ALGEBRA NORMS ON TENSOR-PRODUCTS OF ALGEBRAS, AND THE NORM EXTENSION PROBLEM

We show that, if A is a finite-dimensional -simple associative algebr a with involution (over the field K of real or complex numbers) whose hermitian part H(A, ) is of degree greater than or equal to 3 over it s center, if B is a unital algebra with involution over K, and if para llel to.parallel to is an algebra norm on H(A x B, ), then there exis ts an algebra norm on A x B whose restriction to H(A x B, ) is equiva lent to parallel to.parallel to Applying zel'manovian techniques, we p rove that the same is true if the finite dimensionality of A is relaxe d to the mere existence of a unit for A, but the unital algebra B is a ssumed to be associative. We also obtain results of a similar nature s howing that, for suitable choices of algebras A and B over K, the cont inuity of the natural product of the algebra A x B for a given norm ca n be derived from the continuity of the symmetrized product. (C) 1998 Elsevier Science Inc.