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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 Kadison's 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 \lambda-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 \subseteq A_i \subseteq C_i 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., Comment: 9 pages

Let "HJji e Z) be a sequence of infinite-dimensional Hilbert spaces, and let H := £•{«„ : n e Z}. Write En := P(Hn), Pn := P(• © Hn_, © Hm) where P(S) is the orthogonal projection onto the subspace S of H. Let M be the nest in H consisting of the projections Pn together with 0 and /. The nest algebra, AlgTV, is the set of doubly infinite block operator matrices. Let D = 5Z®Dn be a block diagonal projection, and dn — rankDn. Since the projections Pn are invariant for Alg./V, and the intervals Pn Pm are semi-invariant, it follows that any operator X which factors as ADB for A, B e AlgA/" will satisfy

We show that if is a nest with no isolated atoms of finite multiplicity, then the invertibles in are connected. The key technical ingredient is that in such nest algebras, every operator with zero atomic diagonal part factors through the non-atomic part of . In particular, these results apply for the Cantor nest.

In 1972 Kadison and Kastler [12] asked whether two close von Neumann subalgebras of ~ ( J f ) are unitarily equivalent by a unitary close to the identity. This question was answered affirmatively for certain classes of von Neumann algebras in [5, 4, 6, 15]. The nonselfadjoint case was first considered by Lance who showed in [13] that close nest algebras are similar via a similarity which is close to the identity. For nest algebras, there is a close connection between the question of whether closeness of algebras implies similarity and the classification of nest algebras up to similarity: this connection may be seen in the papers [7] and [14] and we shall discuss it further in Remark 8 below. The desire to extend the classification of nest algebras to the class of reflexive algebras with commutative subspace lattice was one of the motivations for the work on perturbations of matrix algebras begun in [3] and continuing with perturbations of suboolean operator algebras in [8]. In [16], we proved that if ~ ' is a hyperreflexive CSL algebra whose lattice is atomic and satisfies a certain technical condition, then any other CSL algebra whose unit ball is sufficiently close to the unit ball of d has the property that ~ is similar to d . This result was extended in [9] to the class of all hyperreflexive CSL algebras whose lattice is completely distributive. The purpose of the present paper is to remove the hypothesis of hyperreflexivity and complete distributivity from the results described above. We show that the class of all CSL algebras is stable under small perturbations. In particular, we prove Theorem 6 below, which gives a complete answer to Problem 5 of the section entitled "Open Problems" of [11].

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.

Kadison's transitivity theorem implies that, for irreducible representations of C*-algebras, every invariant linear manifold is closed. It is known that CSL algebras have this propery if, and only if, the lattice is hyperatomic (every projection is generated by a finite number of atoms). We show several other conditions are equivalent, including the conditon that every invariant linear manifold is singly generated. We show that two families of norm closed operator algebras have this property. First, let L be a CSL and suppose A is a norm closed algebra which is weakly dense in Alg L and is a bimodule over the (not necessarily closed) algebra generated by the atoms of L. If L is hyperatomic and the compression of A to each atom of L is a C*-algebra, then every linear manifold invariant under A is closed. Secondly, if A is the image of a strongly maximal triangular AF algebra under a multiplicity free nest representation, where the nest has order type -N, then every linear manifold invariant under A is closed and is singly generated., AMS-LaTeX, 15 pages, minor revisions

It has been conjectured that a certain operator T T belonging to the group G \mathcal {G} of invertible elements of the algebra Alg ⁡ Z \operatorname {Alg} {\mathbf {Z}} of doubly infinite upper-triangular bounded matrices lies outside the connected component of the identity in G \mathcal {G} . In this note we show that T T actually lies inside the connected component of the identity of G \mathcal {G} .

In this note we use cohomological techniques to prove that if there is a linear map between two CSL algebras which is close to the identity, then the two CSL algebras are similar. We use our result to show that if 2f is a purely atomic, hyperreflexive CSL with uniform infinite multiplicity which satisfies the 4-cycle interpolation condition, then there are constants δ, C > 0 such that whenever Jt is another CSL such that d(Algi?, MgJt) < δ , then there is an invertible operator S such that S k\gS?S~ι = Alg^ and ||S|| US"1 II < 1 4- Cd(Alg&, 1. Introduction and preliminaries. In this paper, we consider two related types of perturbation questions for CSL algebras. The first deals with the problem of perturbing a linear isomorphism between two algebras to obtain an algebra isomorphism, and the second deals with the problem of deciding whether, for two such algebras, closeness implies isomorphism. Perturbation questions of this kind were considered by Kadison and Kastler for von Neumann algebras in [18] and further studied by many authors, including Christensen ([3, 4, 5]). Johnson ([16]) and Raeburn and Taylor ([21]) obtained results concerning perturbations of closed subalgebras of Banach algebras. Their results show that if $/ is a closed subalgebra of a Banach algebra 3S and certain cohomology groups for sf vanish, then any closed subalgebra of 3S "sufficiently close" to sf is actually isomorphic to stf . The nonself adjoint case was considered first for nest algebras by Lance in [19]. Perturbations of other nonself adjoint operator algebras were considered by Choi and Davidson ([2]), Davidson ([6]). In §2, we prove Theorem 3 which shows that if two CSL algebras s/\ and J^2 are linearly isomorphic via an isomorphism close to the identity, they they are actually spatially isomorphic via an isomorphism which is close to a unitary equivalence. In §3, we introduce the 4-cycle interpolation property, which is closely related to a lattice condition appearing in [12] and to the notion of interpolating lattice introduced in [7]. The main result of §3, Theorem 16, shows that if sfγ is a CSL algebra which is sufficiently close to a purely atomic