2017
Том 69
№ 9

# Structure of Banach algebras of bounded continuous functions in the open disk which contain H∞, Hoffman algebra, and nontangential limits

Іванов О. В.

Абстракт

Let $\mathcal{H}^G_B$ be an algebra of bounded continuous functions in an open disk $\mathbb{D}$, of the form $\mathcal{H}_B \bigcap G$, where $G = C(M(H^{\infty})) \overset{\rm def}{=} \text{alg}(H^{\infty}, \overline{H^{\infty}})$ and $\mathcal{H}_B$ is a closed subalgebra in $C(D)$ which consists of all the functions which have nontangential limits a. e. on $\mathbb{T}$ belonging to the Douglas algebra $B$. The goal of this paper is to describe the maximal ideal space $M(\mathcal{H}^G_B)$ of the algebra $\mathcal{H}^G_B$. We prove that $M(\mathcal{H}^G_B) = M(B) \bigcup M(\mathcal{H}^G_0)$, where $\mathcal{H}^G_0$ is a closed ideal in $G$ which consists of all the functions having non-tangential limits a. e. on $\mathbb{T}$ and these limits are equal to zero. We prove that $H^{\infty}[\overline{z}] = \mathcal{H}^G_{H^{\infty}+C}$ in the disk. We generalize Chang-Marshall theorem on Banach algebras $\mathcal{H}^G_B$ and prove that $\mathcal{H}^G_B = {\rm alg}(\mathcal{H}^G_{H^{\infty}}, \overline{IB})$ for any Douglas algebra $B$, where $IB = \{u_{\alpha}\}_B$ is a set of inner functions such that $\overline{u_{\alpha}} \in B$ on $T$.

Англомовна версія (Springer): Ukrainian Mathematical Journal 45 (1993), no. 7, pp 1023-1030.

Зразок цитування: Іванов О. В. Structure of Banach algebras of bounded continuous functions in the open disk which contain H, Hoffman algebra, and nontangential limits // Укр. мат. журн. - 1993. - 45, № 7. - С. 924–931.

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