The paper contains a description of the maximal ideal spaces (spectra) $\cM_A$ of bi-invariant function algebras $A$ on a compact group $G$. There are natural compatible structures in $\cM_A$: it is a compact topological semigroup with involution, polar decomposition, and analytic structure. The paper contains a description of $\cM_A$ and related results on function algebras; for example, a bi-invariant function algebra on a connected compact Lie group is antisymmetric if and only if the Haar measure of its maximal torus is multiplicative on it. Some results are extended to the case of compact commutative homogeneous spaces. As a consequence, we get an infinite dimensional version of the Hilbert--Mumford criterion for commutative (as homogeneous spaces) orbits of compact connected Lie groups, where one parameter semigroup is replaced by a finite sequence of them.
Keywords: Maximal ideal space, invariant function algebra, complex Lie semigroup, commutative homogeneous space, Hilbert--Mumford criterion
In this paper we study the maximal ideals in a commutative ring of bicomplex numbers and then we describe the maximal ideals in a bicomplex algebra. We found that the kernel of a nonzero multiplicative BC-linear functional in a commutative bicomplex Banach algebra need not be a maximal ideal. Finally, we introduce the notion of bicomplex division algebra and generalize the Gelfand-Mazur theorem for the bicomplex division Banach algebra.
We study the asymptotics at zero of continuous functions on (0, 1] by means of their asymptotic ideals, i.e., ideals in the ring of continuous functions on (0, 1] satisfying a polynomial growth condition at 0 modulo rapidly decreasing functions at 0. As our main result, we characterize maximal and prime ideals in terms of maximal and prime filters.
Given an homogeneous polynomial on a Banach space $E$ belonging to some maximal or minimal polynomial ideal, we consider its iterated extension to an ultrapower of $E$ and prove that this extension remains in the ideal and has the same ideal norm. As a consequence, we show that the Aron-Berner extension is a well defined isometry for any maximal or minimal ideal of homogeneous polynomials. This allow us to obtain symmetric versions of some basic results of the metric theory of tensor products.
Let $PQC$ stand for the set of all piecewise quasicontionus function on the unit circle, i.e., the smallest closed subalgebra of $L^\infty(\mathbb{T})$ which contains the classes of all piecewise continuous function $PC$ and all quasicontinuous functions $QC=(C+H^\infty)\cap(C+\overline{H^\infty})$. We analyze the fibers of the maximal ideal spaces $M(PQC)$ and $M(QC)$ over maximal ideals from $M(\widetilde{QC})$, where $\widetilde{QC}$ stands for the $C^*$-algebra of all even quasicontinous functions. The maximal ideal space $M(\widetilde{QC})$ is decribed and partitioned into various subsets…
We prove when a Banach ideal of linear operators defined, or characterized, by the transformation of vector-valued sequences is maximal. Known results are recovered as particular cases and new information is obtained. To accomplish this task we study a tensor quasi-norm determined by the underlying sequence classes. The duality theory for these tensor quasi-norms is also developed.
We study some topological properties of maximal ideal spaces of certain algebras of almost periodic functions. Our main result is that such spaces are contractible. We present certain corollaries of this result.
Let X be a compact Hausdorf space, let A be a commutative unital Banach algebra, and let C(X,A) denote the algebra of continuous A-valued functions on $X$ equipped with the uniform norm ||f||=sup{||f(x)||:x\in X} for all f in C(X,A). Hausner, in [Proc. Amer. Math. Soc. 8(1957), 246--249], proved that M is a maximal ideal in C(X,A) if and only if there exist a point x in X and a maximal ideal N in A such that M={f in C(X,A) : f(x) in N}. In this note, we give new characterizations of maximal ideals in C(X,A). We also present a short proof of Hausner's result by a different approach.
In 1971, Grauert and Remmert proved that a commutative, complex, Noetherian Banach algebra is necessarily finite-dimensional. More precisely, they proved that a commutative, complex Banach algebra has finite dimension over $\C$ whenever all the closed ideals in the algebra are (algebraically) finitely generated. In 1974, Sinclair and Tullo obtained a non-commutative version of this result. In 1978, Ferreira and Tomassini improved the result of Grauert and Remmert by showing that the statement is also true if one replaces `closed ideals' by `maximal ideals in the Šilov boundary of $A$'. We shal…
We study the structure of the maximal ideal space $M(H^{\infty})$ of the algebra $H^{\infty}=H^{\infty}(\Di)$ of bounded analytic functions defined on the open unit disk $\Di\subset\Co$. Based on the fact that $dim\ M(H^{\infty})=2$ we prove for $H^{\infty}$ the matrix-valued corona theorem. Our results heavily rely on the topological construction describing maximal ideal spaces of certain algebras of continuous functions defined on the covering spaces of compact manifolds.
Let $ω_1$ be the first uncountable ordinal. By a result of Rudin, bounded operators on the Banach space $C([0,ω_1])$ have a natural representation as $[0,ω_1]\times 0,ω_1]$-matrices. Loy and Willis observed that the set of operators whose final column is continuous when viewed as a scalar-valued function on $[0,ω_1]$ defines a maximal ideal of codimension one in the Banach algebra $\mathscr{B}(C([0,ω_1]))$ of bounded operators on $C([0,ω_1])$. We give a coordinate-free characterization of this ideal and deduce from it that $\mathscr{B}(C([0,ω_1]))$ contains no other maximal ideals. We then obt…
We characterize the algebra $H^\infty \circ L_{m}$, where $m$ is a point of the maximal ideal space of $H^\infty$ with nontrivial Gleason part $P(m)$ and $L_{m} : \mathbb{D}\to P(m)$ is the coordinate Hoffman map. In particular, it is shown that for any continuous function $f: P(m) \to \mathbb{C}$ with $f\circ L_{m} \in H^\infty$ there exists $F\in H^\infty$ such that $F|_{P(m)} = f$.
We address the following two questions regarding the maximal left ideals of the Banach algebra $\mathscr{B}(E)$ of bounded operators acting on an infinite-dimensional Banach pace $E$: (Q1) Does $\mathscr{B}(E)$ always contain a maximal left ideal which is not finitely generated? (Q2) Is every finitely-generated, maximal left ideal of $\mathscr{B}(E)$ necessarily of the form \{T\in\mathscr{B}(E): Tx = 0\} (*) for some non-zero $x\in E$? Since the two-sided ideal $\mathscr{F}(E)$ of finite-rank operators is not contained in any of the maximal left ideals given by (*), a positive answer to the se…
The classical Grauert and Ramspott theorems constitute the foundation of the Oka principle on Stein spaces. In this paper we establish analogous results on the maximal ideal space $M(H^\infty)$ of the Banach algebra $H^\infty$ of bounded holomorphic functions on the open unit disk $\mathbb D\subset\mathbb C$. We illustrate our results by some examples and applications to the theory of operator-valued $H^\infty$ functions.
Let $M(H^\infty)$ be the maximal ideal space of the Banach algebra $H^\infty$ of bounded holomorphic functions on the unit disk $\mathbb D\subset\mathbb C$. We prove that $M(H^\infty)$ is homeomorphic to the Freudenthal compactification $γ(M_a)$ of the set $M_a$ of all non-trivial (analytic disks) Gleason parts of $M(H^\infty)$. Also, we give alternative proofs of important results of Suárez asserting that the set $M_s$ of trivial (one-pointed) Gleason parts of $M(H^\infty)$ is totally disconnected and that the Čech cohomology group $H^2(M(H^\infty),\mathbb Z)=0$.
Polyadic arithmetics is a branch of mathematics related to $p$--adic theory. The aim of the present paper is to show that there are very close relations between polyadic arithmetics and the classic theory of commutative Banach algebras. Namely, let $\ms A$ be the algebra consisting of all complex periodic functions on $\Z$ with the uniform norm. Then the polyadic topological ring can be defined as the ring of all characters $\ms A\to\C$ with convolution operations and the Gelfand topology.
In this paper we give precise characterizations of the relation between the Nevanlinna counting function and pull-back measure of an analytic self-map of the unit disk near the boundary. We show that it is quite worth considering these two concepts on the maximal ideal space of the bounded analytic functions.
A recent result of Leung (Proceedings of the American Mathematical Society, to appear) states that the Banach algebra $\mathscr{B}(X)$ of bounded, linear operators on the Banach space $X=\bigl(\bigoplus_{n\in\mathbb{N}}\ell_\infty^n\bigr)_{\ell_1}$ contains a unique maximal ideal. We show that the same conclusion holds true for the Banach spaces $X=\bigl(\bigoplus_{n\in\mathbb{N}}\ell_\infty^n\bigr)_{\ell_p}$ and $X=\bigl(\bigoplus_{n\in\mathbb{N}}\ell_1^n\bigr)_{\ell_p}$ whenever $p\in(1,\infty)$.
We strengthen, in various directions, the theorem of Garnett that every $σ$-compact, completely regular space $X$ occurs as a Gleason part for some uniform algebra. In particular, we show that the uniform algebra can always be chosen so that its maximal ideal space contains no analytic discs. We show that when the space $X$ is metrizable, the uniform algebra can be chosen so that its maximal ideal space is metrizable as well. We also show that for every locally compact subspace $X$ of a Euclidean space, there is a compact set $K$ in some ${\mathbb C}^N$ so that $\hat K \setminus K$ contains a …
Let $ M (X)$ be the ring of all real measurable functions on a measurable space $(X, \mathscr{A})$. In this article, we show that every ideal of $M(X)$ is a $Z^{\circ}$-ideal. Also, we give several characterizations of maximal ideals of $M(X)$, mostly in terms of certain lattice-theoretic properties of $\mathscr{A}$. The notion of $T$-measurable space is introduced. Next, we show that for every measurable space $(X,\mathscr{A})$ there exists a $T$-measurable space $(Y,\mathscr{A}^{\prime})$ such that $M(X)\cong M(Y)$ as rings. The notion of compact measurable space is introduced. Next, we prov…
Stopping-time Banach spaces is a collective term for the class of spaces of eventually null integrable processes that are defined in terms of the behaviour of the stopping times with respect to some fixed filtration. From the point of view of Banach space theory, these spaces in many regards resemble the classical spaces such as $L^1$ or $C(Δ)$, but unlike these, they do have unconditional bases. In the present paper we study the canonical bases in the stopping-time spaces in relation to factorising the identity operator thereon. Since we work exclusively with the dyadic-tree filtration, this …
The space $F^p$ ($1<p<\infty$) consists of all holomorphic functions $f$ on the open unit disk $\Bbb D$ for which $\lim_{r\to 1}(1-r)^{1/q}\log^+M_{\infty}(r,f)=0,$ where $M_{\infty}(r,f)=\max_{\vert z\vert\le r}\vert f(z)\vert$ with $0<r<1$. Stoll [5, Theorem 3.2] proved that the space $F^p$ with the topology given by the family of seminorms $\left\{\Vert \cdot\Vert_{q,c}\right\}_{c>0}$ defined for $f\in F^q$ as $\Vert f\Vert_{q,c}:=\sum_{n=0}^{\infty}\vert a_n\vert\exp\left(-cn^{1/(q+1)} \right)<\infty,$ is a countably normed Fréchet algebra. Notice that for each $p>1$, $F^p$ is the Fréchet …
We classify the weak*-closed maximal left ideals of the measure algebra $M(G)$ for certain Hermitian locally compact groups $G$ in terms of the irreducible representations of $G$ and their asymptotic properties. In particular, we obtain a classification for connected nilpotent Lie groups, and the Euclidean rigid motion groups. We also prove a version of this result for certain weighted measure algebras. We apply our classification to obtain an analogue of Barnes' Theorem on integrable representations for representations vanishing at infinity. We next study the relationship between weak*-closed…
We study the mapping property of the commutator of bilinear Hardy-Littlewood maximal operator in homogeneous Triebel-Lizorkin space. We also show that the commutator of bilinear Hardy-Littlewood maximal operator is a compact operator acting on product of Lebesgue spaces. The results are new even in the linear case.
