The Tensor Product Of Some Operator Algebra

The paper concludes the tensor products of the various types of algebras at first, and shows to us that when tensor product formula: AlgL1(?)AlgL2=Alg(L1(?)L2(1) is valid what conditions L1, and L2should be need. Slice maps was introduced by Tomiyama in [46], then he gave the definition of Fubini products F(S,T). If F(S,T)=S(?)T for any σ-weakly closed subalgebra T of B(K), then S is said to have Property Sσ. Kraus has shown that if one has property Sσ for any sublattices L1and L2, then formula (1) is valid. This is a sufficient condition for formula (1).That is, if Alg L1has Property Sσ, then formula (1) is correct. For this, we do our best to study what conditions the σ-weakly closed subalgebras of B(H) should be need when they have Property Sσ. Kraus showed that if L1is a subspace lattice such that the finite rank operators in AlgL1are σ-weakly dense in Alg L1then AlgL1has property Sσ; hence formula (1) holds for any subspace latticesL2. As we all know that AlgL1has identity in his proofs. However, when a σ-weakly closed subalgebra A of B(H) doesn’t have identity, his results are not necessarily true. The main results of this paper is that if A has approximate identities (in σ-weakly closed topology), and the finite rank operators in A are σ-weakly dense in A, then A also has property Sσ. As a result, the conclusions of Kraus in [13] can be as a corollary of our main results. Further, we shown that the compact nest algebras have property...