David R. Pitts

Invariant Subspaces and Hyper-Reflexivity for Free Semigroup Algebras

In this paper, we obtain a complete description of the invariant subspace structure of an interesting new class of algebras which we call free semigroup algebras. This enables us to prove that they are reflexive, and moreover to obtain a quantitative measure of the distance to these algebras in terms of the invariant subspaces. Such algebras are called hyper-reflexive. This property is very strong, but it has been established in only a very few cases. Moreover the prototypes of this class of algebras are the natural candidate for a non-commutative analytic Toeplitz algebra on n variables. The case we make for this analogy is very compelling. In particular, in this paper, the key to the invariant subspace analysis is a good analogue of the Beurling theorem for invariant subspaces of the unilateral shift. This leads to a notion of inner–outer factorization in these algebras. In a sequel to this paper [13], we add to this evidence by showing that there is a natural homomorphism of the automorphism group onto the group of conformal automorphisms of the ball in Cn. A free semigroup algebra is the weak operator topology closed algebra generated by a set S1, . . . , Sn of isometries with pairwise orthogonal ranges. These conditions are described algebraically by S∗ i Sj = δijI for 1 ≤ i, j ≤ n; (F)

Nevanlinna-Pick interpolation for non-commutative analytic Toeplitz algebras

The non-commutative analytic Toeplitz algebra is the WOT-closed algebra generated by the left regular representation of the free semigroup onn generators. We obtain a distance formula to an arbitrary WOT-closed right ideal and thereby show that the quotient is completely isometrically isomorphic to the compression of the algebra to the orthogonal complement of the range of the ideal. This is used to obtain Nevanlinna-Pick type interpolation theorems

Factorization problems for nests: Factorization methods and characterizations of the universal factorization property

Let M be a von Neumann algebra and let β be a nest in M. We consider the problem of factoring an invertible element S of M as S = WA, where W is a unitary element in M and both A and A−1 are elements of M which also belong to Alg(β). It is known that this is not always possible, however, using a variant of the LU decomposition for matrices, we show that if β is an injective nest and S is an invertible operator, then there exists an isometry W such that both S−1W and W∗S belong to Alg(β). We characterize when an invertible operator factors with respect to an injective nest. We also prove a result which simultaneously generalizes results of Arveson and Gohberg and Krein. Finally, we give a number of characterizations of those nests within a factor such that every invertible operator factors with respect to the nest.

Idempotents in nest algebras

Abstract The algebraic equivalence and similarity classes of idempotents within a nest algebra Alg β are completely characterized. We obtain necessary and sufficient conditions for two idempotents to be equivalent or similar. Our criterion yields examples illustrating pathology and also shows that to each equivalence class of idempotents there corresponds a “dimension function” from β × β into N ∪ { ∞ }. We complete the characterization of the algebraic equivalence classes by proving that any dimension function corresponds to an equivalence class of idempotents. Also, to each sequence of dimension functions, there corresponds a commuting sequence of idempotents. A criterion is obtained for when an idempotent is similar to a subidempotent of another. The mapping which sends an equivalence class (or idempotent) to its associated dimension function plays a role in the nest algebra theory analogous to the role played by the mapping sending a projection in a Type I W ∗ -algebra to its center valued trace. We prove that almost commuting, similar idempotents are homotopic; this contrasts with the situation in certain C ∗ -algebras. Using this, we show that similar, simultaneously diagonalizable idempotents are homotopic, and in the continuous nest case, every diagonal idempotent is homotopic to a core projection.

Norming algebras and automatic complete boundedness of isomorphisms of operator algebras

We combine the notion of norming algebra introduced by Pop, Sinclair and Smith with a result of Pisier to show that if A 1 and A 2 are operator algebras, then any bounded epimorphism of A 1 onto A 2 is completely bounded provided that A 2 contains a norming C*-subalgebra. We use this result to give some insights into Kadisons Similarity Problem: we show that every faithful bounded homomorphism of a C*-algebra on a Hilbert space has completely bounded inverse, and show that a bounded representation of a C*-algebra is similar to a *-representation precisely when the image operator algebra A-norms itself. We give two applications to isometric isomorphisms of certain operator algebras. The first is an extension of a result of Davidson and Power on isometric isomorphisms of CSL algebras. Secondly, we show that an isometric isomorphism between subalgebras A i of C*-diagonals (C i ,D i ) (i = 1, 2) satisfying D i C A i C C, extends uniquely to a *-isomorphism of the C*-algebras generated by A 1 and A 2 ; this generalizes results of Muhly-Qiu-Solel and Donsig-Pitts.

Approximate unitary equivalence of completely distributive commutative subspace lattices

In this note, we classify completely distributive CSLs up to approximate unitary equivalence. Our proof uses a new characterization of complete distributivity and leads to a generalization of a result of Arveson on ordered group lattices. As consequences of our results, we obtain a similarity theorem for hyperreflexive, completely distributive CSLs and some new perturbation results.

A note on the connectedness problem for nest algebras

It has been conjectured that a certain operator T belonging to the group W of invertible elements of the algebra Alg Z of doubly infinite uppertriangular bounded matrices lies outside the connected component of the identity in S . In this note we show that T actually lies inside the connected component of the identity of W. Let T be the unit circle in the complex plane with normalized Lebesgue measure. For 1 < p < 00, let HP be the usual Hardy space of all functions in LP(T) that have analytic extensions to the open unit disk D. Let X = L2(T) and let (X) be set of all bounded linear operators on X. Let W E (X) be the shift operator: (Wf)(eiO) = eiOf(eiO). In this paper, we consider the nest { WnH2: n E Z} of subspaces of L2(T), and its associated nest algebra, AlgZ = {T E (): TWnH2 C WnH2 for all n E Z}. A question which has been unanswered for several years is the following: Question. Is the group of invertible elements of the Banach algebra AlgZ connected in the norm topology? It is frequently conjectured that the answer to this question is no. The reason for conjecturing a negative answer is because of a strong analogy between nest algebras and analytic function theory. We refer the reader to the book by Davidson [1] for details and more background on this question. For each f E LOO(T), let Mf E (X) be the multiplication operator, Mfq

VON NEUMANN ALGEBRAS AND EXTENSIONS OF INVERSE SEMIGROUPS

=fq

The algebraic structure of non-commutative analytic Toeplitz algebras

, bE L2(T). Note that for f E H??, we have Mf E Alg Z. Let a be a positive real number and set h(z)= 9!log(+Z) Received by the editors July 19, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 47D25; Secondary 46B35.

Global existence and non-existence theorems for nonlinear wave equations

In the 1970s, Feldman and Moore classified separably acting von Neumann algebras containing Cartan MASAs using measured equivalence relations and 2-cocycles on such equivalence relations. In this paper, we give a new classification in terms of extensions of inverse semigroups. Our approach is more algebraic in character and less point-based than that of Feldman-Moore. As an application, we give a restatement of the spectral theorem for bimodules in terms of subsets of inverse semigroups. We also show how our viewpoint leads naturally to a description of maximal subdiagonal algebras.