Lyudmila Turowska

Operator synthesis. I. Synthetic sets, bilattices and tensor algebras

Abstract The interplay between the invariant subspace theory and spectral synthesis for locally compact abelian group discovered by Arveson (Ann. of Math. (2) 100 (1974) 433) is extended to include other topics as harmonic analysis for Varopoulos algebras and approximation by projection-valued measures. We propose a “coordinate” approach which nevertheless does not use the technique of pseudo-integral operators, as well as a coordinate free one which allows to extend to non-separable spaces some important results and constructions of Arveson. We solve some problems posed in Arveson (1974).

Spectral Synthesis and Operator Synthesis for Compact Groups

LetG be a compact group andC(G )b e theC-algebra of continuous complex-valued functions onG. The paper constructs an imbedding of the Fourier algebra A(G )o fG into the algebra V(G )= C(G) h C(G) (Haagerup tensor product) and deduces results about parallel spectral synthesis, generalizing a result of Varopoulos. It then characterizes which diagonal sets in GG are sets of operator synthesis with respect to the Haar measure, using the denition of operator synthesis due to Arveson. This result is applied to obtain an analogue of a result of Froelich: a tensor formula for the algebras associated with the pre-orders dened by closed unital subsemigroups of G.

Multidimensional operator multipliers

We introduce multidimensional Schur multipliers and characterise them, generalising well-known results by Grothendieck and Peller. We define a multidimensional version of the two-dimensional operator multipliers studied recently by Kissin and Shulman. The multidimensional operator multipliers are defined as elements of the minimal tensor product of several C*-algebras satisfying certain boundedness conditions. In the case of commutative C * algebras, the multidimensional operator multipliers reduce to continuous multidimensional Schur multipliers. We show that the multipliers with respect to some given representations of the corresponding C * -algebras do not change if the representations are replaced by approximately equivalent ones. We establish a non-commutative and multidimensional version of the characterisations by Grothendieck and Peller which shows that universal operator multipliers can be obtained as certain weak limits of elements of the algebraic tensor product of the corresponding C * -algebras.

Operator synthesis II. Individual synthesis and linear operator equations

Abstract The second part of our work on operator synthesis deals with individual operator synthesis of elements in some tensor products, in particular in Varopoulos algebras, and its connection with linear operator equations. Using a developed technique of “approximate inverse intertwining” we obtain some generalizations of the Fuglede and the Fuglede-Weiss theorems and solve some problems posed in [ Open Problems, Proc. Fourth Conf. Operator Theory (Timişoara/Herculane 1979), Univ. Timişoara and Nat. Inst. Sci. Tech. Creation, Timişoara (1980), 335–342.], [Gary Weiss, An extension of the Fuglede-Putnam theorem modulo the Hilbert-Schmidt class to operators of the form ΣM n XN n, Trans. Amer. Math. Soc. 278 (1983), no. 1, 1–20.], [ Gary Weiss, The Fuglede commutativity theorem modulo the Hilbert-Schmidt class and generating functions for matrix operators. II, J. Oper. Th. 5 (1981), no. 1, 3–16. ]. Additionally, we give some applications to spectral synthesis in Varopoulos algebras and to partial differential equations.

Representations of a q-analogue of the *-algebra Pol(Mat2,2)

Bounded Hilbert space *-representations are studied for a q-analogue of the *-algebra Pol(Mat2,2) of polynomials on the space Mat2,2 of complex 2×2 matrices.

Compactness properties of operator multipliers

Abstract We continue the study of multidimensional operator multipliers initiated in [K. Juschenko, I.G. Todorov, L. Turowska, Multidimensional operator multipliers, Trans. Amer. Math. Soc., in press]. We introduce the notion of the symbol of an operator multiplier. We characterise completely compact operator multipliers in terms of their symbol as well as in terms of approximation by finite rank multipliers. We give sufficient conditions for the sets of compact and completely compact multipliers to coincide and characterise the cases where an operator multiplier in the minimal tensor product of two C ∗ -algebras is automatically compact. We give a description of multilinear modular completely compact completely bounded maps defined on the direct product of finitely many copies of the C ∗ -algebra of compact operators in terms of tensor products, generalising results of Saar [H. Saar, Kompakte, vollstandig beschrankte Abbildungen mit Werten in einer nuklearen C ∗ -Algebra, Diplomarbeit, Universitat des Saarlandes, Saarbrucken, 1982].

*-Representation type of *-doubles of finite dimensional algebras

We determine when the *-double of a finite-dimensional complex algebra is *-finite, *-tame and *-wild.

Unitarizable representations and fixed points of groups of biholomorphic transformations of operator balls

Abstract We show that the open unit ball of the space of operators from a finite-dimensional Hilbert space into a separable Hilbert space (we call it “operator ball”) has a restricted form of normal structure if we endow it with a hyperbolic metric (which is an analogue of the standard hyperbolic metric on the unit disc in the complex plane). We use this result to get a fixed point theorem for groups of biholomorphic automorphisms of the operator ball. The fixed point theorem is used to show that a bounded representation in a separable Hilbert space which has an invariant indefinite quadratic form with finitely many negative squares is unitarizable (equivalent to a unitary representation). We apply this result to find dual pairs of invariant subspaces in Pontryagin spaces. In Appendix A we present results of Itai Shafrir about hyperbolic metrics on the operator ball.

Unbounded Representations of q-Deformation of Cuntz Algebra

We study a deformation of the Cuntz–Toeplitz C*-algebra determined by the relations