Algebra norms on tensor products of algebras, and the norm extension problem

Abstract

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 K , 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 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.