Maximal subalgebra of Douglas algebra.
It is proved that the Fréchet algebra has exactly three closed subalgebras which contain nonconstant functions and which are invariant, in the sense that whenever and is a biholomorphic map of the open unit ball of onto . One of these consists of the holomorphic functions in , the second consists of those whose complex conjugates are holomorphic, and the third is .
Let A be a complex Banach algebra with a unit e, let T, φ be continuous functionals, where T is linear, and let F be a nonlinear entire function. If T ∘ F = F ∘ φ and T(e) = 1 then T is multiplicative.
Let A be a complex Banach algebra with a unit e, let F be a nonconstant entire function, and let T be a linear functional with T(e)=1 and such that T∘F: A → ℂ is nonsurjective. Then T is multiplicative.
Let Ω be an open subset of a real Banach space E and, for 1 ≤ m ≤, let Cm(Ω) denote the algebra of all m-times continuously Fréchet differentiable real functions defined on Ω. We are concerned here with the question as to wether every nonzero algebra homomorphism φ: Cm(Ω) → R is given by evaluation at some point of Ω, i.e., if there exists some a ∈ Ω such that φ(f) = f(a) for each f ∈ Cm(Ω). This problem has been considered in [1,4,5] and [6]. In [6], a positive answer is given in the case that...
L. de Branges has originated a viewpoint one of whose repercussions has been the detailed analysis of certain Hilbert spaces of holomorphic functions contained within the Hardy space H2 of the unit disk (...).
We generalize, to the setting of Arveson’s maximal subdiagonal subalgebras of finite von Neumann algebras, the Szegő -distance estimate and classical theorems of F. and M. Riesz, Gleason and Whitney, and Kolmogorov. As a byproduct, this completes the noncommutative analog of the famous cycle of theorems characterizing the function algebraic generalizations of from the 1960’s. A sample of our other results: we prove a Kaplansky density result for a large class of these algebras, and give a necessary...
We continue the study of the completeness and completions of normed algebras of differentiable functions Dⁿ(K) (where K is a perfect, compact plane set), initiated by Bland, Dales and Feinstein [Studia Math. 170 (2005) and Indian J. Pure Appl. Math. 41 (2010)]. We prove new characterizations of the completeness of D¹(K) and results concerning the semisimplicity of the completion of D¹(K). In particular, we prove that semi-rectifiability is necessary for the completion of D¹(K) to be semisimple in...