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 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 , and if • is an algebra norm on H(A B, *), then there exists an algebra norm on A B whose restriction to H(A B, *) is equivalent to • . Applying zelʹmanovian 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 , 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.