On the Radius in Cayley-Dickson Algebras
aa r X i v : . [ m a t h . R A ] M a y ON THE RADIUS IN CAYLEY–DICKSON ALGEBRAS
MOSHE GOLDBERG AND THOMAS J. LAFFEY
Abstract.
In the first two sections of this paper we provide a brief account of the Cayley–Dickson algebras and prove that the radius on these algebras is given by the Euclidean norm.With this observation we resort to three related topics: a variant of the Gelfand formula,stability of subnorms, and the functional power equation. The Cayley–Dickson algebras
The Cayley–Dickson algebras constitute a familiar series of algebras, A , A , A , . . . over thereals, with dim A n = 2 n .The first five algebras in this series are the reals R , the complex numbers C , the quaternions H , the octonions O , and the sedenions S . While R and C are both commutative and associative, H is no longer commutative, and O and S are not even associative. As it is, O is alternative,and S is merely power-associative.As usual, we call an algebra A alternative if the subalgebra generated by any two elementsis associative. Further, A is called power-associative if the subalgebra generated by any oneelement is associative. It thus follows that if A is power-associative, then the powers of eachelement in A are unambiguously defined .It is well known that the Cayley–Dickson algebras can be inductively obtained from eachother by the following Cayley–Dickson doubling process (e.g., [W1]). We initiate this processby setting A = R and defining a ∗ , the conjugate of a real number a , to equal a . Then, assumingthat A n − , n ≥
1, has been determined, we define A n to be the set of all ordered pairs A n = { ( a, b ) : a, b ∈ A n − } , such that addition and scalar multiplication are taken componentwise on the Cartesian product A n − × A n − , conjugation is determined by( a, b ) ∗ = ( a ∗ , − b ) , and multiplication is given by ( a, b )( c, d ) = ( ac − d ∗ b, da + bc ∗ ) . With this definition, each element in A n is of the form a = ( α , . . . , α n ), α j ∈ R . Moreover,it readily follows that the distributive laws hold, and that the conjugate of a and the unit Mathematics Subject Classification.
Primary 16P10, 17A05, 17A35, 17D05, 39B22.
Key words and phrases.
Cayley–Dickson algebras, power-associative algebras, radius of an element in afinite-dimensional power-associative algebra, subnorms, the Gelfand formula, stability of subnorms, the powerequation. element in A n are given, respectively, by a ∗ = ( α , − α , . . . , − α n ) and n = (1 , , . . . , . For example, we have A = { ( α, β ) : α, β ∈ R } , where ( α, β ) ∗ = ( α ∗ , − β ) , ( α, β )( γ, δ ) = ( αγ − δβ, δα + βγ ) , = (1 , A with C upon writing z = α + iβ as ( α, β ).By construction, A n − can be viewed as a subalgebra of A n . Since the octonions are non-associative, it follows that A n is non-associative for all n ≥
3. Similarly, since the sedenions arenot alternative, A n is not alternative for n ≥
4. It is known, however, that all Cayley–Dicksonalgebras are power-associative —a fact that will be revisited in Theorem 1.1 below.Carrying on with our short account, we shall now state the following known result whoseproof is provided for the reader’s convenience:
Lemma 1.1.
Each a = ( α , α , . . . , α n ) ∈ A n is annihilated by the quadratic polynomial p a ( t ) = t − α t + | a | , where | a | = q α + · · · + α n is the Euclidean norm.Proof. Let us show by induction that(1.1) a ∗ a = | a | n , a ∈ A n . The case n = 0 is trivial, so assuming the assertion for n − a ∈ A n as a = ( b, c ) , b, c ∈ A n − , we get a ∗ a = ( b, c ) ∗ ( b, c ) = ( b ∗ , − c )( b, c ) = ( b ∗ b + c ∗ c, cb ∗ − cb ∗ )= ( b ∗ b + c ∗ c,
0) = ( | b | + | c | )( n − ,
0) = | a | n . We also have,(1.2) a + a ∗ = ( α , α , . . . , α n ) + ( α , − α , . . . , − α n ) = (2 α , , . . . ,
0) = 2 α n . So by (1.1) and (1.2), a − α a + | a | n = a − ( a + a ∗ ) a + a ∗ a = 0 , and we are done. (cid:3) Aided by this lemma one can prove that A n is power-associative. Since this property lies atthe heart of our paper, and since the proof is short and not readily available in the literature,we take the liberty of posting it here: N THE RADIUS IN CAYLEY–DICKSON ALGEBRAS 3
Theorem 1.1.
The Cayley–Dickson algebras are power-associative.Proof.
By Lemma 3 in [A], A n is power-associative if and only if a a = aa and a a = ( a a ) a for all a ∈ A n . By Lemma 1.1, each a ∈ A n satisfies a relation of the form(1.3) a = αa + β n α, β ∈ R . So,(1.4) a a = ( αa + β n ) a = αa + βa = a ( αa + β n ) = aa . Furthermore, by (1.3) and (1.4), a a = ( αa + β n ) a = αaa + βa = αa a + βa = ( αa + βa ) a = (( αa + β n ) a ) a = ( a a ) a, which yields the desired result. (cid:3) We mention, in passing, that it is a simple matter to verify that the sedenions are notalternative. Indeed, denoting by e j , j = 1 , . . . ,
16, the basis element of S whose j th entryis 1 and all others are zero, we put o = e + e and o = e − e . Hence, consulting themultiplication table for the sedenions (e.g., [W2]), we find that o ( o o ) = 0 while o o = − o ;so the option of alternativity is shattered.2. The radius on the Cayley–Dickson algebras
We begin this section by considering an arbitrary finite-dimensional power-associative algebra A over a field F .As is customary, by a minimal polynomial of an element a in A , we mean a monic polynomialof lowest positive degree with coefficients in F that annihilates a .With this familiar definition, we recall: Theorem 2.1 ([G1, Theorem 1.1]) . Let A be a finite-dimensional power-associative algebraover a field F . Then: (a) Every element a ∈ A possesses a unique minimal polynomial. (b) The minimal polynomial of a divides every other polynomial that annihilates a . Restricting attention to the case where the base field of our algebra is either R or C , anddenoting the minimal polynomial of an element a ∈ A by q a , we recall, [G1], that the radius of a is defined to be the nonnegative quantity r ( a ) = max {| λ | : λ ∈ C , λ is a root of q a } . The radius has been computed for elements in several well-known finite-dimensional power-associative algebras. For instance, it was shown in [G1, Section 2 and page 4072] that the
MOSHE GOLDBERG AND THOMAS J. LAFFEY radius on the low dimensional Cayley–Dickson algebras C , H , and O is given, in each case, bythe corresponding Euclidean norm.Another example of considerable interest emerged in [G1], where it was observed that if A is an arbitrary matrix algebra over R or C (with the usual matrix operations), then the radiusof a matrix A ∈ A is given by the classical spectral radius, ρ ( A ) = max {| λ | : λ ∈ C , λ is an eigenvalue of A } . With this last example in mind, we register the following theorem which tells us that theradius retains some of the most basic properties of the spectral radius not only for matrices,but in the general case as well.
Theorem 2.2 ([G1, Theorems 2.1 and 2.4]) . Let A be a finite-dimensional power-associativealgebra over a field F , either R or C . Then: (a) The radius r is a nonnegative function on A . (b) The radius is homogeneous, i.e., for all a ∈ A and α ∈ F , r ( αa ) = | α | r ( a ) . (c) For all a ∈ A and all positive integers k , r ( a k ) = r ( a ) k . (d) The radius vanishes only on nilpotent elements of A . (e) The radius is a continuous function on A . Having stated this theorem which shows that the radius is a natural extension of the spectralradius, we are now ready to readdress the Cayley–Dickson algebras.
Theorem 2.3.
The radius r of an element a = ( α , . . . , α n ) ∈ A n is the Euclidean norm, i.e., r ( a ) = | a | , a ∈ A n . Proof.
Select a in A n . By Lemma 1.1, the second order monic p a ( t ) = t − α t + | a | annihilates a . Hence p a is divisible by the minimal polynomial of a . Further, since the roots of p a are t ± = α ± q α − | a | = α ± q − α − · · · − α n , it follows that | t ± | = | a | ; so r ( a ) = | a | and the proof is complete. (cid:3) Since the radius on the Cayley–Dickson algebras is a norm, it automatically satisfies proper-ties (a), (b) and (e) of Theorem 2.2. Moreover, since a norm vanishes only on the zero element,Theorem 2.2(d) implies that the Cayley–Dickson algebras are void of nonzero nilpotent elements.
Combining Theorems 2.2(c) and 2.3, we realize that | a k | = | a | k for all a ∈ A n and k = 1 , , , . . . Naturally, a real-valued function on a finite-dimensional algebra A is said to be continuous if it is continuouswith respect to the unique finite-dimensional topology on A . N THE RADIUS IN CAYLEY–DICKSON ALGEBRAS 5
So if a = 0, then | a k | = | a | k = 0 , which is another way of showing that the Cayley–Dickson algebras are void of nonzero nilpo-tents. 3. Subnorms and a formula for the radius
Again, let A be a finite-dimensional power-associative algebra over a field F , either R or C .Following [GL2], we call a real-valued function f : A → R a subnorm if for all a ∈ A and α ∈ F , f ( a ) > , a = 0 ,f ( αa ) = | α | f ( a ) . We recall that a real-valued function N is a norm on A if for all a, b ∈ A and α ∈ F , N ( a ) > , a = 0 ,N ( αa ) = | α | N ( a ) ,N ( a + b ) ≤ N ( a ) + N ( b );hence, a norm is a subadditive subnorm. We also recall that in our finite-dimensional settings,a norm is a continuous function on A .With the above definition of a subnorm, we may now state a variant of a result which iswell known in the context of complex Banach algebras (e.g., [R, Theorem 18.9], [L, Chapter17, Theorem 4]), and which is often referred to as the Gelfand formula. Theorem 3.1 ([G2, Theorem 2.1]) . Let f be a continuous subnorm on a a finite-dimensionalpower-associative algebra A over R or C . Then: (3.1) lim k →∞ f ( a k ) /k = r ( a ) for all a ∈ A . In particular, for the Cayley–Dickson algebras we get:
Corollary 3.1. If f is a continuous subnorm on A n , then (3.2) lim k →∞ f ( a k ) /k = | a | for all a ∈ A n . For example, we observe that for each fixed p , 0 < p ≤ ∞ , the function(3.3) | a | p = ( | α | p + · · · + | α n | p ) /p , a = ( α , . . . , α n ) ∈ A n , is a continuous subnorm on A n (a norm precisely when 1 ≤ p ≤ ∞ ). Thus, we getlim k →∞ | a k | /kp = | a | for all a ∈ A n . MOSHE GOLDBERG AND THOMAS J. LAFFEY
We point out that contrary to norms, a subnorm on an algebra A over R or C with dim A ≥ f be a continuous subnorm on suchan algebra. Fix an element a = 0 in A , and consider V = { αa : α ∈ F } , the linear subspace of A generated by a . For each real κ , κ >
1, define(3.4) g κ ( a ) = (cid:26) κf ( a ) , a ∈ V ,f ( a ) , a ∈ A r V . Then evidently, g κ is a subnorm on A . Further, g κ is discontinuous at a sincelim a → a a/ ∈ V g κ ( a ) = lim a → a a/ ∈ V f ( a ) = lim a → a f ( a ) = f ( a ) = g κ ( a ) . Motivated by the above example, we shall next show that a subnorm on a finite-dimensionalpower-associative algebra over R or C may satisfy formula (3.1) without being continuous .To this end, we recall that two subnorms f and g are equivalent on an algebra A if thereexist constants µ > ν >
0, such that µf ( a ) ≤ g ( a ) ≤ νf ( a ) for all a ∈ A . With this familiar definition, we may now register:
Proposition 3.1 ([G2, Theorem 3.1]) . Let g be a subnorm on a finite-dimensional power-associative algebra A over R or C . If g is equivalent to a continuous subnorm on A , then g satisfies formula (3.1) . To illustrate Proposition 3.1, let A be a finite-dimensional power-associative algebra over R or C with dim A = 2, and let us fall back on the discontinuous subnorm g κ in (3.4). Since κ >
1, we have(3.5) f ( a ) ≤ g κ ( a ) ≤ κf ( a ) , a ∈ A ;so (as indicated in Section 3 of [G2]), our proposition yieldslim k →∞ g κ ( a k ) /k = r ( a ) for all a ∈ A . In contrast with the last example, we shall now show that not all discontinuous subnormson Cayley–Dickson algebras satisfy formula (3.2). In order to support this assertion, considerthe familiar Cauchy equation(3.6) ϕ ( x + y ) = ϕ ( x ) + ϕ ( y ) , x, y ∈ R , whose (real) solutions have been discussed in the literature for over a century (e.g., [H], [HLP,Section 3.20], [HR], [B, Section 20], and [GL3, Section 2]). It is well known that every solutionof (3.6) satisfies(3.7) ϕ ( γx ) = γϕ ( x ) for all rational γ and real x. N THE RADIUS IN CAYLEY–DICKSON ALGEBRAS 7
Hence, the only continuous solutions of (3.6) are of the form ϕ ( x ) = xϕ (1) , x ∈ R , where ϕ (1) is an arbitrary real value. It is also known that equation (3.6) has discontinuoussolutions, and that all such solutions are discontinuous everywhere and unbounded (both frombelow and above) on any interval in R . Moreover, given a positive number c , one may select adiscontinuous solution ϕ with ϕ ( c ) = 0. Consequently,(3.8) ϕ ( x + c ) = ϕ ( x ) + ϕ ( c ) = ϕ ( x ) , x ∈ R , so ϕ can be chosen to be c -periodic. In fact, (3.8) tells us that c is a period of ϕ if and only if ϕ ( c ) = 0; thus if c is a period then, by (3.7), so is every rational multiple of c .With these remarks, we may quote the following result. Theorem 3.2 ([GL3, Theorems 2.1(a,c) and 2.2(a)]) . Let f be a continuous subnorm on C ,and let ϕ be a discontinuous π -periodic solution of the Cauchy equation (3.6) . Then: (a) The function (3.9) g ϕ ( z ) = f ( z ) e ϕ (arg z ) , z ∈ C , where arg z denotes the principal argument of z , ≤ arg z < π , and where arg 0 = 0 , is asubnorm on C . (b) g ϕ is discontinuous everywhere in C . (c) For every z ∈ C , (3.10) lim k →∞ g ϕ ( z k ) /k = | z | e ϕ (arg z ) . Now, as indicated in Section 3 of [G2], equation (3.10) implies that the utterly discontinuoussubnorm g ϕ satisfies formula (3.2) only for those values of z for which(3.11) ϕ (arg z ) = 0 . As ϕ is unbounded on any subinterval of [0 , π ), we infer that the set of points where thesubnorm g ϕ violates formula (3.2) is dense in C .We further note that since our ϕ is π -periodic, the remarks preceding Theorem 3.2 implythat (3.11) holds whenever arg z is a rational multiple of π . Hence the set of complex pointswhere g ϕ does satisfy formula (3.2) is also dense in C .We close this section by appealing to Theorems 3.1(a,c) and 3.2(a) in [GL3], by which weacquire the following analog of Theorem 3.2 for the quaternions. Theorem 3.3.
Let f be a continuous subnorm on H , and let ϕ be a discontinuous π -periodicsolution of the Cauchy equation (3.6) . Then: (a) The function (3.12) h ϕ ( q ) = f ( q ) e ϕ (arg q ) , q ∈ H , MOSHE GOLDBERG AND THOMAS J. LAFFEY where arg q = arg( α + i p β + γ + δ ) , q = α + iβ + jγ + kδ ∈ H , is a subnorm on H . (b) h ϕ is discontinuous everywhere in H . (c) For every q ∈ H , lim k →∞ h ϕ ( q k ) /k = | q | e ϕ (arg q ) . With this result at hand, we may echo the argument which followed Theorem 3.2 and observethat the set on which h ϕ satisfies formula (3.2) and the one on which h ϕ violates this formulaare both dense in H .4. Stability of subnorms and radial dominance
In accordance with routine nomenclature for norms, we follow [GL2] and say that a subnorm f on a finite-dimensional power-associative algebra A over R or C is stable if there exists apositive constant σ such that(4.1) f ( a k ) ≤ σf ( a ) k for all a ∈ A and k = 1 , , , . . . Furthermore, we say that f is radially dominant on A if f majorizes the radius, i.e., f ( a ) ≥ r ( a ) , a ∈ A . With these notions of stability and radial dominance, we may now cite one of the main resultsin [G1].
Theorem 4.1 ([G1, Theorem 3.4]) . Let f be a continuous subnorm on a finite-dimensionalpower-associative algebra A over R or C , and let A be void of nonzero nilpotents. Then f isstable if and only if f is radially dominant on A . By Theorem 2.2, in the absence of nonzero nilpotents the radius is a continuous subnorm.Hence, Theorem 4.1 instantly implies:
Corollary 4.1.
Let A , a finite-dimensional power-associative algebra over R or C , be void ofnonzero nilpotents. Then the radius is the smallest of all stable continuous subnorms on A . Another immediate consequence of Theorem 4.1 reads:
Corollary 4.2.
Let f be a subnorm on the Cayley–Dickson algebra A n . Then f is stable ifand only if f majorizes the Euclidean norm on A n . We illustrate the last corollary by noticing that the subnorm in (3.3) is stable if and only if | a | p ≥ | a | for all a ∈ A n , which holds, of course, only when 0 < p ≤ N THE RADIUS IN CAYLEY–DICKSON ALGEBRAS 9
With this example and with Theorem 3.1 in the bag, we see that while continuity of asubnorm is enough to force formula (3.1), it is not enough to force stability, not even in theabsence of nonzero nilpotents .To further illustrate Corollary 4.2, let w = ( u, v ) be a fixed vector of two positive entries,and consider the weighted sup norm on C , k z k w, ∞ = max { u | α | , v | β |} , z = α + iβ ∈ C . By the last corollary, k · k w, ∞ is stable on C if and only ifmax { u | α | , v | β |} ≥ p α + β for all α, β ∈ R ;which, by Theorem 3.1 in [GL2], holds precisely when u v ≥ u + v . A similar example is obtained by considering the weighted ℓ -norm,(4.2) k z k w, = u | α | + v | β | , z = α + iβ ∈ C , where again, w = ( u, v ) is a fixed 2-vector of positive entries. By Corollary 4.2, k · k w, is stableon C if and only if u | α | + v | β | ≥ p α + β for all α, β ∈ R ;which, by Theorem 3.3 in [GL2], is valid exactly for u ≥ , v ≥ . To put Theorem 4.1 in perspective, we shall now register a simple result which implies that under certain conditions stability and radial dominance may prevail even when continuity doesnot.
Proposition 4.1 ([G3, Proposition 5.1(b)]) . Let g be a stable subnorm on a finite-dimensionalpower-associative algebra A over R or C . If g is equivalent to a continuous subnorm on A ,then g is radially dominant. Indeed, let A be a finite-dimensional power-associative algebra over R or C with dim A ≥ f be a continuous stable subnorm on A , and let g κ be the discontinuous subnorm in (3.4).By (3.5) and (4.1), g κ ( a k ) ≤ κf ( a k ) ≤ κσf ( a ) k ≤ κσg ( a ) k , a ∈ A , k = 1 , , , . . . So g κ is stable and, by Proposition 4.1, g κ is radially dominant a well.In view of this example, we proceed to show that not all stable subnorms on Cayley–Dicksonalgebras are radially dominant . We begin by citing: Theorem 4.2 ([GL3, Theorem 2.2(b)]) . Let f be a continuous stable subnorm on C , let ϕ bea discontinuous π -periodic solution of equation (3.6) , and let g ϕ be the discontinuous subnormin (3.9) . Then g ϕ is stable on C . Now, letting f , ϕ and g κ be as in Theorem 4.2, and walking in the footsteps of Section 4 in[G2], we fix z ∈ C , z = 0, and select ε , 0 < ε < | z | . Since f is continuous on C , f is boundedon the closed disc D ε = { z : | z − z | ≤ ε } . Moreover, since ϕ is unbounded from below on[0 , π ), the exponential function e ϕ (arg z ) attains arbitrary small positive values in D ε ( z ); sothere exists a point z ε ∈ D ε ( z ) such that g ϕ ( z ε ) = f ( z ε ) e ϕ (arg z ε ) < | z | − ε. We thus obtain(4.3) g ϕ ( z ε ) < | z ε | , and it follows that the ubiquitously discontinuous stable subnorm g ϕ is not radially dominanton C .As a matter of fact, since f ( αz ) = | α | f ( z ) for all z ∈ C and α ∈ R , and since every rationalmultiple of π ia a period of ϕ , it is not hard to conclude from (4.3) that the set of complexnumbers where g ϕ fails to be radially dominant is dense in C .Furthermore, since f vanishes only at a = 0, f is bounded away from zero on D ε ( z ); andsince ϕ is unbounded from above on open subintervals of [0 , π ), we can find a point w ε ∈ D ε ( z )for which g ϕ ( w ε ) = f ( w ε ) e ϕ (arg w ε ) > | z | + ε > | w ε | . Therefore, the set where the subnorm g ϕ majorizes the radius is also dense in C , showing thatno inequality between g ϕ and the radius is possible.To obtain an analog of Theorem 4.2 for the quaternions, we resort to: Theorem 4.3 ([GL3, Theorem 3.2(b)]) . Let f be a continuous stable subnorm on H , let ϕ be adiscontinuous π -periodic solution of the Cauchy equation (3.6) , and let h ϕ be the discontinuoussubnorm in (3.12) . Then h ϕ is stable on H . Rephrasing the argument that followed Theorem 4.2, we can now gather that if f , ϕ and h ϕ are as in Theorem 4.3, then h ϕ neither majorizes nor is majorized by the radius on H .We end this section by recalling, [GL2], that a subnorm f on a finite-dimensional power-associative algebra A over R or C is strongly stable if (4.1) holds with σ = 1 , that is, f ( a k ) ≤ f ( a ) k for all a ∈ A and k = 1 , , , . . . To illustrate this definition, we allude to Theorem 2.2 and observe that if A , a finite-dimensional power-associative algebra over R or C , is void of nilpotent elements, then theradius is a strongly stable subnorm on A . Another example is obtained by referring to Theorem 3.4 in [GL2], which tells us that thenorm k · k w, in (4.2) is strongly stable on C if and only if v ≥ u ≥
1. Since we already knowthat k · k w, is stable only when u ≥ v ≥
1, we deduce that k · k w, is stable but notstrongly stable precisely for u ≥ v ≥ N THE RADIUS IN CAYLEY–DICKSON ALGEBRAS 11
In light of this observation, it would be useful to obtain a characterization of strongly stablesubnorms on the Cayley–Dickson algebras or, better yet, on general finite-dimensional power-associative algebras over R or C . 5. The power equation
Let A be a finite-dimensional power-associative algebra over R or C and let f : A → R be a real-valued function on A . We say that f is a solution of the power equation on A if(5.1) f ( a k ) = f ( a ) k for all a ∈ A and k = 1 , , , . . . If, in addition, f is a subnorm, we say that f is a submodulus on A .Obviously, the only constant solutions of (5.1) are f = 0 and f = 1. When the base fieldof A is the real or the complex numbers, all we have to do in order to produce a non-trivialsolution of (5.1), is to invoke Theorem 2.2 which yields: Corollary 5.1.
Let A be a finite-dimensional power-associative algebra over R or C . Then theradius r is a nonnegative, homogeneous, and continuous solution of the power equation whichvanishes only on nilpotent elements. In general, the radius is not a submodulus, nor even a subnorm, since it vanishes on nonzeronilpotents. Barring such nilpotents, we may obtain the following, more subtle version of Corol-lary 5.1:
Theorem 5.1 ([G1, Theorem 3.3(b)]) . Let A , a finite-dimensional power-associative algebraover R or C , be void of nonzero nilpotent elements. Then the radius is the only continuoussubmodulus on A . Since the Cayley–Dickson algebras are void of nonzero nilpotents, this last theorem allowsus to record:
Corollary 5.2.
The radius is the only continuous submodulus on A n . We remark that the definition of submodulus gives rise to yet another simple result whichholds for finite-dimensional as well as for infinite-dimensional algebras:
Proposition 5.1 ([GGL, Proposition 3]) . If A , a power-associative algebra over R or C , con-tains nonzero nilpotents, then A has no submodulus. With corollary 5.2 in mind, and recalling that the radius on the Cayley–Dickson algebras isthe Euclidean norm, we appeal again to two of the results in [GL3] which tell us that solutionsof the power equation on C and H may fail to have even a shred of continuity: Theorem 5.2 ([GL3, Theorems 2.1(b,c) and 3.1(b,c)]) . Let ϕ be a discontinuous π -periodicsolution of the Cauchy equation (3.6) . Then: (a) The function g ϕ ( z ) = | z | e ϕ (arg z ) , z ∈ C , is a submodulus on C which is discontinuous everywhere. (b) Analogously, the function h ϕ ( z ) = | q | e ϕ (arg q ) , q ∈ H , is a submodulus on H which is discontinuous everywhere. Whether continuous or not, we maintain that all solutions of the power equation on A n , n ≥
1, are nonnegative. To establish this fact, we begin with the following lemma which seemsto be of independent interest.
Lemma 5.1.
Every element in A n , n ≥ , is a square; i.e., for each a ∈ A n there is an element b ∈ A n such that a = b .Proof. Select a = ( α , . . . ,
0) in A n with α = 0. A simple induction on n ≥ α ≥
0, then ( √ α , , . . . ,
0) is a square root of a , and if α <
0, then (0 , √− α , , . . . ,
0) willdo.Next, select a = ( α , . . . , α n ) with α j = 0 for some j = 1. Then | a | > | α | ; so α + | a | > b = 1 p α + 2 | a | ( a + | a | n )is well defined.By Lemma 1.1, a = 2 α a − | a | n . So, b = 12 α + 2 | a | ( a + | a | n ) = 12 α + 2 | a | ( a + 2 | a | a + | a | n )= 12 α + 2 | a | (2 α a − | a | n + 2 | a | a + | a | n ) = a, and the assertion is secured. (cid:3) We can now post:
Theorem 5.3 (Compare [GL1, Proposition 3.1]) . All solutions of the power equation on A n , n ≥ , are nonnegative.Proof. Let f be a solution of the power equation on A n , n ≥
1. Select an element a ∈ A n . Soby Lemma 5.1, a = b for some b ∈ A n . Whence, f ( a ) = f ( b ) = f ( b ) ≥ N THE RADIUS IN CAYLEY–DICKSON ALGEBRAS 13 and the proposition follows. (cid:3)
The function f ( a ) = a, a ∈ R , shows, of course, that Theorem 5.3 is false for A .We comment that with the help of the solutions of the power equation which are already inour possession, it is easy to construct a plethora of others. For instance, if f is a solution, thenso is f κ for any fixed positive constant κ . Also, if f and g are solutions, then so are max { f, g } and min { f, g } , as well as f κ g τ for any pair of fixed positive κ and τ .We conclude this paper by recalling that on R , C , H and O , the radius is multiplicative ; i.e.,(5.2) | ab | = | a || b | , a, b ∈ A n , n = 0 , , , , where for the octonions this identity follows from the Eight Square Theorem, [D]. This sufficesto imply, of course, that the Euclidean norm is a solution of the power equation on the reals,the complex numbers, the quaternions, and the octonions. For n ≥
4, however, formula (5.2)fails since the Cayley–Dickson algebras contain zero divisors. To confirm this known fact, weuse once more, as we did at the end of Section 1, the octonions o = e + e and o = e − e ,and notice that o o = 0. References [A] A. A. Albert,
Power-associtive rings , Trans. Amer. Math. Soc. (1948), 552–593.[B] R. P. Boas Jr., A Primer of Real Functions , Amer. Math. Soc., Providence, Rhode Island, 1960.[D] L. E. Dickson,
On quaternions and their generalization and the history of the Eight Square Theorem ,Ann. of Math. (1918–1919), 155–171.[G1] M. Goldberg, Minimal polynomials and radii of elements in finite-dimensional power-associative alge-bras , Trans. Amer. Math. Soc. (2007), 4055–4072.[G2] M. Goldberg,
Radii and subnorms on finite-dimensional power-associative algebras , Linear MultilinearAlgebra, (2007), 405–415.[G3] M. Goldberg, Stable subnorms on finite-dimensional power-associative algebras , Electron. J. LinearAlgebra (2008), 359–75.[GGL] M. Goldberg, R. Guralnick and W. A. J. Luxemburg, Stable subnorms II , Linear Multilinear Algebra (2003), 209–219.[GL1] M. Goldberg and E. Levy, The power equation , Electron. J. Linear Algebra (2011), 810–821.[GL2] M. Goldberg and W. A. J. Luxemburg, Stable subnorms , Linear Algebra Appl. (2000), 89–101.[GL3] M. Goldberg and W. A. J. Luxemburg,
Discontinuous subnorms , Linear Multilinear Algebra (2001),1–24.[H] G. Hamel, Eine Basis aller Zahlen und die unstetigen L¨osungen der Functionalgleichung: f ( x + y ) = f ( x ) + f ( y ), Math. Ann. (1905), 450–462.[HLP] G. H. Hardy, J. E. Littlewood and G. P´olya, Inequalities , Cambridge Univ. Press, Cambridge, 1934.[HR] H. Hahn and A. Rosenthal,
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Real and Complex Analysis , 3rd edition, McGraw-Hill, New York, 1987.[W1] http://en.wikipedia.org/wiki/Cayley-Dickson construction[W2] http://en.wikipedia.org/wiki/Sedenions
Department of Mathematics, Technion – Israel Institute of Technology, Haifa 32000, Israel
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