That A is often a C -subalgebra of A. As is customary, we create A for a. If : A B is often a homomorphism of ordinary C -algebras, we let : A aB( a )Mathematics 2021, 9,four ofSince homomorphisms are norm-contracting, the map is well-defined. Furthermore, it is simple to confirm that it really is a homomorphism. All of the above assumptions and notations are in force all through this paper. Similarly to the above, 1 defines the Tianeptine sodium salt Neuronal Signaling nonstandard hull H of an internal Hilbert space H. It is actually a straightforward verification that H is definitely an ordinary Hilbert space with respect towards the common part of the inner solution of H. ML-SA1 Cancer Additionally, 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 single a A can 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.) Hence we can regard A as a C -subalgebra of B( H ). 3. Three Recognized Benefits The results in this section may be rephrased in ultraproduct language and can be proved by utilizing the theory of ultraproducts. The nonstandard proofs that we present under show how you can apply the nonstandard strategies in mixture with the nonstandard hull construction. three.1. Infinite Dimensional Nonstandard Hulls Fail to be von Neumann Algebras In [8] [Corollary 3.26] it is actually proved that the nonstandard hull B( H ) of your in internal algebra B( H ) of bounded linear operators on some Hilbert space H more than C is a von Neumann algebra if and only if H is (common) finite dimensional. Basically, this outcome may be quickly enhanced by showing that no infinite dimensional nonstandard hull is, as much as isometric isomorphism, a von Neumann algebra. It is actually well-known that, in any infinite dimensional von Neumann algebra, there is certainly an infinite sequence of mutually orthogonal non-zero projections. Hence one particular could need to apply [8] [Corollary 3.25]. Albeit the statement of your latter is correct, its proof in [8] is wrong inside the final part. As a result we begin by restating and reproving [8] [Corollary three.25] with regards to rising sequences of projections. We denote by Proj( A) the set of projections of a C -algebra A. Lemma 1. Let A be an internal C -algebra and let ( pn )nN be an growing sequence of projections in Proj( A ). Then there exists an growing 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 choose 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 3.22(vi)], we are able to further assume that pn1 = r, for some r Proj( A ). By [11] [II.3.3.1], we’ve got rqn = qn , namely rqn qn . Therefore, by Transfer of [11] [II.3.3.5], for all k N there’s rk Proj( A) such that qn rk and r – rk 1/k. By Overspill, there is q Proj( A) such that qn q and q r. We let qn1 = q. Then we right away 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 just isn’t a von Neumann algebra. Proof. From ( pn )nN , we get an escalating sequence ( pn )nN of projections within a by letting pn = p0 pn , for all n N. By Lemma 1, there exists an increasing sequence (qn )nN of projections in a. From the latter we get a sequence (qn )nN of non-zero mutually orthogonal projections,.
