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For an abstract
-algebra the GNS construction provides a concrete representation of its elements as bounded operators on
a Hilbert space.
For non-selfadjoint algebras there is, hitherto, no analogue in the framework of classical functional analysis.
But endowed with an operator space structure (which is compatible with the multiplicative structure), these
non-selfadjoint algebras do have a representation in some
( theorem of Ruan type
for
operator algebras).
The so-called operator modules (over algebras)
are also characterized by Axioms of Ruan type;
here, matrices whose entries are algebra elements take the place of the scalar ones.
The corresponding morphisms are the
completely bounded
module homomorphisms , the most important properties of which make their appearance in Representation, Decomposition and
Extension Theorems.
Subsections
Prof. Gerd Wittstock
2001-01-07