Showing papers on "Idempotence published in 1989"

Arveson nests and operator factorization along commutative subspace lattices

Abstract: Similar commutative subspace lattices (CSL's) are shown to be unitarily equivalent if certain sublattices (which may be taken to be nests!) are unitarily equivalent and a technical condition is satisfied. This result provides a connection between existing results for arbitrary similarities of countable CSL's and similarities of general CSL's by operators near the identity. One consequence is the generalizaton to CSL's of a theorem of David Pitts on the relationship between similarity and unitary equivalence of nests he calls "injective." H denotes a separable Hilbert space with real or complex scalars. B(H) is the space of all bounded, linear operators on H. The word "projection" always refers to an operator of orthogonal projection. For any A in B(H), rp(A) denotes the projection of H on the closure of the range of A. R(A) is the range of A, and I is the identity operator. If D is a linear operator on B(H) satisfying (i) for all X E B(H) such that X > 0, ?(X) ? 0, (ii) ?(X*) = ?D(X)* for all X E B(H), (iii) D is idempotent, and (iv) 'D(X)'D(Y) = ?D(X(?D(Y)) for all X and Y in B(H), then D is a conditional expectation from B(H) onto R(:D). Remarks. 1. (ii) is a consequence of (i) when the scalars are complex. 2. The identity ?(X)?(Y) = ?(DID(X)Y) is easily derived from (iv) and (ii). 3. R(D) is the set of fixed points of (D. 4. [11; Chapter II] is a good reference for the properties of conditional expectation operators. Let F c 7 c B(H). A function (D from 7 into itself is F-homogeneous if D(CX) = C'D(X) and D(XC) = 'D(X)C for all C in W and X in F. (Thus we do not require any linearity or continuity.) Of course, conditional expectation operators are examples of homogeneous maps, and the significance of Received by the editors September 6, 1988. Presented to the Great Plains Operator Theory Seminar in Indianapolis, Indiana in May, 1988. 1980 Mathemnatics Subject Classification ( 1985 Revision). Primary 47C05, 47A 1 5, 47A68. Ket' words and phrases. Commutative subspace lattice algebra, operator factorization, nest, conditional expectations on von Neumann algebras. (? 1989 American Mathematical Society 0002-9939/89 $1.00 + $.25 per page

Derative of analytic elements on infraconnected clopen sets

Abstract: Let K be an algebraically closed field of characteristic 0 provided with an ultrametric absolute value 1·1 for which it is complete. For all set D in K we will denote by R(D) the K-algebra of the rational function h(x) E K(x) with no pole in D. When D is closed and bounded, the algebra R(D) is provided with the norm of the uniform convergence on D denoted by IIII D (Ed that makes it a normed K-algebra. Its completion for that norm is then a K-Banach algebra denoted by H(D), the elements of which are called the analytic elements on D (Ks, A, E!, E2) . A set D is said to be infraconnected if for all a ED, the closure of the set {!x-al!aED} in IR is an interval. We know that a bounded closed set Dis infraconnected if and only if H(D) has no trivial idempotent (E 2 ) . The main problem we will consider here is whether an analytic element whose derivative is identically null on an infraconnected clopen (bounded) set D is necessarily a constant function in D. We will also have to study when the derivative is an analytic element. If it is so, is its Mittag-Leffler series the deriv­ ative series of the Mittag-Leffler series of the considered element? Asserting the theorems requires to introduce a lot of definitions and notations.

On the existence of idempotent liftings

Abstract: An existence theorem for idempotent liftings is proved. This implies that every compact measure space with full support and separable measure algebra admits an idempotent lifting.

lsometries of Nest Algebras

Abstract: In [S], Richard Kadison characterized all linear isometric maps of one C*-algebra, d, onto another one, .!3. In case d and 93 are von Neumann algebras, his result assumes a particularly simple and pleasing form. Where should one look for generalizations of Kadison’s results? One possibility is to investigate isometries acting on reflexive operator algebras, which represent a natural generalization of von Neumann algebras. In this paper, we begin such a study by considering nest algebras, which in some sense lie at the “opposite pole” (among reflexive algebras) from von Neumann algebras. To fix terminology and notation, we will throughout the paper denote by S a complex, separable, infinite-dimensional Hilbert space. An operator is a continuous linear transformation on 2, and the set of all such is &9(X)). A projection on SF is a self-adjoint idempotent operator in a(X). There is an obvious correspondence between projections and their ranges, which are always norm-closed subspaces of 2”. It will from time to time be convenient to blur the distinction between a projection and its range; thus, we might speak of a projection as being invariant for an operator. A lattice 9 of projections (or subspaces) is a collection of projections closed under the operations A and v , where E A F is the projection whose range is (range E)