Algebra Norms on Tensor Products of Algebras, and the Norm Extension Problem*

We show that, if A is a finite-dimensional *-simple associative algebra with involution (over the field K of real or complex numbers) whose hermitian part H( A, * > is of degree > 3 over its center, if B is a unital algebra with involution over 06, and if (I.11 is an algebra norm on H( A @ B, * 1, then there exists an algebra norm on A @ B whose restriction to H(A @ B, *> is equivalent to 11 . 11. Applying zel’manovian techniques, we prove that the same is true if the finite dimensionality of A is relaxed to the mere existence of a unit for A, but the unital algebra B is assumed to be associative. We also obtain results of a similar nature showing that, for suitable choices of algebras A and B over K, the continuity of the natural product of the algebra A @ B for a given norm can be derived from the continuity of the symmetrized product.