On the Calkin Representations of B(H)

Abstract

Irreducible representations of B(N), with ) a not necessarily separable Hilbert space, are constructed and analyzed along the lines of a similar study of Reid for separable Hilbert spaces. Here we construct and study certain representations of B()), the algebra of bounded operators on the Hilbert space 3X. 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(1), X 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 { I s E A} c 3X, the range projection onto the linear span of the ,, s E A, will be denoted by P = (4, I s E A). Let 3( be a Hilbert space of dimension a. Then the proper closed two-sided ideals of B(Y) are just the i-compact operators I,, w < X, < a. I, is generated by all projections of dimension strictly less than ic. Thus I, is the usual ideal of compact operators and for B(Y) we have the composition series {?}1 C Iw C ..C Inc C ..C Iae C B (Y). An irreducible representation wr of B(1) will thus have one of these ideals as the kernel. The Calkin representations which we are going to study are all extensions of representations of the maximal abelian subalgebra lJ(o(a) of B(Y), dim ( = a. Remember that we identify the cardinal (ordinal) a with the corresponding set. Let Y = l1x(a, 3X) be the system of bounded 3X-valued sequences indexed by a. If p is an ultrafilter on a, we can define an inner product on Y by (1) (d, rj)1, = lim((8, rs) for ( = (b), r = (n) p With )p = {f E I ((, ()p = 11(11p = 0}, 9p = Hc/)p becomes a Hilbert space. B(31) operates on Y9' coordinatewise by (2) (T()s = T(S. Received by the editors June 3, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 46L05.