The results in the case where (mathcal{K}_{C mathcal{A}}) is a left (mathcal{G} -family, ((X,T)) is left (mathcal {G} -familyible in a point (w^{0}in X,T and ((X,T^{[k]})) is left (mathcal {G} -admissibleA}})-closed on X now follow at once from Steps I-III.
The results in the case where (mathcal{K}_{C mathcal{A}}) is a left (mathcal{W} -family, ((X,T)) is left (mathcal {W} -familyible in a point (w^{0}in X,T and ((X,T^{[k]})) is left (mathcal {M}_{C;mathcal{A}})-closed on X noW} -admissiblece from Steps V-VII.
Then a nonzero finite-dimensional linear subspace (W subset {mathcal A}) is called a left ((mathcal {F}, varepsilon )) -Følner subspace if it satisfies begin{aligned} frac{dim (a W +W)}{dim (W }le 1+varepsilon, quad mathrm {for~all}quad ain mathcal {F}.
Let ((A_{n})_{ninmathbb{N}}) be a left K-Cauchy sequence in ((mathcal{A},d)).
Let ((X,mathcal{P}_{C mathcal{A}})) be a quasi-triangular space, and let (mathcal{J}_{C mathcal{A}}) be a left (right) family generated by (mathcal{P}_{C mathcal{A}}).
A subalgebra (mathcal{U}) of an algebra (mathcal{A}) is a right ideal if (mathcal{U}mathcal{A}={ua: uinmathcal{U}, ainmathcal{A}}subsetmathcal{A}) and it is a left ideal if (mathcal{A} mathcal{U}={au:uinmathcal{U}, ain mathcal{A}}subsetmathcal{A}); it is a two-sided ideal if it is both left and right ideal.
If (lambdainmathbb{C}), (xinmathbb{C}^{m}backslash{0}), and (yin mathbb{C}^{n}backslash{0}) are solutions of textstylebegin{cases} mathcal{A}x^{p-1}y^{q}=lambda x^{[l-1]}, mathcal{A}x^{p}y^{q-1}=lambda y^{[l-1]}, end{cases} then we say that λ is a singular value of (mathcal{A}), x and y are a left and a right eigenvectors of (mathcal{A}), associated with λ.
Let ({mathcal G} = ({mathcal U} cup {mathcal V}, {mathcal E})) be an unbalanced left regular bipartite graph with (|{mathcal U}| = n geq |{mathcal V}| = m) and left degree h.
Let ({mathcal G}) be an unbalanced (left) expander bipartite graph as defined in (21).
Let (mathcal{K}_{C mathcal{A}}) be the left (right) (mathcal {G} -family generated by (mathcal{G} -familycal{A}}).
