For C*-subalgebras A and B of a C*-algebra C we study the relation A ~ B, which means that for any a in A, there exists an operator b in B such that []a-b]]
NEAR INCLUSIONS OF C*-ALGEBRAS BY E R I K CHRISTENSEN University of Copenhagen Copenhagen, Denmark
1. Introduction For C*-subalgebras A and B of a C*-algebra C we study the relation A ~ B, which means that for any a in A, there exists an operator b in B such that []a-b]] uau*. Questions of this type were discussed in [6] and [7], and it follows that in the situation considered here, we are able to find such a unitary. Therefore we get that uAu* is contained in B for some unitary u close to the identity and we are done. I n order to be able to perform the second and third step, the analysis from [6] and [7] show, that it is important that the algebra A has the property that any operator in C which nearly commutes with all elements in A 1 is close to the commutant of A in C. I n section 2 we recapitulate these concepts in detail, and we show how the results in [4], [8] and [15] can be used to extend the validity of the results in [6] and [7].
2. Preliminaries In their article [18] Kadison and Kastler defined the distance between two von Neumann algebras as the Hausdorff distance between the respective unitballs. I n the articles [5], [6], [7] we used this notion too, but since then we have found it more natural and easier to deal with the distance concept introduced below. The metrics are of course equivalent.
2.1. Definition. Let E and F be subspaces of a normed space G and let y >0. If for any e with [[eli ~ 0 there exists operators
