Norm Estimates for Multiplication Operators in Hilbert Algebras

In this paper, it is proved that for the bilinear operator defined by the operation of multiplication in an arbitrary associative algebra $\mathbf V$ with unit $\mathbf e_0$ over the fields $\mathbb R$ or $\mathbb C$, the infimum of its norms with respect to all scalar products in this algebra (with $||\mathbf e_0||=1$) is either infinite or at most $\sqrt {4/3}$. Sufficient conditions for this bound to be not less than $\sqrt {4/3}$ are obtained. The finiteness of this bound for infinite-dimensional Grassmann algebras was first proved by Kupsh and Smolyanov (this was used for constructing a functional representation for Fock superalgebras).