# Allan Donsig's Research Interests

My research interests are in operator algebras and operator theory. This subject is a mix of algebra and analysis. It has important applications in physics, such as quantum mechanics and quantum field theory. There is a long standing theme in mathematics of understanding spaces in terms of the functions on those spaces. The commutative self-adjoint operator algebras are the natural function algebras on a topological space or a measure space. Topological or measure-theoretic questions about the space can be translated into algebraic questions about the commutative operator algebra. Motivated partly by this and partly by other topics, the study of non-commutative (selfadjoint) operator algebras can be viewed as `non-commutative topology' and `non-commutative measure theory'.

My own work is focuses on non-selfadjoint operator algebras. These correspond, in the sense of the previous paragraph, to the study of analytic functions. One interest of mine is limit algebras, which are infinite dimensional algebras obtained as limits of sequences of finite dimensional algebras, each embedded inside the next.

### Conferences

Great Plains Operator Theory Symposium, 2006

### Papers & Preprints (most recent first)

Because of copyright restrictions, I can only make available some of published papers. To read the preprints and papers, you may need the Adobe Acrobat Reader.

Norms of Schur Multipliers (with Ken Davidson),
Abstract: A subset P of N x N is called Schur bounded if every infinite matrix with bounded entries which is zero off of P yields a bounded Schur multiplier on B(H). Such sets are characterized as being the union of a subset with at most k entries in each row with another that has at most k entries in each column, for some finite k. If k is optimal, there is a Schur multiplier supported on the pattern with norm O(k^(1/2)), which is sharp up to a constant.

The same techniques give a new, more elementary proof of results of Varopoulos and Pisier on Schur multipliers with given matrix entries of random sign.

We consider the Schur multipliers for certain matrices which have a large symmetry group. In these examples, we are able to compute the Schur multiplier norm exactly. This is carried out in detail for a few examples including the Kneser graphs.

This paper is available on the arXiv preprint server as Math.OA 0506073.

Analytic Partial Crossed Products (with Alan Hopenwasser), Houston Journal of Mathematics 31 (2005) 495-527.

Abstract: Partial actions of discrete abelian groups can be used to construct both groupoid C*-algebras and partial crossed product algebras. In each case there is a natural notion of an analytic subalgebra. We show that for countable subgroups of R and free partial actions, these constructions yield the same C*-algebras and the same analytic subalgebras. We also show that under suitable hypotheses an analytic partial crossed product preserves all the information in the dynamical system in the sense that two analytic partial crossed products are isomorphic as Banach algebras if, and only if, the partial actions are conjugate.

This paper is available on the arXiv preprint server as Math.OA 0305337.

The Jacobson Radical of Analytic Crossed Products (with Aristides Katavolos and Antonios Manoussos), Journal of Functional Analysis 187 (2001) 129-145, Math Review 2002k:46170.

Abstract: We characterise the (Jacobson) radical of the analytic crossed product of C_0(X) by the non-negative integers (Z_+), answering a question first raised by Arveson and Josephson in 1969. In fact, we characterise the radical of analytic crossed products of C_0(X) by (Z_+)^d. The radical consists of all elements whose `Fourier coefficients' vanish on the recurrent points of the dynamical system (and the first one is zero). The multi-dimensional version requires a variation of the notion of recurrence, taking into account the various degrees of freedom.

This paper is available on the arXiv preprint server as Math.OA 0010142.

Algebraic Isomorphisms and the Spectra of Limit Algebras (with Steve Power and David Pitts) Indiana University Mathematics Journal 50 (2001) 1131-1147, Math Review 2002k:47148.

Abstract:We show that the spectrum of a triangular regular limit algebra (TAF algebra) is an invariant for algebraic isomorphism. Combining this with previous results provides a striking rigidity property: two triangular regular limit algebras are algebraically isomorphic if and only if they are isometrically isomorphic. A consequence of spectral invariance is a structure theorem for isomorphisms between limit algebras. The proof of the main theorem makes use of a characterization of the completely meet irreducible ideals of a TAF algebra and a dual space formulation of the spectrum.

Automatic closure of invariant linear manifolds for operator algebras (with Alan Hopenwasser and David Pitts), Illinois Journal of Mathematics 45 (2001) 787-802.

Abstract: 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.

This paper is available on the arXiv preprint server as Math.OA 0005159.

The Classification of Limits of 2n-cycle Algebras (with Steve Power), Indiana University Mathematics Journal 48 (1999) 411-427, Math Review 2000j:47127.

Abstract: We obtain a complete classification of the locally finite algebras and the operator algebras, given as algebraic inductive limits and Banach algebraic inductive limits, respectively, of systems:

A_1 contained in A_2 contained in A_3 and so on.

Here the A_k are 2n-cycle algebras, where n is at least 3 and the inclusions are of rigid type. The complete isomorphism invariant is essentially the triple (K_0(A), H_1(A), Sigma(A)) where K_0(A) is viewed as a scaled ordered group, H_1(A) is a partial isometry homology group and Sigma(A), contained in the direct sum of K_0(A) and H_1(A) is the 2n-cycle joint scale.

This paper is available in PDF format.

Algebraic Isomorphisms of Limit Algebras (with Tim Hudson and Elias Katsoulis), Transactions of the American Math. Society 353 (2000) 1169-1182.

Abstract: 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 of Power (see the Notes to Chapter 8 in Power's book) and to an open problem (see Problem 7.8 in Power's book). As a further consequence, we describe epimorphisms between various classes of limit algebras.

Meet Irreducible Ideals in Direct Limit Algebras (with Alan Hopenwasser, Tim Hudson, Michael Lamoureux, and Baruch Solel), Mathematica Scandinavia 87 (2000) 27-62, Math Reviews 2001f:47112.

Abstract: This paper studies meet-irreducible ideals, that is, ideals I so that if I is the intersection of two ideals, then one of the two ideals must equal I, in certain direct limit algebras. The limit algebras are usually strongly maximal triangular subalgebras of AF C*-algebras.

We give descriptions of these ideals in terms of coordinates. Meet-irreducible ideals are interesting because they are closely related to nest representations; in particular, we show that each meet-irreducible ideal is the kernel of a nest representation. We obtain a distance formula for ideals analogous to Arveson's distance formula for nest algebras. Finally, the subset of completely meet-irreducible ideals (they satisfy the same property for an arbitrary intersection) is shown to be isomorphic to the spectrum of the limit algebra, for certain limit algebras.

Algebraic Orders and Chordal Limit Algebras, Proc. Edinburgh Math. Soc. 41 (1998) 465-485, Math Reviews 2000d:47103.

Abstract: We develop an isomorphism invariant for limit algebras: an extension of Power's strong algebraic order on the scale of the K_0-group (Power, J. Operator Theory 27 1992, 87--106). This invariant is complete for a certain family of limit algebras: inductive limits of digraph algebras (a.k.a.\ finite dimensional CSL algebras) satisfying two conditions: (1) the inclusions of the digraph algebras respect the order-preserving normalisers, and (2) the digraph algebras have chordal digraphs. The first condition is also used to show that the invariant depends only on the limit algebra and not the direct system. We give an intrinsic characterisation of the limit algebras satisfying both~(1) and~(2).

Dilations of Limit Algebras and Interpolating Spectrum, Pacific J. of Math. 184 (1998) 75-93, Math Reviews 99f:47053.

Abstract: We show that limit algebras having interpolating spectrum are characterized by the property that all locally contractive representations have *-dilations. This extends a result for digraph algebras by Davidson. It is an open question if such a limit algebra is the limit of a direct system of digraph algebras with interpolating digraphs, although a positive answer would allow one to obtain one direction of our result directly from Davidson's. Instead, we give a `local' construction of digraph algebras with interpolating digraphs and use this to extend representations.

Tree algebras (in the sense of Davidson, Paulsen, and Power) have been characterized by a commutant lifting property among digraph algebras with interpolating digraphs. We show that the analogous result holds for limit algebras, i.e., limit algebras with the analogous spectral condition are characterized by the same commutant lifting property among the limit algebras with interpolating spectrum.

Norm-closed Bimodules of Nest Algebras, (with Ken Davidson and Tim Hudson) J. Operator Theory 39 (1998) 59-87, Math Reviews 99e:47053.

Abstract: We study the support and essential support functions of a norm--closed bimodule of a nest algebra. An allowable support function pair determines a maximal bimodule. There is also a natural candidate for the minimal bimodule for a given support function pair. We determine precisely when this candidate is the minimal element. In the other cases, this module is still the intersection of all bimodules with a given support function pair, but it is not in this class itself.

Homology for Operator Algebras IV: On the Regular Classification of Limits of 4-Cycle Algebras (with Steve Power), J. of Functional Analysis 150 (1997) 240-287, Math Reviews 99g:47109.

Abstract: A 4-cycle algebra is a finite dimensional digraph algebra (CSL algebra) whose reduced digraph is a 4-cycle. A rigid embedding between such algebras is a direct sum of certain nondegenerate multiplicity one star-extendible embeddings. A complete classification is obtained for the regular isomorphism classes of direct systems \$A\$ of 4-cycle algebras with respect to rigid embeddings. The critical invariant is a binary relation in the direct sum of K0 A and H1 A, generalising the scale of the \$K0\$-group, called the joint scale. The joint scale encapsulates other invariants and compatibility conditions of regular isomorphism. These include the scale of H1 A, the scale of the direct sum of H0 A and H1 A sign compatibility, congruence compatibility and H0 H1 coupling classes. These invariants are also important for lifting (K0 direct sum H1) isomorphisms to algebra isomorphisms; we resolve this lifting problem for various classes of 4-cycle algebra direct systems.

On Derivations of Semi-nest Algebras (with B.E. Forrest and L.W. Marcoux), Houston J. of Mathematics 22 (1996) 375-398, Math Reviews 97h:47042.

Abstract: A seminest algebra can be decomposed a two by two block matrix, where the (1,1) entry is a nest algebra, the (4,4) entry is a reflexive algebra and the (1,2) entry is a reflexive subspace. In this paper, we study the derivations from a semi-nest algebra into itself and show that such derivations are always continuous. If the algebra is also a CSL algebra, we scharacterize when the first homology group of the algebra is contained in the first homology group of the (4,4) entry; in these cases, the only obstruction to a derivation being inner arises from the (4,4) entry. In particular, the H1-group of the algebra vanishes if the (4,4) entry is a direct sum of nest algebras or is selfadjoint. If the algebra is not a CSL algebra, then this fails; for example, if the (4,4) entry is multiples of In, n>1, we give outer derivations on the algebra inspite of the (4,4) entry being selfadjoint.

The Failure of Approximate Inner Conjugacy for Standard Diagonals in Regular Limit Algebras (with Steve Power), Canadian Math. Bulletin, 39 (1996) 420-428, Math Reviews 97i:46096.

Abstract: AF C*-algebras contain natural AF masas which, here, we call standard diagonals. Standard diagonals are unique, in the sense that two standard diagonals in an AF C*-algebra are conjugate by an approximately inner automorphism. We show that this uniqueness fails for non-selfadjoint AF~operator algebras. Precisely, we construct two standard diagonals in a particular non-selfadjoint AF~operator algebra which are not conjugate by an approximately inner automorphism of the non-selfadjoint algebra.

Nest Algebras with Locally Constant Cocycles, (with J.R. Peters) J. of the London Math. Society 55 (1997) 569-587, Math Reviews 98d:47095.

Abstract: The first examples of triangular AF algebras to be studied were the refinement algebra and the standar algebra. Both are analytic algebras with the property that the cocycle can be taken to be constant on the matrix units of the algebra. The latter property is quite special and still quite ill-understood. In the present paper, we weamine the clase of nest algebras T in AF C*-algebras which share the distinctive properties of the refinement algebra: (1) T is a nest algebra in which the nest generated the diagonal, (2) T admits a locally constant cocycle. There are many such algebras and we classify them in terms of source-ordered Bratteli diagrams.

This paper is available in PDF format.

On the Lattice of Ideals of Triangular AF Algebras (with Tim Hudson), J. of Functional Analysis 138 (1996) 1-39, Math Reviews 97e:47068.

Abstract: We study triangular AF (TAF) algebras in terms of their lattices of closed two-sided ideals. Not (isometrically) isomorphic algebras can have isomorphic lattices of ideals; indeed, there is an uncountable family of pairwise non-isomorphic algebras, all with isomorphic lattices of ideals. In the positive direction, if A and B are strongly maximal TAF algebras with isomorphic lattices of ideals, then there is a bijective isometry between the subalgebras of A and B generated by their order-preserving normalizers. This bijective isometry is the sum of an algebra isomorphism and an anti-isomorphism. Using this, we show that if the TAF algebras are generated by their order-preserving normalizers and are triangular subalgebras of primitive C*-algebras, then the lattices of ideals are isomorphic if and only if the algebras are either (isometrically) isomorphic or anti-isomorphic. Finally, we use complete distributivity to show that there are TAF algebras whose lattices of ideals can not arise from TAF algebras generated by their order preserving normalizers. Our techniques rely on constructing a topological binary relation based on the lattice of ideals; this realtion is closely connected to the spectrum or fundamental relation (also a topological binary relation) of the TAF algebra.

Order Preservation in Limit Algebras (with Alan Hopenwasser), J. of Functional Analysis 133 (1995) 342-394, Math Reviews 96k:46099.

Abstract: The matrix units of a digraph algebra, A, induce a relation, known as the diagonal order, on the projections in a masa in the algebra. Normalizing partial isometries in A act on these projections by conjugation; they are said to be order preserving when they repect the diagonal order. Order preserving embeddings, in turn, are those embeddings which carry order preserving normalizers to order preserving normalizers. This paper studies operatre algebras which are direct limits of finite dimensional algebras with order preserving embeddings. We give a complete classification of direct limits of full triangular matrix algebras with order preserving embeddings. We also investigate the problem of characterizing algebras with order preserving embeddings.

Semisimple Triangular AF Algebras, J. of Functional Analysis 111 (1993) 323-349, Math Reviews 94b:46084.

Abstract: We give several necessary and sufficient conditions for a triangular AF algebra to be semisimple. In particular, a triangular AF algebra which can be written using the standard embedding infinitely often is semisimple; we also give a semisimple triangular AF algebra which does not have a presentation of this form. If two triangular AF algebras have the same Peters-Poon-Wagner diagonal invariant, then either both are semisimple or both are not. However, we give two algebras with the same diagonal invariant where one has Jacobson radical equal to the strong radical and the other does not. Semisimplicity can be characterized in terms of Power's fundamental relation. We give generalizations of these results to triangular sublagebras of groupoid C*-algebras. It is shown that there is a unique maximal bimodule over the diagonal which does not intersect the radical. Using this, we show that the Wedderburn principal theorem does not hold for triangular AF algebras. A necessary condition for the theorem to hold is that the above mentioned unique bimodule is a subalgebra.