Geodesic
Norm Estimates for Multiplication Operators in Hilbert Algebras
Matematičeskie zametki, Tome 72 (2002) no. 2, pp. 283-291.

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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). 
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     author = {A. N. Urinovskii},
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A. N. Urinovskii. Norm Estimates for Multiplication Operators in Hilbert Algebras. Matematičeskie zametki, Tome 72 (2002) no. 2, pp. 283-291. http://geodesic.mathdoc.fr/item/MZM_2002_72_2_a11/

[1] Kupsh J., Smolyanov O. G., “Hilbert norms for graded algebras”, Proc. Amer. Math. Soc., 128:6 (2000), 1647–1653 | DOI | MR