Extensions of pure states of C*-algebras
Let A be a C*-algebra and B be a C*-subalgebra of A. B is said to have the extension property in A if every pure state of B has unique state extension to A. Such pairs (A, B) are considered where B is a maximal abelian self-adjoint subalgebra (masa) of A with the extension property. For A unital and n-homogeneous, results are obtained for the structure of (A, B) as fibre bundles over the primitive ideal space of A (2.2). For A a Type I C -algebra, necessary and sufficient conditions are given for B" to be maximal abelian in A" for all representations of A (2.3). The problem of whether an atomic masa of JL(H) has the extension property in i(H) is related to questions about other pairs (A,B) (3.4.2). A subalgebra of 3t(H) is constructed in which the atomic masa does have the extension property (3.2). Let Fn denote the free group on n generators, A = C*(Fn) and B be the masa of A generated by the image of one of the group generators. It is shown that states of A which restrict to pure states of B are pairwise non-equivalent (3.4.3). A similar result is obtained for states of the Calkin algebra Q which restrict to pure states of a masa of Q constructed by Anderson (3.4.4). In particular, Anderson's masa is not the image of an atomic masa of (H). Conditions of a dynamical nature are given for B to have the extension property in A for certain cases where B is a masa of a C*-crossed product algebra A (4.2). Let A be the W*-crossed product algebra for a W*-dynamical system (M,G,?). For G an amenable group, certain G-invariant states of M are shown to induce conditional expectations from A onto the group von Neumann algebra (G) (4.3, 4.4). This result generalises that of Anderson concerning the existence of many conditional expectations from (H) onto a continuous masa of (H). In conclusion it must be said that the results of this thesis throw little light on the question of whether pure states of an atomic masa of have unique state extensions to (H). The limitations of an analytic approach are illustrated by the failure of Chapter 4 to provide results for actions by compact groups. In view of the remarks of 3.3.5 it seems likely that a solution of the atomic masa problem will require hard results of a geometrical or combinatorial nature for finite dimensional algebras to settle the question of uniform compressibility in ? ? Mn.