Abstract
If mathcal{A} is a finite-dimensional commutative associative real algebra with norm | cdot| then we say that the rth submultiplicative constant of mathcal{A} is the smallest constant m_{r}(mathcal{A}) for which | x_{1} x_{2} cdots x_{r} | leq m_{r}(mathcal {A}) | x_{1} | | x_{2} | cdots | x_{r} |. For a product algebra, we show that there exist zero divisors where equality is attained in the inequality defining m_{r}( mathcal{A}). We also study rho_{mathcal{A}} = limsup_{r rightarrowinfty} sqrt[r]{m_{r}(mathcal{A})}. We explain how rho_{mathcal{A}} appears in the generalization of the CauchyâHadamard criterion for hypercomplex power series. We find the submultiplicative constants and rho_{mathcal{A}} for the real group algebra of the cyclic group of order n as well as the complicated numbers mathcal{C}_{n} = { a_{1}+a_{2}k+ cdots+ a_{n} k^{n-1} | a_{i} inmathbb{R}, k^{n} = -1 } with Euclidean norm. Submultiplicative constants for the n-dual numbers Delta_{n} with Euclidean norm are also calculated or conjectured for n leq6. We show, for n geq2, rho_{Delta_{n}} = 1 for Delta_{n} given the p-norm.
Highlights
Suppose V1, . . . , Vr are finite-dimensional spaces over either R or C, and X is a normed space
It is known that any bounded multilinear map T : V1 Ă V2 à · · · Ă Vr â X is continuous
Proof If x y †mA x y for all x, y â A by repeated application of the inequality for the norm of the product of x1, x2, . . . , xr we find x1 x2 · · · xr †mrAâ1 x1 x2 · · · xr
Summary
Suppose V1, . . . , Vr are finite-dimensional spaces over either R or C, and X is a normed space. Vr are finite-dimensional spaces over either R or C, and X is a normed space. Vr) : vi †1, 1 †i †r} is attained by evaluation of T at some point in the Cartesian product of the unit-balls of V1, . Vr. Since r-fold multiplication on an algebra A provides an A-valued r-linear map over A, we have the following result: Theorem 1.1 If A is a finite-dimensional unital associative algebra (with multiplication denoted as ) over R or C with a norm · there exists a smallest constant mr(A) â R for each r â N, r †2 such that x1 x2 · · · xr †mr(A) x1 x2 · · · xr for all x1, x2, . This improves the less sharp use of mA in [1]
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