That A is usually a C -subalgebra of A. As is customary, we write

That A is usually a C -subalgebra of A. As is customary, we write A for any. If : A B is often a homomorphism of ordinary C -algebras, we let : A aB( a )Mathematics 2021, 9,4 ofSince homomorphisms are norm-contracting, the map is well-defined. In addition, it truly is straightforward to confirm that it is a homomorphism. Each of the above assumptions and notations are in force all through this paper. Similarly towards the above, one particular defines the nonstandard hull H of an internal Hilbert space H. It is a straightforward verification that H is an ordinary Hilbert space with respect to the common part of the inner item of H. In addition, let B( H ) be the internal C -algebra of bounded linear operators on some internal Hilbert space H and let A be a subalgebra of B( H ). Every a A might be regarded as an element of B( H ) by letting a( x ) = a( x ), for all x H of finite norm. (Note that a( x ) is properly defined because a is norm inite.) Thus we can regard A as a C -subalgebra of B( H ). three. Three Identified Final results The outcomes in this section might be rephrased in ultraproduct language and may be proved by utilizing the theory of ultraproducts. The nonstandard proofs that we present beneath show how you can apply the nonstandard strategies in combination using the nonstandard hull building. three.1. Infinite Dimensional Nonstandard Hulls Fail to be von Neumann Algebras In [8] [C2 Ceramide custom synthesis Corollary 3.26] it can be proved that the nonstandard hull B( H ) of the in internal algebra B( H ) of bounded linear operators on some Hilbert space H over C is actually a von Neumann algebra if and only if H is (normal) finite dimensional. Truly, this result is often easily enhanced by displaying that no infinite dimensional nonstandard hull is, up to isometric isomorphism, a von Neumann algebra. It really is well-known that, in any infinite dimensional von Neumann algebra, there is certainly an infinite sequence of mutually orthogonal non-zero projections. Hence 1 may possibly desire to apply [8] [Corollary 3.25]. Albeit the statement on the latter is appropriate, its proof in [8] is incorrect within the final portion. Hence we start by restating and reproving [8] [Corollary three.25] when it comes to rising sequences of projections. We denote by Proj( A) the set of PF-06873600 Autophagy projections of a C -algebra A. Lemma 1. Let A be an internal C -algebra and let ( pn )nN be an increasing sequence of projections in Proj( A ). Then there exists an escalating sequence of projections (qn )nN in Proj( A) such that, for all n N, pn = qn . Proof. We recursively define (qn )nN as follows: As q0 we pick any projection r Proj( A) such that p0 = r. (See [8] [Theorem 3.22(vi)].) Then we assume that q0 qn in Proj( A) are such that pi = qi for all 0 i n. Once more by [8] [Theorem three.22(vi)], we can additional assume that pn1 = r, for some r Proj( A ). By [11] [II.three.three.1], we’ve got rqn = qn , namely rqn qn . Therefore, by Transfer of [11] [II.three.3.5], for all k N there’s rk Proj( A) such that qn rk and r – rk 1/k. By Overspill, there is certainly q Proj( A) such that qn q and q r. We let qn1 = q. Then we promptly get the following: Corollary 1. Let A be an internal C -algebra of operators and let ( pn )nN be a sequence of non-zero mutually orthogonal projections in Proj( A ). Then A will not be a von Neumann algebra. Proof. From ( pn )nN , we get an growing sequence ( pn )nN of projections in a by letting pn = p0 pn , for all n N. By Lemma 1, there exists an rising sequence (qn )nN of projections within a. In the latter we get a sequence (qn )nN of non-zero mutually orthogonal projections,.