I I n and all b i=1 Bi( j) such that i (1) = i (two) = . . . = i (n) and (bi(1) ) = = (bi(n) ) = 0 it holds that (bi(1) bi(n) ) = 0. We tension that freeness will depend on the state . Hence, in the previous definition, it will be more proper to say that the loved ones ( Bj ) j I is free with respect to . Generally it really is the context that prevents any ambiguity. Notice that Definition 2 makes sense also for any family members of unital -subalgebras of A. Notational convention. A household ( Bj ) j I of C -algebras is actually a ML-SA1 manufacturer function B defined on I. Thus we denote its nonstandard extension B, which is a function defined on I, byMathematics 2021, 9,12 of( Bj ) j I . For notational Nimbolide Apoptosis simplicity we create Bj for Bj . With out loss of generality, we are able to further assume that I I. The above notation and assumption are in force throughout this section.The chain of equivalences in the following result offers a nonstandard characterization of freeness and proves that the latter is preserved and reflected by the nonstandard hull building. Proposition 11. Let ( A, ) be an ordinary C ps and let ( Bj ) j I be a household of C -subalgebras of A. The following are equivalent: (1) (2)(3)( Bj ) j I is totally free; there exists some N N \ N for which the following holds: For all M N, all internal M i ( I ) M and all internal b j=1 Bi( j) such that i (1) = i (two) = . . . = i ( M ) if (bi(1) ) = = (bi( M) ) = 0 then (bi(1) bi( M) ) = 0; ( Bj ) j I is no cost with respect to .Proof. (1) (two) is actually a consequence of Transfer. Concerning (2) (3), we repair N as in (two). Let 0 m N, i ( I )m and b m j=1 Fin( Bi( j) ) be such that i (1) = i (two) = . . . = i (m) and (bi(1) ) = = (bi(m) ) = 0. Then (bi( j) ) 0 for all 1 j m. Let di( j) = bi( j) – (bi( j) )1. Therefore di( j) bi( j) and (di( j) ) = 0, for all 1 j m. It follows by assumption that (di(1) di(m) ) = 0. Since bi(1) bi(m) di(1) di(m) , we get (bi(1) bi(m) ) (di(1) di(m) ). Therefore 0 = ((bi(1) bi(m) )) = (bi(1) bi(m) ). The proof of (3) (1) is straightforward from Bj Bj , for all j I. The proof with the preceding proposition naturally leads to formulating a nonstandard variant in the notion of freeness. Definition three. Let ( A, ) be an internal C ps. A household ( Bj ) j I (not necessarily internal) of internal C -subalgebras of A is nearly free if, for all n N, all i I n and all b n=1 Fin( Bi( j) ), whenever j i (1) = i (two) = . . . = i (n) and (bi(1) ) 0, . . . , (bi(n) ) 0 then (bi(1) bi(n) ) 0. Proposition 12. Let ( A, ) be an ordinary C ps and let ( Bj ) j I be a family members of subalgebras of A. The following are equivalent: (1) (two)( Bj ) j I is cost-free. ( Bj ) j I is practically totally free.Proof. (1) (two). Let n N, i ( I )n and b n=1 Fin( Bi( j) ) be such that i (1) = j i (2) = . . . = i (n) and (bi(1) ) 0, . . . , (bi(n) ) 0. Since (bi( j) ) – (bi( j) )1) = 0, for all 1 j n, then (n=1 (bi( j) – (bi( j) )1)) = 0. We notice that n=1 (bi( j) – (bi( j) )1)) = j j bi(1) bi(n) S, exactly where S is really a normal finite sum of terms each having infinitesimal norm. Hence (bi(1) bi(n) ) 0, as expected. (2) (1). The following chain of implications holds: ( Bj ) j I is practically absolutely free ( Bj ) j I is free of charge ( Bj ) j I is absolutely free ( Bj ) j I is totally free. The leftmost implication is simple and the middle 1 holds by Proposition 11. The rightmost implication holds by Transfer. Corollary 2. Let ( A, ), ( Bj ) j I be as in Proposition 12. Then ( Bj ) j I is free if and only if ( Bj ) j I is pretty much totally free. Let ( A, ) be an internal C p.

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