Topological Stable Rank Research Articles

On a Banach algebra of entire functions with a weighted Hadamard multiplication

New algebraic-analytic properties of a previously studied Banach algebra A(p) of entire functions are established. For a given fixed sequence (p(n))n≥0 of positive real numbers, such that limn→∞⁡p(n)1n=∞, the Banach algebra A(p) is the set of all entire functions f such that f(z)=∑n=0∞fˆ(n)zn (z∈C), where the sequence (fˆ(n))n≥0 of Taylor coefficients of f satisfies fˆ(n)=O(p(n)−1) for n→∞, with pointwise addition and scalar multiplication, a weighted Hadamard multiplication ⁎ with weight given by p (i.e., (f⁎g)(z)=∑n=0∞p(n)fˆ(n)gˆ(n)zn for all z∈C), and the norm ‖f‖=supn≥0⁡p(n)|fˆ(n)|. The following results are shown:•The Topological stable rank of A(p) is 1.•The Bass stable rank of A(p) is 1.•A(p) is a Hermite ring.•A(p) is not a projective-free ring.•Idempotents in A(p) are described.•Exponentials in A(p) are described, and it is shown that every invertible element of A(p) has a logarithm, so that the first Čech cohomology group H1(M(A(p)),Z) with integer coefficients of the maximal ideal space M(A(p)) is trivial.•A generalised necessary and sufficient ‘corona-type condition’ on the matricial data (A,b) with entries from A(p) is given for the solvability of Ax=b with x also having entries from A(p).•The Krull dimension of A(p) is infinite.•A(p) is neither Artinian nor Noetherian.•A(p) is coherent.•The special linear group over A(p) is generated by elementary matrices.

Topological stable rank of mathcal {E}'(mathbb {R})

The set mathcal {E}'(mathbb {R}) of all compactly supported distributions, with the operations of addition, convolution, multiplication by complex scalars, and with the strong dual topology is a topological algebra. In this article, it is shown that the topological stable rank of mathcal {E}'(mathbb {R}) is 2.

The Bass and topological stable ranks for algebras of almost periodic functions on the real line, II

Let Λ be either a subgroup of the integers Z, a semigroup in N, or Λ=Q (resp., Q+). We determine the Bass and topological stable ranks of the algebras APΛ={f∈AP:σ(f)⊆Λ} of almost periodic functions on the real line and with Bohr spectrum in Λ. This answers a question in the first part of this series of articles under the same heading, where it was shown that, in contrast to the present situation, these ranks were infinite for each semigroup Λ of real numbers for which the Q-vector space generated by Λ had infinite dimension.

When is multiplication in a Banach algebra open?

We develop the theory of Banach algebras whose multiplication (regarded as a bilinear map) is open. We demonstrate that such algebras must have topological stable rank 1, however the latter condition is strictly weaker and implies only that products of non-empty open sets have non-empty interior. We then investigate openness of convolution in semigroup algebras resolving in the negative a problem of whether convolution in ℓ1(N0) is open. By appealing to ultraproduct techniques, we demonstrate that neither in ℓ1(Z) nor in ℓ1(Q) convolution is uniformly open. The problem of openness of multiplication in Banach algebras of bounded operators on Banach spaces and their Calkin algebras is also discussed.

Unitary stable ranks and norm-one ranks

ABSTRACTIn the context of commutative C*-algebras, we solve a problem related to a question of M. Rieffel by showing that the all-units rank and the norm-one rank coincide with the topological stable rank. We also introduce the notion of unitary M-stable rank for an arbitrary commutative unital ring and compare it with the Bass stable rank. In case of uniform algebras, a sufficient condition for norm-one reducibility is given.

SOME CURIOSITIES OF THE ALGEBRA OF BOUNDED DIRICHLET SERIES

It is shown that the algebra H ∞ of bounded Dirichlet series is not a coherent ring, and has infinite Bass stable rank. As corollaries of the latter result, it is derived that H ∞ has infinite topological stable rank and infinite Krull dimension.

On the Bézout equation in the ring of periodic distributions

Abstract A corona type theorem is given for the ring D'A(Rd) of periodic distributions in Rd in terms of the sequence of Fourier coefficients of these distributions,which have at most polynomial growth. It is also shown that the Bass stable rank and the topological stable rank of D'A(Rd) are both equal to 1.

On the Topological Stable Rank of a Noncommutative Version of the Disc Algebra

Based on a bounded approximate version of LU decomposition, we calculate the topological stable rank of \({{\mathcal {A}}}_{{\mathcal {N}}}\), which is a noncommutative version of the disc algebra. In addition, a general characterization of the maximal two-sided ideals of \({{\mathcal {A}}}_{{\mathcal {N}}}\) will be obtained.

The Bass and topological stable ranks for algebras of almost periodic functions on the real line

Let $\Lambda$ be a sub-semigroup of the reals. We show that the Bass and topological stable ranks of the algebras $\textrm {AP}_\Lambda =\{f\in \textrm {AP}: \sigma (f)\subseteq \Lambda \}$ of almost periodic functions on the real line and with Bohr spectrum in $\Lambda$ are infinite whenever the algebraic dimension of the $\mathbb {Q}$-vector space generated by $\Lambda$ is infinite. This extends Suárez’s result for $\textrm {AP}_\mathbb {R}=\textrm {AP}$. Also considered are general subalgebras of AP.

The Bass and Topological Stable Ranks of the Bohl Algebra Are Infinite

The Bohl algebra B is the ring of linear combinations of functions tke?t on the real line, where k is any nonnegative integer, and ? is any complex number, with pointwise operations. We show that the Bass stable rank and the topological stable rank of B (where we use the topology of uniform convergence) are infinite.

The stable rank of $\cstar$-modules

We prove equality between the Topological Stable Rank and the Bass Stable Rank for finitely generated projective left modules over a unital $C^{\ast}$-algebra. In order to do so, the concept of Stable Rank of a Hilbert module is introduced.

RANK PRESERVING IN INTEGRAL EXTENSIONS OF COMMUTATIVE $C^*$-ALGEBRAS

Let $A$, $B$ be two regular commutative unital Banach algebras such that $B$ is integral over $A$. In 2003, Dawson and Feinstein showed that the topological stable rank $\operatorname{tsr}(B) = 1$ whenever $\operatorname{tsr}(A) = 1$. In this note, we investigate whether we will have $\operatorname{tsr}(A) = \operatorname{tsr}(B)$ in general. For instance, when $A$ is a commutative unital $C^*$-algebra, we show that $\operatorname{tsr}(A) \leq \operatorname{tsr}(B)$, and the equality holds at least when the integral extension is separable. In general, $A$ and $B$ have the same Bass stable ranks $\operatorname{Bsr}(A) = \operatorname{Bsr}(B)$.

RANK PRESERVING IN INTEGRAL EXTENSIONS OF COMMUTATIVE $C^*$-ALGEBRAS

Let $A$, $B$ be two regular commutative unital Banach algebras such that $B$ is integral over $A$. In 2003, Dawson and Feinstein showed that the topological stable rank $\operatorname{tsr}(B)=1$ whenever $\operatorname{tsr}(A)=1$. In this note, we investigate whether we will have $\operatorname{tsr}(A) = \operatorname{tsr}(B)$ in general. For instance, when $A$ is a commutative unital $C^*$-algebra, we show that $\operatorname{tsr}(A) \leq \operatorname{tsr}(B)$, and the equality holds at least when the integral extension is separable. In general, $A$ and $B$ have the same Bass stable ranks $\operatorname{Bsr}(A)=\operatorname{Bsr}(B)$.

The covering dimension of a distinguished subset of the spectrum [formula omitted] of [formula omitted] and the algebra of real-symmetric and continuous functions on [formula omitted

We show that the covering dimension, dim E, of the closure E of the interval ] − 1 , 1 [ in the spectrum of H ∞ equals one. Using Suárezʼs result that dim M ( H ∞ ) = 2 , we then compute the Bass and topological stable ranks of the algebra C ( M ( H ∞ ) ) sym of real-symmetric continuous functions on M ( H ∞ ) .

Approximation by invertible elements and the generalized $E$-stable rank for $A({\boldsymbol D})_{\mathsf R}$ and $C({\boldsymbol D})_{\mathrm{sym}}$

We determine the generalized $E$-stable ranks for the real algebra, $C(\boldsymbol{D})_{\mathrm{sym}}$, of all complex valued continuous functions on the closed unit disk, symmetric to the real axis, and its subalgebra $A(\boldsymbol{D})_{\mathsf R}$ of holomorphic functions. A characterization of those invertible functions in $C(E)$ is given that can be uniformly approximated on $E$ by invertibles in $A(\boldsymbol {D})_{\mathsf R}$. Finally, we compute the Bass and topological stable rank of $C(K)_{\mathrm{sym}}$ for real symmetric compact planar sets $K$.

Real-Symmetric Extensions of Invertible Tuples of Multivariable Continuous Functions

We give necessary and sufficient conditions on pairs of real-symmetric compact sets (K, L) in \({{\mathbb C}^n, K\, {\subseteq}\, L}\), such that every invertible n-tuple of real-symmetric continuous functions on K can be extended to such an invertible tuple on L. We also determine the Bass and topological stable ranks of \({C(K)_{{\rm sym}}}\).

The absolute stable rank of [formula omitted

Let X be a compact Hausdorff space and τ a topological involution on X. Let C ( X , τ ) be the real algebra of all complex-valued continuous functions on X that satisfy f ( τ ( x ) ) = f ( x ) ¯ for every x ∈ X . It is shown that the absolute stable rank of C ( X , τ ) equals the Bass, and hence topological stable rank of C ( X , τ ) .

A note on the topological stable rank of an ideal

We show that if J \mathfrak {J} is an ideal of a Banach algebra A \mathfrak {A} , then the left (right) topological stable rank of J \mathfrak {J} is no greater than the left (right) topological stable rank of A \mathfrak {A} .

Homotopical stable ranks for Banach algebras

The connected stable rank and the general stable rank are homotopy invariants for Banach algebras, whereas the Bass stable rank and the topological stable rank should be thought of as dimensional invariants. This paper studies the two homotopical stable ranks, viz. their general properties as well as specific examples and computations. The picture that emerges is that of a strong affinity between the homotopical stable ranks, and a marked contrast with the dimensional ones.

Stable ranks for the real function algebra $C(X,\tau)$

We determine the Bass and topological stable ranks of the real algebra C(X, ) of all complex-valued continuous functions on the compact Hausdor space X that satisfy f = f, where is a topological involution on X.