A notion of weak invertibility in a unital associative algebra A and a corresponding notion of strong spectrum of an element of A is defined. It is shown that many relationships between the Jacobson radical, the group of invertibles and the spectrum have analogues relating the strong radical, the set of weakly invertible elements and the strong spectrum. The nonunital case is also discussed. A characterization is given of all (submultiplicative) norms on A in which every modular maximal ideal M...

Let E be a real normed linear space with unit ball B and unit sphere S. The classical modulus of convexity of J. A. Clarkson [2] δE(ε) = inf {1 - 1/2||x + y||: x,y ∈ B, ||x - y|| ≥ ε} (0 ≤ ε ≤ 2)is well known and it is at the origin of a great number of moduli defined by several authors. Among them, D. F. Cudia [3] defined the directional, weak and directional weak modulus of convexity of E, respectively, asδE(ε,g) = inf {1 - 1/2||x + y||: x,y ∈ B, g(x-y) ≥ ε}δE(ε,f) = inf {1 - 1/2 f(x,y): x,y...

For a function algebra A let ∂A be the Shilov boundary, δA the Choquet boundary, p(A) the set of p-points, and |A| = |f|: f ∈ A. Let X and Y be locally compact Hausdorff spaces and A ⊂ C(X) and B ⊂ C(Y) be dense subalgebras of function algebras without units, such that X = ∂A, Y = ∂B and p(A) = δA, p(B) = δB. We show that if Φ: |A| → |B| is an increasing bijection which is sup-norm-multiplicative, i.e. ||Φ(|f|)Φ(|g|)|| = ||fg||, f,g ∈ A, then there is a homeomorphism ψ: p(B) → p(A) with respect...

Let G be a finite group. Consider the algebra A of all complex functions on G (with pointwise product). Define a coproduct Δ on A by Δ(f)(p,q) = f(pq) where f ∈ A and p,q ∈ G. Then (A,Δ) is a Hopf algebra. If G is only a groupoid, so that the product of two elements is not always defined, one still can consider A and define Δ(f)(p,q) as above when pq is defined. If we let Δ(f)(p,q) = 0 otherwise, we still get a coproduct on A, but Δ(1) will no longer be the identity in A ⊗ A. The pair (A,Δ)...

It is proved that a Köthe sequence space is weakly orthogonal if and only if it is order continuous. Criteria for weak property ($\beta$) in Orlicz sequence spaces in the case of the Luxemburg norm as well as the Orlicz norm are given.

We give sufficient conditions for subsets of compact operators to be weakly precompact. Let $L_{w*}(E*,F)$ (resp. $K_{w*}(E*,F)$) denote the set of all w* - w continuous (resp. w* - w continuous compact) operators from E* to F. We prove that if H is a subset of $K_{w*}(E*,F)$ such that H(x*) is relatively weakly compact for each x* ∈ E* and H*(y*) is weakly precompact for each y* ∈ F*, then H is weakly precompact. We also prove the following results: If E has property (wV*) and F has property (V*), then $K_{w*}(E*,F)$ has property (wV*). Suppose...

We show that if ϕ is a continuous homomorphism between weighted convolution algebras on ℝ⁺, then its extension to the corresponding measure algebras is always weak* continuous. A key step in the proof is showing that our earlier result that normalized powers of functions in a convolution algebra on ℝ⁺ go to zero weak* is also true for most measures in the corresponding measure algebra. For some algebras, we can determine precisely which measures have normalized powers converging to zero weak*. We...

Köthe and Toeplitz introduced the theory of sequence spaces and established many of the basic properties of sequence spaces by using methods of classical analysis. Later many of these same properties of sequence spaces were reestablished by using soft proofs of functional analysis. In this note we would like to point out that an improved version of a classical lemma of Schur due to Hahn can be used to give very short proofs of two of the weak sequential completeness results of Köthe and Toeplitz....

In this paper we consider the following Dirichlet problem for elliptic systems: $$\begin{array}{ccc}\hfill \overline{DA(x,u(x),Du(x\left)\right)}=& B(x,u(x),Du(x\left)\right),x\in \Omega ,u\left(x\right)=\hfill & \hfill 0,x\in \partial \Omega ,\end{array}$$ where $D$ is a Dirac operator in Euclidean space, $u\left(x\right)$ is defined in a bounded Lipschitz domain $\Omega$ in ${ℝ}^n$ and takes value in Clifford algebras. We first introduce variable exponent Sobolev spaces of Clifford-valued functions, then discuss the properties of these spaces and the related operator theory in these spaces. Using the Galerkin method, we obtain the existence of weak solutions to the scalar part of the above-mentioned...

Let G be a locally compact group, K a compact subgroup of G and A(G/K) the Fourier algebra of the coset space G/K. Applying results from [E. Kaniuth, Weak spectral synthesis in commutative Banach algebras, J. Funct. Anal. 254 (2008), 987-1002], we establish injection and localization theorems relating weak spectral sets and weak Ditkin sets for A(G/K) to such sets for A(H/H ∩ K), where H is a closed subgroup of G. We also prove some results towards the analogue of Malliavin's theorem for weak spectral...