A Class of C ∗ -Algebras
Horst Behncke, Wolfgang Bös · Proceedings of the American Mathematical Society · 1973
In this paper we present a construction of C*- algebras based on a countable system of subsets of a countable set. The properties of such algebras are described in terms of this system of sets. In (1) Behncke, Krauss and Leptin suggested a generalization of their construction of C*-algebras. In this note this program is carried through and all results of (1) are extended to this more general case. This note is organized as follows: In the first part we construct for a given system M of subsets of a countable set m0 a C*-algebra 2l(M). Then the properties ofH(M) for a finite m0 are studied. These results are the key to the remainder. With their aid the general case can be investigated. We conclude this note with two examples. Throughout all sets will be countable and all Hilbert spaces will be complex separable spaces. If H is a Hilbert space, the algebra of all compact (bounded) linear operators will be denoted by R(H) (23(£f)). The term ideal always means closed two sided ideal. If m is a set, the com- plement of m will be denoted by m'.