Timothy D. Hudson

Algebraic isomorphisms of limit algebras

We prove that algebraic isomorphisms between limit algebras are automatically continuous, and consider the consequences of this result. In particular, we give partial solutions to a conjecture and an open problem by Power. As a further consequence, we describe epimorphisms between various classes of limit algebras. In this paper, we study automatic continuity for limit algebras. Automatic continuity involves algebraic conditions on a linear operator from one Banach algebra into another that guarantee the norm continuity of the operator. This is a generalization, via the open mapping theorem, of the uniqueness of norms problem. Recall that a Banach algebra A is said to have a unique (Banach algebra) topology if any two complete algebra norms on A are equivalent, so that the norm topology determined by a Banach algebra is unique. Uniqueness of norms, automatic continuity, and related questions have played an important and long-standing role in the theory of Banach algebras [6, 29, 28, 10, 11, 2]. Limit algebras, whose theory has grown rapidly in recent years, are the nonselfadjoint analogues of UHF and AF C-algebras. We first prove that algebraic isomorphisms between limit algebras are automatically continuous (Theorem 1.4). This proof uses the ideal theory of limit algebras as well as key results from the theory of automatic continuity for Banach algebras. Combining this with [23, Theorem 8.3] verifies Powers conjecture that the C-envelope of a limit algebra is an invariant for purely algebraic isomorphisms, for limits of finite dimensional nest algebras, and in particular, for all triangular limit algebras (Corollary 1.6). In [5], the first two authors studied triangular limit algebras in terms of their lattices of ideals. By combining automatic continuity with this work, we show that within the class of algebras generated by their order preserving normalizers (see below for definitions), algebraically isomorphic algebras are isometrically isomorphic (Theorem 2.5). This shows that the spectrum, or fundamental relation [20], a topological binary relation which provides coordinates for limit algebras and is a useful tool in classifications, is a complete algebraic isomorphism invariant for this class (Corollary 2.6). In recent work, the second two authors studied primitivity for limit algebras [9], showing that a variety of limit algebras are primitive. These results, together with automatic continuity, give descriptions of epimorphisms between various classes of limit algebras, namely lexicographic algebras (Theorem 3.2) and Z-analytic algebras (Theorem 3.3). Received by the editors April 6, 1998 and, in revised form, October 7, 1999. 2000 Mathematics Subject Classification. Primary 47D25, 46K50, 46H40. Research partially supported by an NSF grant. ()2000 American Mathematical Society

LIE IDEALS IN TRIANGULAR OPERATOR ALGEBRAS

We study Lie ideals in two classes of triangular operator algebras: nest algebras and triangular UHF algebras. Our main results show that if 2 is a closed Lie ideal of the triangular operator algebra A, then there exist a closed associative ideal IC and a closed subalgebra Or of the diagonal A ni A* so that K C 2 C IC + Oc. INTRODUCTION Let A be an associative complex algebra. Under the Lie multiplication [x, y] xy yx, A becomes a Lie algebra. A Lie ideal in A is a linear manifold Z in A for which [a, k] c ? for every a E A and k c 2. In many instances, there is a close connection between the Lie ideal structure and the associative ideal structure of A. This connection has been investigated for prime rings in [6], in [3] for B(

Extreme points in triangular UHF algebras

5) the set of bounded operators on a Hilbert space Si, and in [10] for certain von Neumann algebras. (See also [9, 4, 11].) In this paper we pursue this line of investigation for two classes of triangular operator algebras, namely nest algebras anld triangular UHF algebras. The authors would like to thank Frank A. Zorzitto for many helpful conversations. 1. WEAKLY CLOSED LIE IDEALS IN NEST ALGEBRAS Recall that a nest JK on a Hilbert space Si is a chain of closed subspaces of Si which is closed under the operations of arbitrary intersections and closed linear spans, and which includes {O} and S5. The nest algebra T(KJ) is the algebra of all operators on Se leaving every member of JK invariant. This is always closed in the weak operator topology. The diagonal D(KJ) of a nest algebra T(JK) is the von Neumann subalgebra 7(K) n T(K)*. If E, F E KV with E < F, then F E is called an interval of the nest. The nonzero minimal intervals are called atoms. A nest is atomic if the atoms of the nest span 55. We refer the reader to [1] for more information on nest algebras. Our main result, Theorem 12, shows that for every weakly closed Lie ideal Z in 7(JK), there exist a corresponding weakly closed associative ideal IC and a von Neumann subalgebra SK of O(A/) such that

Norms of inner derivations of limit algebras

We examine the strongly extreme point structure of the unit balls of triangular UHF algebras. The semisimple triangular UHF algebras are characterized as those for which this structure is minimal in the sense that every strongly extreme point belongs to the diagonal. In contrast to this, for the class of full nest algebras we prove a Krein-Milman type theorem which asserts that every operator in the open unit ball of the algebra is a convex combination of strongly extreme points. Results concerning the geometry of the unit ball have a long history both in Banach space theory and in the theory of operator algebras. This geometry can be affiliated with structural and algebraic properties. Moreover, differences in geometric properties can prove useful in classification problems. Two fundamental results are the Russo-Dye Theorem and Kadison’s Theorem on isometries. More recently, there has been interest in the unit balls of nonselfadjoint operator algebras, especially nest algebras [1, 2, 3, 4, 15, 16]. This paper concerns the unit balls of triangular UHF algebras. These and the larger class of triangular AF algebras are nonselfadjoint analogues of the UHF and AF C*-algebras studied by Glimm and Bratteli. Their theory has grown rapidly, cf. [8, 9, 18, 19, 20]. We focus on the extreme point structure, and our results have a different flavor than those for nest algebras. Specifically, we study the strongly extreme points, those boundary points whose “stable character” with respect to approximations makes them behave well under direct limits. Triangular UHF algebras are direct limits of full upper triangular matrix algebras. Unit balls embed into unit balls in the direct limit scheme, and some types of embeddings respect the extreme point structure while others do not. This leads to structural differences in the limit algebras. The geometric structures of the unit balls of different triangular UHF algebras can be very dissimilar. The convex hull of the strongly extreme points, even without closure, always contains the unit ball of the diagonal. Theorem 7 shows that the two coincide if and only if the algebra is semisimple. This is a characterization of a purely geometric property in terms of a purely algebraic one. In contrast to Received by the editors January 11, 1996 and, in revised form, March 28, 1996. 1991 Mathematics Subject Classification. Primary 47D25, 46K50, 46B20.

Ideals in Triangular Af Algebras

Let A be a strongly maximal TAF-algebra. It is shown that ½ Orc(A) ≤ K(A) ≤ 4/√3 Orc(A), where K(A) and Orc(A) are constants determined by the norms of inner derivations of A, and by the hull-kernel topology on the space of meet-irreducible ideals of A, respectively. It follows that the set of inner derivations of A is closed in the Banach space of all bounded derivations of A if and only if Orc(A) < oo. These results are analogous to those for C * -algebras.