Abstract
We introduce and develop a theory of tensor products of JC-algebras and JW-algebras. Since JC-algebras and JW-algebras are so close to C*-algebras and W*-algebras, respectively, one can expect that the C*-algebra and W*-algebra tensor product theory will be actively involved. Chapter I is just preliminary, in which we have collected the basic definitions and theory which will be needed in the sequel. In chapter II, the definition of the tensor product JC(A $\sbsp{\lambda}{\otimes}$ B) of two JC-algebras A, B is introduced. Though the universal enveloping real C*-algebra R*(JC(A $\sbsp{\lambda}{\otimes}$ B)) can be identified with R*(A) $\sbsp{\lambda}{\otimes}$ R*(B), it is proved that C*(A) $\sbsp{\lambda}{\otimes}$ C*(B) is not the universal enveloping complex C*-algebra, in general, but under certain circumstances it can be. The notion of nuclear JC-algebra is given in chapter III, and connection of nuclearity of A and C*(A) is established. Also, the investigation of the Type of JC(A $\sbsp{\lambda}{\otimes}$ B), with a Jordan analogue of some results in the C*-algebra theory are contained in chapter III. A theory of the JW-tensor product is initated in chapter IV. Here, the key technical theorem which relates C*(M), and W*(M), where M is a JW-algebra, is proved. The Von Neumann tensor product W*(M) $\bar\otimes$ W*(N) turns out to be the universal enveloping Von Neumann algebra of the JW-tensor product JW(M $\bar\otimes$ N) of M and N, when JW(M $\bar\otimes$ N) is universally reversible. In chapter V, the type decomposition of JW(M $\bar\otimes$ N) is determined in terms of the type decomposition of the JW-algebras M and N, which essentially rely on the relationship between the types of the JW-algebra M and the types of its universal enveloping Von Neumann algebra W*(M).