In the bimodule setting, we show that relative operator 1-biflatness of A(G) is equivalent to the existence of a contractive approximate indicator for the diagonal GΔ in the Fourier Stieltjes algebra B(G×G), thereby establishing the converse to a result of Aristov, Runde, and Spronk [3].
This establishes the converse.
By direct computation, one can establish the converse of this lemma.
To establish the converse, let (g_{n}) be in the range of (mathbf{l}_{alpha_{1},alpha_{2}}), (forall ninmathbb{N}).
To establish the converse, for each φ ∈ C 0 ∞ , let φ+ be the positive part and φ- be the negative part of φ.
In order to establish the converse inclusion, we fix an arbitrary pair ((u^{ast},y^{ast})) in (operatorname{cl}_{mu} Xi_{Delta}).
Combining Lemma 4 with the trivial point-to-point bounds establishes that the region (mathcal {D}_{mathsf {FD}}), given in Theorem 1, is an outer bound on any achievable degrees-of-freedom pair, thus establishing the converse part of Theorem 1.
To establish the converse part of Theorem 1, we must show that the region (mathcal {D}_{mathsf {FD}}), which we have already shown is achievable, is also an outer bound on the degrees-of-freedom, i.e., we want to show that if an arbitrary degree-of-freedom pair (d 1,d 2) is achievable, then ((d_{1},d_{2}) in mathcal {D}_{mathsf {FD}}).
To establish the converse, assume that e={ u, v}∈ E(V| W) is not contained in any minimum spanning tree.
We also establish that the converse may not be true.
