Spaces of operators that are left and right modules over maximal abelian selfadjoint algebras (masa bimodules for short) are natural generalizations of algebras with commutative subspace lattices. This paper is concerned with density properties of finite rank operators and of various classes of compact operators in such modules. It is shown that every finite rank operator of a norm closed masa bimodule M is in the trace norm closure of the rank one subspace of M. An important consequence is that the rank one subspace of a strongly reflexive masa bimodule (that is, one which is the reflexive hull of its rank one operators) is dense in the module in the weak operator topology. However, in contrast to the situation for algebras, it is shown that such density need not hold in the ultraweak topology.A new method of representing masa bimodules is introduced. This uses a novel concept of an |-topology. With the appropriate notion of |-support, a correspondence is established between reflexive masa bimodules and their |-supports. It is shown that, if a C 2 -closed masa bimodule contains a trace class operator then it must contain rank one operators; indeed, every such operator is in the C 2 -norm closure of the rank one subspace of the module. Consequently the weak closure of any masa bimodule of trace class operators is strongly reflexive. However, the trace norm closure of the rank one subspace need not contain all trace class operators of the module. Also, it is shown that there exists a CSL algebra which contains no trace class operators yet contains an operator belonging to C p for all p>1. From this it follows that a transitive bimodule spanned by the rank one operators it contains need not be dense in C p for 1 p< .As an application, it is shown that there exists a commutative subspace lattice L such that L is non-synthetic but every weakly closed algebra which contains a masa and has invariant lattice L coincides with Alg L.

We develop a symbol calculus for normal bimodule maps over a masa that is the natural analogue of the Schur product theory. Using this calculus we are able to easily give a complete description of the ranges of contractive normal bimodule idempotents that avoids the theory of J*-algebras. We prove that if P is a normal bimodule idempotent and P < 2/ √ 3 then P is a contraction. We finish with some attempts at extending the symbol calculus to non-normal maps.

The Fourier binest algebra is defined as the intersection of the Volterra nest algebra on L2(ℝ) with its conjugate by the Fourier transform. Despite the absence of nonzero finite rank operators this algebra is equal to the closure in the weak operator topology of the Hilbert–Schmidt bianalytic pseudo-differential operators. The (non-distributive) invariant subspace lattice is determined as an augmentation of the Volterra and analytic nests (the Fourier binest) by a continuum of nests associated with the unimodular functions exp(−isx2/2) for s>0. This multinest is the reflexive closure of the Fourier binest and, as a topological space with the weak operator topology, it is shown to be homeomorphic to the unit disc. Using this identification the unitary automorphism group of the algebra is determined as the semi-direct product ℝ2×κℝ for the action κt(λ, μ) =(etλ, e−t μ).

We characterise the Jacobson radical of an analytic crossed product C 0 (X) × f Z + , answering a question first raised by Arveson and Josephson in 1969. In fact, we characterise the Jacobson radical of analytic crossed productsThis consists of all elements whose ''Fourier coefficients'' vanish on the recurrent points of the dynamical system (and the first one is zero). The multidimensional version requires a variation of the notion of recurrence, taking into account the various degrees of freedom.

This paper is concerned with weak* closed masa-bimodules generated by A(G)-invariant subspaces of VN(G). An annihilator formula is established, which is used to characterise the weak* closed subspaces of B(L 2 (G)) which are invariant under both Schur multipliers and a canonical action of M (G) on B(L 2 (G)) via completely bounded maps. We study the special cases of extremal ideals with a given null set and, for a large class of groups, we establish a link between relative spectral synthesis and relative operator synthesis.Key words and phrases. Fourier algebra, masa-bimodule, invariant subspaces. arXiv:1402.0721v2 [math.OA] 12 Jul 2014 2 M. ANOUSSIS, A. KATAVOLOS AND I. G. TODOROVis that these two operations have the same outcome; in other words, the diagramis commutative. The proof uses the techniques developed by J. Ludwig, N. Spronk and L. Turowska in [31] and [20]. Some of the results in Section 3 also appear in the aforementioned papers; we have chosen to present complete arguments in order to clarify some details.Using this result, we present a unified approach to some problems of Harmonic Analysis on G. In particular, in Section 5, we look at the special cases where J is the minimal, or the maximal, ideal of A(G) with a given null set E ⊆ G. The main result here is Theorem 5.3; as a corollary, we obtain the result established in [20] that if A(G) possesses an approximate identity then a closed set E ⊆ G satisfies spectral synthesis if and only if the set E * = {(s, t) : ts −1 ∈ E} satisfies operator synthesis.The connection between spectral synthesis and operator synthesis was discovered by W. B. Arveson in [1]. The above result is due to J. Froelich [11] for G abelian and to N. Spronk and L. Turowska [31] for G compact. J. Ludwig and L. Turowska [20] show that a closed subset E of a locally compact group G satisfies local spectral synthesis if and only if E * satisfies operator synthesis; local spectral synthesis coincides with spectral synthesis when A(G) has an approximate identity.Spectral synthesis relative to a fixed A(G)-invariant subspace of VN(G) was introduced for locally compact groups by E. Kaniuth and A.T. Lau in [17]. In [26], the authors define relative operator synthesis for subsets of G × G, where G is compact, and link it to relative spectral synthesis. In Section 6, using our results, and assuming that the A(G)-invariant subspace of VN(G) is weak* closed we prove an analogous relation for locally compact groups for which A(G) possesses an approximate identity. We note that this class contains, but is larger than, the class of amenable groups.As another application, we are able to identify the weak* closed subspaces of B(L 2 (G)) that are invariant under both Schur multiplication and an action of the measure algebra M (G). More precisely, let Γ : M (G) → B(B(L 2 (G))) be the representation of M (G) given by). This action was studied by F. Ghahramani, M. Neufang, Zh.-J. Ruan, R. Smith, N. Spronk and E. Størmer in [12], [21], [22], [29], [32], among others.The maps Γ(µ) are precisely thos...