Norm Estimates for Multiplication Operators in Hilbert Algebras

Abstract

In this paper, it is proved that for the bilinear operator defined by the operation of multiplication in an arbitrary associative algebra $V$ with unit $e_0$ over the fields $\mathbb{R}$ or $\mathbb{C}$, the infimum of its norms with respect to all scalar products in this algebra (with $||e_0 ||{\text{ = 1}}$) 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).