On the tensor products of $C^{\ast}$-algebras

Abstract

introduced a tensor product A x ® a A 2 of two C^-algebras A 1 and A 2> which is the C*-algebra obtained as the completion of the ^-algebraic tensor product A ί QA 2 of A x and A 2 with respect to the tf-norm || || α .As Wulfsohn [7] established, the tf-norm has the property: for every faithful representations 7Γχ of A x and π 2 of A 2 .It was observed in [5] that the tf-norm is not necessarily the unique compatible norm in A X QA 2 and that it is the least one among the all compatible norms.On the other hand, A. Guichardet [4] gave, with the corresponding tensor product, the greatest compatible norm || || y in A λ QA 2 the z/-norm.These arguments will bring forward many interesting problems on the relations between compatible norms in AiOA 2 and corresponding tensor products, and some of them will be considered in this paper.We shall discuss on ΰ^-norms in A λ QA 2 in Theorems 1 and 2, and on the enveloping C~*-algebras of ^-Banach algebras in Theorem 3. The auther wishes to express his hearty thanks to Prof. M. Fukamiya and Dr. M. Takesaki for their many valuable suggestions.