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This is an expository paper on the importance and applications of GB*-algebras in the theory of unbounded operators, which is closely related to quantum field theory and quantum mechanics. After recalling the definition and the main examples of GB*-algebras we exhibit their most important properties. Then, through concrete examples we are led to a question concerning the structure of the completion of a given C*-algebra 𝓐₀[||·||₀], under a locally convex *-algebra topology τ, making the multiplication...
We obtain characterizations of left character amenable Banach algebras in terms of the existence of left ϕ-approximate diagonals and left ϕ-virtual diagonals. We introduce the left character amenability constant and find this constant for some Banach algebras. For all locally compact groups G, we show that the Fourier-Stieltjes algebra B(G) is C-character amenable with C < 2 if and only if G is compact. We prove that if A is a character amenable, reflexive, commutative Banach algebra, then A...
A topological algebra A is said to be fundamental if there exists b > 1 such that for every sequence (xn) in A, (xn) is Cauchy whenever the sequence bn(xn − xn-1) tends to zero as n → ∞. Let A be a complex unital fundamental F-algebra with bounded elements such that A* separates the points on A. Then we prove that the spectrum σ(a) of every element a ∈ A is nonempty compact. Moreover, if A is a division algebra, then A is isomorphic to the complex numbers ℂ. This result is a generalization of...
2000 Mathematics Subject Classification: Primary 46H05, 46H20; Secondary 46M20.Every unital "combinatorially regular" commutative uniform complete locally m-convex algebra is local.
It is a survey talk concerning locally bounded algebras.
Multi-dimensional generalizations of the Wiener-Żelazko and Lévy-Żelazko theorems are obtained.
We investigate the weak amenability of the Banach algebra ℬ(X) of all bounded linear operators on a Banach space X. Sufficient conditions are given for weak amenability of this and other Banach operator algebras with bounded one-sided approximate identities.
For a completely contractive Banach algebra , we find conditions under which the completely bounded multiplier algebra is a dual Banach algebra and the operator amenability of is equivalent to the operator Connes-amenability of . We also show that, in this case, these are equivalent to the existence of a normal virtual operator diagonal.
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