Extension Problems and Stable Ranks

Mortini, Raymond; Rupp, Rudolf

doi:10.1007/978-3-030-73872-3

Cited by 14 publications

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“…(1) This is well known (see e.g. [5,Theorem 12.42]). The idea is that the C 1function h WD u x i u y satisfies the Cauchy-Riemann equations due to u xx C u yy D 0.…”

mentioning

confidence: 83%

The Milne-Thomson formula for the harmonic conjugate and its associated holomorphic function

Mortini

2021

Elem. Math.

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“…(1) This is well known (see e.g. [5,Theorem 12.42]). The idea is that the C 1function h WD u x i u y satisfies the Cauchy-Riemann equations due to u xx C u yy D 0.…”

mentioning

confidence: 83%

The Milne-Thomson formula for the harmonic conjugate and its associated holomorphic function

Mortini

2021

Elem. Math.

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“…The notion of the Bass stable rank was introduced in algebraic Ktheory by Bass [4], but it has since proved useful in other branches of mathematics as well, for example, in analysis (see e.g. [20]).…”

Section: Bass Stable Rank Of A(p) Ismentioning

confidence: 99%

“…A(p) is a Hermite ring. It is known that a commutative unital ring having Bass stable rank ≤ 2 is Hermite (see e.g., [20,Corollary 36.17]). In light of this result and Theorem 3.2, we have:…”

Section: 1mentioning

confidence: 99%

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

Sasane¹

2023

Preprint

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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 lim n ∞ p(n)∞, with pointwise addition and scalar multiplication, a weighted Hadamard multiplication * with weight given by p (i.e., (f * g)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 H 1 (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.

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“…If 𝐾 ⊆ ℂ and 𝑀 ⊆ 𝐾, then 𝑀 𝐾 and 𝜕 𝐾 𝑀 are the closure, respective boundary of 𝑀 inside the topological space 𝐾. If 𝐾 is closed in ℂ, then 𝑀 𝐾 , the relative closure of 𝑀 in 𝐾, coincides with 𝑀, the closure of 𝑀 in ℂ (see [7,Proposition 1.31]). If  is the set of holes of a compact set 𝐾 ⊆ ℂ (i.e., the set of bounded components of the open set ℂ ⧵ 𝐾), then K denotes the topological hull of 𝐾; that is,…”

Section: Polynomial Convex Hullmentioning

confidence: 99%

“…Note that each hole is a non-void open set again (see [7,Proposition 1.144]). It is well known that in ℂ, K coincides with the polynomial convex hull…”

Section: Polynomial Convex Hullmentioning

confidence: 99%

Which sets are images of disk‐algebra functions?

Mortini

Pflug

2023

Mathematische Nachrichten

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We give a characterization of those compact sets in the plane with finitely many holes that are images of disk-algebra functions. We also show that the image of the closed unit disk via a polynomial is, in general, not polynomially convex. K E Y W O R D Sadmissible compacta, crosscuts, images of disk-algebra functions, Jordan domains, locally connected continua, polynomial convexity, polynomials, winding number M S C ( 2 0 2 0 ) 30H05 Primary, 30C10, 30C20 Secondary INTRODUCTIONAt the Mini-Workshop Complex Approximation and Universality of the MFO in 2008 the first author asked for a characterization of the images of the disk-algebra functions (see [4, p. 344] and [6]). More precisely, let 𝐴(𝐃) be the space of functions continuous on the closed unit disk 𝐃 and holomorphic in its interior 𝔻. Given a compact set 𝐾 of the plane, under which conditions on 𝐾 there exists 𝑓 ∈ 𝐴(𝐃) such that 𝑓(𝐃) = 𝐾? By the Hahn-Mazurkiewicz theorem (see, e.g., [7, Theorem 13.21]) a compact metric space 𝑋 is a curve (that is, the continuous image via a map ℎ of [0,1]) if and only if 𝑋 is a locally path-connected continuum or a singleton. Hence, the map 𝑓(𝑧) ∶= ℎ(|𝑧|) gives a continuous map of 𝐃 onto 𝑋. Holomorphy puts additional restrictions on 𝐾. For example, the interior 𝐾 • must of course be nonvoid if 𝐾 is not a singleton. In Proposition 4.1, several easy necessary conditions will be provided. 1 Our main result is to give necessary and sufficient conditions on 𝐾 within the class of planar compacta with a finite number (or none) of holes for the existence of 𝑓 ∈ 𝐴(𝐃) with 𝑓(𝐃) = 𝐾. We also answer the question whether the image of 𝐃 with respect to a polynomial 𝑝 ∈ ℂ[𝑧] is polynomially convex, or whether 𝑝(𝐃) may have holes. KNOWN TOPOLOGICAL TOOLSIn this preliminary section, we present for the reader's convenience several known topological results in order to avoid to look up too many different sources, some of them not being directly available. As it is quite often the case with topological results involving connectedness, many of these results are intuitively "clear," but the formal proofs in this abstract setting are quite involved (e.g., see, for instance, [7, Proposition 1.171] or [7, Theorem 12.71]).Recall that a topological space 𝑋 is locally connected if for every 𝑥 ∈ 𝑋 and every neighborhood 𝑁(𝑥) ⊆ 𝑋 of 𝑥 there exists a connected open set 𝐶 with 𝑥 ∈ 𝐶 ⊆ 𝑁(𝑥) and 𝑋 is locally path-connected if for every 𝑥 ∈ 𝑋 and every neighborhood 𝑈 ⊆ 𝑋 of 𝑥 there exists a neighborhood 𝑉 of 𝑥 such that 𝑉 ⊆ 𝑈 and for any pair (𝑢, 𝑣) of points in 𝑉 there is a path from 𝑢 to 𝑣 lying in 𝑈.Dedicated to the memory of H. Garth Dales and Jean Esterle.

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Extension Problems and Stable Ranks

Cited by 14 publications

References 0 publications

The Milne-Thomson formula for the harmonic conjugate and its associated holomorphic function

The Milne-Thomson formula for the harmonic conjugate and its associated holomorphic function

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

Which sets are images of disk‐algebra functions?

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