Unital operator spaces and discrete groups

Abstract We introduce the trivial intersection property for concrete operator spaces and we show that a unital space with this property has no nontrivial boundary ideals. We provide various examples of such spaces, among which are strongly reflexive masa bimodules and completely distributive CSL algebras. We show that unital operator spaces acting on $$\ell ^2(\Gamma )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>ℓ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>Γ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> for any set $$\Gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Γ</mml:mi> </mml:math> , that contain the masa $$\ell ^\infty (\Gamma )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>ℓ</mml:mi> <mml:mi>∞</mml:mi> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>Γ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , possess the trivial intersection property, and we use this to prove that a unital surjective complete isometry between such spaces is a unitary equivalence. Then, we apply these results to $$w^*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>w</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:math> -closed $$\ell ^\infty (G)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>ℓ</mml:mi> <mml:mi>∞</mml:mi> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> -bimodules acting on $$\ell ^2(G)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>ℓ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> for a group G and we relate them to algebraic properties of G .