Abstract
In what follows all vector spaces are over the complex field ℂ, H will be a Hilbert space, and elements of H will usually be denoted in lower case Greek letters: ζ, η, …. We shall also write K. for the compact operators on ℓ2. By a concrete operator algebra we mean a subalgebra A of B(H). We shall assume A is norm closed (although this is not usually necessary), but we shall not assume A is selfadjoint (that is, a C*-algebra). In most of the later sections we shall assume the operator algebras have identity of norm 1 or a contractive approximate identity (c.a.i.). This article is a very brief survey of some of our efforts to study the class of all operator algebras1. In other words, what is the “general theory of operator algebras”? There does not appear to be a text in existence which addresses this topic. A few related questions come to mind:
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What properties does an operator algebra have?
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What are the good examples of operator algebras? (Good examples might include those of classical interest, or those which illustrate typical behaviour, or which are a good source of counterexamples).
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Given an algebra, when is it an operator algebra?
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What are the ‘basic constructions’ with operator algebras? (Such as direct sums, tensor produsts, …)
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What are interesting classes of modules over operator algebras? For instance what should a “projective module” be? It is fairly clear from talk at this conference that there are several quite diffrent notions of “projective”modules over operator algebras, depending on our context and needs.
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What is the good notion/notions of isomorphism of operator algebras?
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Blecher, D.P. (1997). Some General Theory of Operator Algebras and Their Modules. In: Katavolos, A. (eds) Operator Algebras and Applications. NATO ASI Series, vol 495. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-5500-7_2
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