type and a quasi-basis of n elements, then the topological stable rank satisfies tsr(B) ≤ tsr(A) + n - 1. As an application we show that if an inclusion 1 ∈ A ⊂ B of unital C*-algebras has index-finite type and finite depth, and A is a simple unital C * -algebra with tsr(A) = 1 and Property (SP), then B has cancellation. In particular, if α is an action of a finite group G on A, then the crossed product A ×α G has cancellation. For outer actions of Z, we obtain cancellation for A ×α Z under the additional condition that α * = id on K 0 (A). Examples are given.

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 .

Topological stable rank of nest algebras

We establish a general result about extending a right invertible row over a Banach algebra to an invertible matrix. This is applied to the computation of right topological stable rank of a split exact sequence. We also introduce a quantitative measure of stable rank. These results are applied to compute the right (left) topological stable rank for all nest algebras. This value is either 2 or infinity, and rtsr(T(N)) = 2 occurs only when N is of ordinal type less than omega^2 and the dimensions of the atoms grows sufficiently quickly. We introduce general results on `partial matrix algebras' over a Banach algebra. This is used to obtain an inequality akin to Rieffel's formula for matrix algebras over a Banach algebra. This is used to give further insight into the nest case.

STABLE RANK FOR INCLUSIONS OF C*-ALGEBRAS

When a unital C*-algebra A has topological stable rank one (write tsr (A) = 1), we know that tsr (pAp) = 1 for a non-zero projection p ∈ A. When, however, tsr (A) ≥ 2, it is generally false. We prove that if a unital C*-algebra A has a simple unital C*-subalgebra D of A with common unit such that D has Property (SP) and supp ∈ P(D)tsr (pAp) < ∞, then tsr (A) ≤ 2. As an application let A be a simple unital C*-algebra with tsr (A) = 1 and Property (SP), [Formula: see text] finite groups, αkactions from Gkto Aut ((⋯((A × α1G1) ×α2G2)⋯) ×αk-1Gk-1). (G0= {1}). Then [Formula: see text]

On the topological stable rank of non-selfadjoint operator algebras

We provide a negative solution to a question of M. Rieffel who asked if the right and left topological stable ranks of a Banach algebra must always agree. Our example is found amongst a class of nest algebras. We show that for many other nest algebras, both the left and right topological stable ranks are infinite. We extend this latter result to Popescu’s non-commutative disc algebras and to free semigroup algebras as well.

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.

Banach function algebras with dense invertible group

In 2003 Dawson and Feinstein asked whether or not a Banach function algebra with dense invertible group can have a proper Shilov boundary. We give an example of a uniform algebra showing that this can happen, and investigate the properties of such algebras. We make some remarks on the topological stable rank of commutative, unital Banach algebras. In particular, we prove that $\mathrm {tsr}(A) \geq \mathrm {tsr}(C(\Phi _A))$ whenever $A$ is approximately regular.

On the topological stable rank of non-selfadjoint operator algebras

We provide a negative solution to a question of M. Rieffel who asked if the right and left topological stable ranks of a Banach algebra must always agree. Our example is found amongst a class of nest algebras. We show that for many other nest algebras, both the left and right topological stable ranks are infinite. We extend this latter result to Popescu's non-commutative disc algebras and to free semigroup algebras as well.

JB*-Algebras of Topological Stable Rank 1

In 1976, Kaplansky introduced the classJB*-algebras which includes allC*-algebras as a proper subclass. The notion of topological stable rank 1 forC*-algebras was originally introduced by M. A. Rieffel and was extensively studied by various authors. In this paper, we extend this notion to generalJB*-algebras. We show that the complex spin factors are of tsr 1 providing an example of specialJBW*-algebras for which the enveloping von Neumann algebras may not be of tsr 1. In the sequel, we prove that every invertible element of aJB*-algebra𝒥is positive in certain isotope of𝒥; if the algebra is finite-dimensional, then it is of tsr 1 and every element of𝒥is positive in some unitary isotope of𝒥. Further, it is established that extreme points of the unit ball sufficiently close to invertible elements in aJB*-algebra must be unitaries and that in anyJB*-algebras of tsr 1, all extreme points of the unit ball are unitaries. In the end, we prove the coincidence between theλ-function andλu-function on invertibles in aJB*-algebra.

TOPOLOGICAL STABLE RANK OF INCLUSIONS OF UNITAL C*-ALGEBRAS

Let 1 ∈ A ⊂ B be an inclusion of C*-algebras of C*-index-finite type with depth 2. We try to compute the topological stable rank of B (= tsr (B)) when A has topological stable rank one. We show that tsr (B) ≤ 2 when A is a tsr boundedly divisible algebra, in particular, A is a C*-minimal tensor product UHF ⊗ D with tsr (D) = 1. When G is a finite group and α is an action of G on UHF, we know that a crossed product algebra UHF ⋊α G has topological stable rank less than or equal to two. These results are affirmative data to a generalization of a question by Blackadar in 1988.

Tracially quasidiagonal extensions and topological stable rank

It is known that a unital simple $C^*$-algebra $A$ with tracial topological rank zero has real rank zero and topological stable rank one. In this note we construct unital $C^*$-algebras with tracial topological rank zero but topological stable rank two.

Trivial Gleason parts and the topological stable rank of H ∞

We show that the subset of the maximal ideal space of H ∞ formed by the trivial Gleason parts is totally disconnected. A pair ( ƒ, g ) ∈ H ∞ × H ∞ is called a corona pair if [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /][inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /] > 0, where [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="03i" /] is the open unit disc. We prove that the set U 2 ( H ∞ ) of corona pairs is dense in H ∞ × H ∞ .

The algebra of almost periodic functions has infinite topological stable rank

The algebra of almost periodic functions has infinite topological stable rank