On the Calkin Representations of B(H)

H. Behncke
1987 Proceedings of the American Mathematical Society  
Irreducible representations of B()i), with )i a not necessarily separable Hubert space, are constructed and analyzed along the lines of a similar study of Reid for separable Hubert spaces. Here we construct and study certain representations of B(M), the algebra of bounded operators on the Hubert space M. These have been studied previously by Calkin [4] for separable Hilbert spaces and in the general case by Barnes [3]. A more systematic study of these irreducible Calkin representations of B(M),
more » ... sentations of B(M), M separable, has been undertaken by Reid [7], but not much seems to be known in the nonseparable case beyond [1] . In this paper we extend the results of Reid to the nonseparable case. Since such a study involves some cardinal arithmetic we shall assume Zorn's Lemma and the generalized continuum hypothesis throughout. For the notation and terminology regarding ordinals, cardinals, and filters we refer the reader to the book of Comfort and Negrepontis [5]. Our notation and terminology regarding C*algebras will be standard; i.e., that of [6] . If {Çs | s € A} C #, the range projection onto the linear span of the Çs, s € A, will be denoted by P -(fs | s € A). Let M be a Hilbert space of dimension a. Then the proper closed two-sided ideals of B(M) are just the /«-compact operators IK, oj < k < a. IK is generated by all projections of dimension strictly less than k. Thus Iu is the usual ideal of compact operators and for B()l) we have the composition series
doi:10.2307/2046558 fatcat:rtyue3ekgfeb3hha525cs4jfda