Prime ideals and automatic continuity problems for Banach algebras

Conditions are given for Banach algebras $\text{U}$ and commutative Banach algebras $\text{B}$ which insure that every homomorphism v from $\text{U}$ into $\text{B}$ is continuous. Similar results are obtained for derivations which either map the algebra $\text{U}$ into itself or map the algebra into a suitable $\text{U}$-module.     

The relationship between a commutative Banach algebra and its maximal ideal space

Our main result is an extension of a theorem due to Novodvorskii and Taylor; we give some special cases. Let A be a commutative Banach algebra with identity, and let Δ be its maximal ideal space. Let B be a Banach algebra with identity; let B^?1 denote the invertible group in B and id B denote the set of idempotents in B. Let [$\text{(A}\text{}\text{?}\text{bo B)}{}^{\mathrm{?1}}$] denote the set of path components of $\text{(A}\text{}\text{?}\text{bo B)}{}^{\mathrm{?1}}$, and [Δ, B^?1] denote the set of homotopy classes of continuous maps of Δ into B^?1. We prove that the Gelfand transform on A induces a bijection of [$\text{(A}\text{}\text{?}\text{bo B)}{}^{\mathrm{?1}}$] onto [Δ, B^?1], and extend this result to prove a theorem of Davie. We show that the Gelfand transform induces a bijection of [$\text{id}\text{(A}\text{}\text{?}\text{bo B)}$] onto [Δ, id B], and investigate consequences of this result for specific examples of the Banach algebra B.     

Symplectic groups and the Klein-Gordon field

In this paper a Cohen factorization theorem x = a^t · x_t (t > 0) is proved for a Banach algebra A with a bounded approximate identity, where t ? a^t is a continuous one-parameter semigroup in A. This theorem is used to show that a separable Banach algebra B has a bounded approximate identity bounded by 1 if and only if there is a homomorphism θ from L¹($\text{R}$⁺) into B such that ∥ θ ∥ = 1 and θ(L¹($\text{R}$⁺)). B = B = B · θ(L¹($\text{R}$⁺)). Another corollary is that a separable Banach algebra with bounded approximate identity has a commutative bounded approximate identity, which is bounded by 1 in an equivalent algebra norm.     

Un contre-exemple pour les formes invariantes sur un corps fini

Let A be a Jordan algebra over the reals which is a Banach space with respect to a norm satisfying the requirements: (i) ∥ a ° b ∥ ≤ ∥ a ∥ ∥ b ∥, (ii) ∥ a² ∥ = ∥ a ∥², (iii) ∥ a² ∥ ≤ ∥ a² + b² ∥ for a, b?A. It is shown that A possesses a unique norm closed Jordan ideal J such that $\text{A}\text{J}$ has a faithful representation as a Jordan algebra of self-adjoint operators on a complex Hilbert space, while every “irreducible” representation of A not annihilating J is onto the exceptional Jordan algebra M₃⁸.     

Interpolating sequences and the Shilov boundary of H∞(Δ)

Let H^∞(Δ) denote the Banach algebra of bounded analytic functions on the open unit disc, let $\text{M}$ denote its maximal ideal space, and let ? denote its Shilov boundary. D. J. Newman has shown that a homomorphism ? in $\text{M}$ will be in ? if and only if ? is unimodular on all Blaschke products. We answer a question of K. Hoffman by showing that ? will be in ? if and only if ? is unimodular on every Blaschke product whose zero set is an interpolating sequence. Our method is based on a construction due to L. Carleson, originally developed for the proof of the Corona theorem.     

Spectral representations for abelian automorphism groups of von Neumann algebras

Let $\text{A}$ be a von Neumann algebra, let σ be a strongly continuous representation of the locally compact abelian group G as ¹-automorphisms of $\text{A}$. Let M(σ) be the Banach algebra of bounded linear operators on $\text{A}$ generated by ∝ σ_tdμ(t) (μ?M(G)). Then it is shown that M(σ) is semisimple whenever either (i) $\text{A}$ has a σ-invariant faithful, normal, semifinite, weight (ii) σ is an inner representation or (iii) G is discrete and each σ_t is inner. It is shown that the Banach algebra L(σ) generated by $\text{∝ ?(t)σ}{}_{\mathrm{t}}\text{dt (? ? L}{}^1\text{(G))}$ is semisimple if a is an integrable representation. Furthermore, if σ is an inner representation with compact spectrum, it is shown that L(σ) is embedded in a commutative, semisimple, regular Banach algebra with isometric involution that is generated by projections. This algebra is contained in the ultraweakly continuous linear operators on $\text{A}$. Also the spectral subspaces of σ are given in terms of projections.     

Automatic continuity over Moore groups

LetG be a Moore group, letB be a Banach algebra, and let :L ¹(G)B be a homomorphism. We show that is continuous if and only if its restriction to the center ofL ¹(G) is continuous. As a consequence, we obtain that (i) every homomorphism fromL ¹(G) orC ^*(G) onto a dense subalgebra of a semisimple Banach algebra, and (ii) every epimorphism fromC ^*(G) onto a Banach algebra is automatically continuous.     

On the Schur multiplier norm of matrices

The Schur product of two n×n complex matrices A=(a_ij), B=(b_ij) is defined by A°B=(a_ijb_ij. By a result of Schur [2], the algebra of n×n matrices with Schur product and the usual addition is a commutative Banach algebra under the operator norm (the norm of the operator defined on $\text{C}$ⁿ by the matrix). For a fixed matrix A, the norm of the operator B?A°B on this Banach algebra is called the Schur multiplier norm of A, and is denoted by ∥A∥^m. It is proved here that $\text{∥A∥=∥U}{}^1\text{AU∥}{}_{\mathrm{m}}$ for all unitary U (where ∥·∥ denotes the operator norm) iff A is a scalar multiple of a unitary matrix; and that ∥A∥_m=∥A∥ iff there exist two permutations P, Q, a p×p (1?p?n) unitary U, an (n?p)×(n?p)1 contraction C, and a nonnegative number λ such that

$\text{A=λP}\text{U}\text{0}\text{0}\text{C}\text{Q;}$

and this is so iff $\text{∥A°}\text{A}\text{?}\text{∥=∥A∥}{}^2$, where ā is the matrix obtained by taking entrywise conjugates of A.     

The automorphisms of convolution algebras of C∞ functions

We find the automorphisms and the spectra of several different topological convolution algebras of C^∞-functions on the real line. Starting with the convolution algebra of compactly supported C^∞-functions, equipped with the usual LF-topology, we define a corresponding convolution algebra of C^∞-functions of arbitrarily fast exponential decay at ∞; and convolution algebras of a given finite degree r of exponential decay at ∞. These algebras may be described topologically as “hyper Schwartz spaces.” With a natural Frechet topology, which we define, they get a structure as locally m-convex algebras. The continuous automorphisms and spectra of these algebras are described completely. We show that the algebra of C^∞-functions of infinitly fast exponential decay at ∞, $\text{H J}$, on the one hand, and the algebra of C^∞-functions of only a finite degree $\text{e}{}^{\mathrm{?r¦x¦}}$ decay at ∞, $\text{J}$_r⁰, on the other hand, have quite different automorphisms, although $\text{H J}$ = ∩_r$\text{J}$_r⁰. As an application, we show that the conformal group is canonically represented as the full group of automorphisms of $\text{J}$_r⁰, and that this representation does not extend to a representation on the Banach algebra L¹($\text{R}$).     

A classification theorem for essentially binormal operators

An essentially binormal operator on Hilbert space is an operator which is unitarily equivalent to a 2 × 2 matrix of essentially commuting, essentially normal operators. A natural invariant of essentially binormal operators up to unitary equivalence in the Calkin Algebra is the reducing essential 2 × 2 matricial spectrum. A nonempty compact subset X of the set of 2 × 2 matrices is called hypoconvex, if it is the reducing essential 2 × 2 matricial spectrum of an operator on Hilbert space. The set EN₂(X) is then defined to be the set of all equivalence classes (up to unitary equivalence in the Calkin algebra) of essentially binormal operators whose reducing essential 2 × 2 matricial spectrum coincides with X. The aim of this paper is to prove a result that enables one to compute EN₂(X) in terms of the topological structure of the space $\text{X}\text{?}$ of unitary orbits of X. Indeed, it is shown that for every hypoconvex subset X of the set of 2 × 2 matrices, there exists a natural homomorphism from $\text{Ext}\text{(}\text{X}\text{?}\text{)}$ onto EN₂(X). Also, a six term cyclic exact sequence is obtained, which produces a characterization of the kernel of the above-mentioned homomorphism.     

An addendum to “M-ldeal structure in Banach algebras”

M-ideals in a commutative Banach algebra A are shown to correspond to certain hermitian central projections in $\text{A}{}^7$, and thus possess bounded approximate identities. This leads to a new characterization of M-ideals in function algebras.     

When does continuity on the center imply continuity?

LetA be a Banach algebra. We give a condition forA which forces a homomorphism fromA into a Banach algebra to be continuous if the closure of its continuity ideal has finite codimension, and if its restriction to the center ofA is continuous. We apply this result to answer the question in the title for centralC ^*-algebras,AW ^*-algebras, andL ¹ (G) for certain [FIA]^?-groupsG.     

Grothendieck's theorem for noncommutative C1-algebras,with an Appendix on Grothendieck's constants

We study a conjecture of Grothendieck on bilinear forms on a C¹-algebra $\text{Ol}$. We prove that every “approximable” operator from $\text{Ol}$ into $\text{Ol}$¹ factors through a Hilbert space, and we describe the factorization. In the commutative case, this is known as Grothendieck's theorem. These results enable us to prove a conjecture of Ringrose on operators on a C¹-algebra. In the Appendix, we present a new proof of Grothendieck's inequality which gives an improved upper bound for the so-called Grothendieck constant k_G.     

Semi-crossed products of C1-algebras

Given a C¹-algebra $\text{U}$ and endomorphim α, there is an associated nonselfadjoint operator algebra $\text{Z}$⁺ X_α$\text{U}$, called the semi-crossed product of $\text{U}$ with α. If α is an automorphim, $\text{Z}$⁺ X_α$\text{U}$ can be identified with a subalgebra of the C¹-crossed product $\text{Z}$⁺ X_α$\text{U}$. If $\text{U}$ is commutative and α is an automorphim satisfying certain conditions, $\text{Z}$⁺ X_α$\text{U}$ is an operator algebra of the type studied by Arveson and Josephson. Suppose S is a locally compact Hausdorff space, φ: S → S is a continuous and proper map, and α is the endomorphim of U=C₀(S) given by α(?) = ? ō φ. Necessary and sufficient conditions on the map φ are given to insure that the semi-crossed product Z⁺X_αC₀(S) is (i) semiprime; (ii) semisimple; (ii) strongly semisimple.     

The continuity of derivations of Banach algebras

This paper investigates conditions on a semisimple Banach algebra $\text{U}$ and a Banach $\text{U}$-module $\text{M}$ which insure that every derivation from $\text{U}$ into $\text{M}$ is necessarily a bounded linear operator.     

Extreme points in sets of positive linear maps on B(H)

Three main results are obtained: (1) If $\text{D}$ is an atomic maximal Abelian subalgebra of $\text{B}$($\text{H}$), $\text{P}$ is the projection of $\text{B}$($\text{H}$) onto $\text{D}$ and h is a complex homomorphism on $\text{D}$, then h ° $\text{P}$ is a pure state on $\text{B}$($\text{H}$). (2) If {P_n} is a sequence of mutually orthogonal projections with rank(P_n) = n and ∑ P_n = I, $\text{P}$ is the projection of $\text{B}$($\text{H}$) onto {P_n}″ given by P(T)=∑trace_n(T)P_n and h is a homomorphism on {P_n}″ such that h(P_n) = 0 for all n then h ° $\text{P}$ induces a type II_∞ factor representation of the Calkin algebra. (3) If $\text{M}$ is a nonatomic maximal Abelian subalgebra of $\text{B}$($\text{H}$) then there is an atomic maximal Abelian subalgebra $\text{D}$ of $\text{B}$($\text{H}$) and a large family {Φ_α} of ¹-homomorphisms from $\text{D}$ onto $\text{M}$ such that for each α, Φ_α ° $\text{P}$ is an extreme point in the set of projections from $\text{B}$($\text{H}$) onto $\text{M}$. (Here $\text{P}$ denotes the projection of $\text{B}$($\text{H}$) onto $\text{D}$.)     

Green's and Dirichlet spaces associated with fine Markov processes

This is the second paper in a series devoted to Green's and Dirichlet spaces. In the first paper, we have investigated Green's space $\text{K}$ and the Dirichlet space $\text{H}$ associated with a symmetric Markov transition function p_t(x, B). Now we assume that p is a transition function of a fine Markov process X and we prove that: (a) the space $\text{H}$ can be built from functions which are right continuous along almost all paths; (b) the positive cone $\text{K}$⁺ in $\text{K}$ can be identified with a cone M of measures on the state space; (c) the positive cone $\text{H}$⁺ in $\text{H}$ can be interpreted as the cone of Green's potentials of measures μ?M. To every measurable set B in the state space E there correspond a subspace $\text{K}$(B) of $\text{K}$ and a subspace $\text{H}$(B) of $\text{H}$. The orthogonal projections of $\text{K}$ onto $\text{K}$ and of $\text{H}$ onto $\text{H}$(B) can be expressed in terms of the hitting probabilities of B by the Markov process X. As the main tool, we use additive functionals of X corresponding to measures μ?M.     

The Euclidean distance construction of order homomorphisms

The Euclidean distance technique of Arrow and Hahn is used to construct an upper semicontinuous order homomorphism (partial utility function) from (X, ≻) to ($\text{R}$, >), where X is a closed, convex subset of $\text{R}$^N and ≻ is a continuous strict partial order on X. It is also shown that the order homomorphism is upper semicontinuous as a function on $\text{X×}\text{P}\text{(X)}$, where$\text{P}\text{(X)}$ is the set of continuous strict partial orders on X, taken with the topology of closed convergence.     

Uniqueness of analytic continuation: Necessary and sufficient conditions

Necessary and sufficient conditions for uniqueness of analytic continuation are investigated for a system of m ? 1 first-order linear homogeneous partial differential equations in one unknown, with complex-valued $\text{b}$^∞ coefficients, in some connected open subset of $\text{R}$^k, k ? 2. The type of system considered is one for which there exists a real k-dimensional, $\text{b}$^∞, connected C-R submanifold M^k of $\text{C}$ⁿ, for k, n ? 2, such that the system may be identified with the induced Cauchy-Riemann operators on M^k. The question of uniqueness of analytic continuation for a system of partial differential equations is thus transformed to the question of uniqueness of analytic continuation for C-R functions on the manifold M^k ? Cⁿ. Under the assumption that the Levi algebra of M^k has constant dimension, it is shown that if the excess dimension of this algebra is maximal at every point, then M^k has the property of uniqueness of analytic continuation for its C-R functions. Conversely, under certain mild conditions, it is shown that if M^k has the property of uniqueness of analytic continuation for all $\text{b}$^∞ C-R functions, and if the Levi algebra has constant dimension on all of M^k, then the excess dimension must be maximal at every point of M^k.     

Coherence of some rings of functions

To say that a commutative ring $\text{R}$ with unit is coherent amounts to saying, in case $\text{R}$ has no divisors of zero, that the intersection of two finitely generated ideals in $\text{R}$ is finitely generated. We prove that the ring H^∞ of bounded analytic functions in the unit disc is coherent, while the disc algebra A is not coherent. For any positive measure μ, L^∞(μ) is coherent.