On the Ranges of Bimodule Projections

Abstract

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.

Herz–Schur multipliers of dynamical systems

“…The map S ϕ is defined on the space B(H 1 , H 2 ) of all bounded linear operators from H 1 into H 2 by taking the second dual of the constructed map on K(H 1 , H 2 ). A characterisation of measurable Schur multipliers, extending Grothendieck's result, was obtained in [13] and [28] (see also [17] and [38]). Namely, a function ϕ ∈ L ∞ (X × Y ) was shown to be a Schur multiplier if and only if ϕ coincides almost everywhere with a function of the form Among the large number of applications of Schur multipliers is the description of the space M cb A(G) of completely bounded multipliers (also known as Herz-Schur multipliers) of the Fourier algebra A(G) of a locally compact group G, introduced by J. de Cannière and U. Haagerup in [7].…”

“…Note that Schur C-multipliers coincide with the classical (measurable) Schur multipliers [28], [17].…”

Ideals of the Fourier algebra, supports and harmonic operators

“…The operators of the form S w , w ∈ S(G), are precisely the bounded weak* continuous D-bimodule maps on B(L 2 (G)) (see [7], [21], [18] and [14]). A weak* closed subspace U of B(L 2 (G)) is invariant under the maps S w , w ∈ S(G), if and only if it is invariant under all left and right multiplications by elements of…”

Equivariant maps and bimodule projections

“…For another proof of this fact see [9]. Solel also conjectured that the same result would hold even when Φ was not weak * -continuous.…”