A -algebra of operators on some Hilbert space may be viewed as a non-commutative generalization of a function algebra acting as multiplication operators on some -space associated with a measure on the locally compact space . The space being compact corresponds naturally to the case where the algebra is unital. In the non-unital case any embedding of as an essential ideal in some larger unital -algebra (i.e., the annihilator of in is zero) can be viewed as an analogue of a compactification of the locally compact Hausdorff space . Thus, the one-point compactification of corresponds to the unitization of the algebra . The analogue of the maximal compactification — the Stone–Čech compactification — is the algebra of multipliers of , defined by R.C. Busby in 1967 [a4] and studied in more detail in [a2]. It is defined simply as the idealizer of in (assuming that or, equivalently, that no non-zero vector in is annihilated by ).
Linear operators and on are called left and right centralizers if and for all , in . They are automatically bounded. A double centralizer is a pair of left, right centralizers such that (whence ), and the closed linear spaces of double centralizers becomes a -algebra when product and involution are defined by and (where ). As shown by B.E. Johnson, [a8], there is an isomorphism between the abstractly defined -algebra of double centralizers of and the concrete -algebra . This, in particular, shows that is independent of the given representation of on .
The strict topology on is defined by the semi-norms on with in , [a4]. It is used as an analogue of uniform convergence on compact subsets of in function algebras. Thus, it can be shown that is the strict completion of in and that the strict dual of equals the norm dual of , [a16].
If is the universal Hilbert space for (the orthogonal sum of all Hilbert spaces obtained from states of via the Gel'fand–Naimark–Segal construction), then has a more constructive characterization: Let denote the space of self-adjoint operators in that can be obtained as limits (in the strong topology) of some increasing net of self-adjoint elements from the unitized algebra (cf. also Net (directed set); Self-adjoint operator). Similarly, let be the space of limits of decreasing nets. Then
Thus, for every self-adjoint multiplier there are nets and in , one increasing, the other decreasing, such that . If is -unital, i.e. contains a countable approximate unit, in particular if is separable (cf. also Separable algebra), these nets can be taken as sequences, [a2], [a12], p. 12. In the commutative case, where , whence , this expresses the well-known fact that a bounded, real function on is continuous precisely when it is both lower and upper semi-continuous.
For any -algebra containing as an ideal there is a natural morphism (i.e. a -homomorphism) , defined by , that extends the identity mapping of onto . If is essential in , one therefore obtains an embedding . Any morphism between -algebras and extends uniquely to a strictly continuous morphism , provided that is proper (i.e. maps an approximate unit for to one for ). Such morphisms are the analogues of proper continuous mappings between locally compact spaces. If is -unital and is a quotient morphism, i.e. surjective, then is also surjective. This result may be viewed as a non-commutative generalization of the Tietze extension theorem, [a2], [a13] (cf. also Extension theorems).
The corona of a -algebra is defined as the quotient -algebra , [a13]. The commutative analogue is the compact Hausdorff space (the corona of the locally compact space , [a6]), but the pre-eminent example of such algebras is the Calkin algebra , obtained by taking as the algebra of compact operators on (whence ). Corona -algebras are usually non-separable and cannot even be represented on separable Hilbert spaces, [a14]. Nevertheless, they have important roles in the formulation of G. Kasparov's KK-theory and the later variation known as E-theory. The foremost application, however, is to the theory of extensions: An extension of -algebras and is any -algebra that fits into a short exact sequence (cf. also Exact sequence)
Thus, contains as an ideal, and is simply the quotient morphism. In particular, may be regarded as an extension of by , and in fact a maximal such. Namely, any other extension will give rise to a commutative diagram
Here is the morphism defined above and the induced morphism is known as the Busby invariant for . This invariant determines up to an obvious equivalence, because the right square in the diagram above describes as the pull-back of and over , i.e.
For any quotient morphism between -algebras one may ask whether an element in with specific properties is the image of some in with the same properties. This is known as a lifting problem, and is the non-commutative analogue of extension problems for functions. Many lifting problems have positive (and easy) solutions: If or or , one can find counter-images in with the same properties. However, the properties (being idempotent) and (being normal) are not liftable in general. It follows that the more general commutator relation is not liftable either. But the orthogonality relation is liftable (even in the -fold version ). Using this one may show that the nilpotency relation is liftable, [a1], [a11], [a9].
As advocated by T.A. Loring, lifting problems may with advantage be replaced by -algebra problems concerning projectivity. A -algebra is projective if any morphism into a quotient -algebra can be factored as for some morphism , [a3]. This means that one is lifting a whole -subalgebra and not just some elements. Projective -algebras are the non-commutative analogues of topological spaces that are absolute retracts, but since the category of -algebras is vastly larger than the category of locally compact Hausdorff spaces, projectivity is a rare phenomenon. However, the cone over the -matrices, i.e. the algebra
Corona -algebras form an indispensable tool for more complicated lifting problems, because by Busby's theory, mentioned above, it suffices to solve the lifting for quotient morphisms of the form . Thus, one may utilize the special properties that corona algebras have. A brief outline of these follows.
In topology, a compact Hausdorff space is called sub-Stonean if any two disjoint, open, -compact sets have disjoint closures. Exotic as this may sound, it is a property that any corona set will have, if is locally compact and -compact. In such a space, every open, -compact subset is also regularly embedded, i.e. it equals the interior of its closure in , [a6]. The non-commutative generalization of this is the fact that if is a -unital -algebra, then every -unital hereditary -subalgebra of its corona algebra equals its double annihilator, i.e. , [a13]. The analogue of the sub–Stonean property, sometimes called the -condition, is even more striking: For any two orthogonal elements and in (say ) there is an element in with , such that and . Even better, if and are separable subsets of such that commutes with and annihilates , then the element can be chosen with the same properties, [a11], [a14]. Note that if could be taken as a projection, e.g. the range projection of , this would be a familiar property in von Neumann algebra theory. The fact that corona algebras will never be von Neumann algebras (if is non-unital and -unital) indicates that the property (first established by G. Kasparov as a "technical lemma" ) is useful. Actually, a potentially stronger version is true: If and are monotone sequences of self-adjoint elements in , one increasing, the other decreasing, such that for all , and if and are separable subsets of , such that all commute with and annihilate , then there is an element in such that for all , and commutes with and annihilates , [a11]. This has as a consequence that if is any -unital -subalgebra of , commuting with and annihilating , as above, then for any multiplier in there is an element in the idealizer of in , still commuting with and annihilating , such that for every in , [a5], [a15]. In other words, the natural morphism (with ) is surjective. This indicates the size of corona algebras, even compared with large multiplier algebras.
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