Radii of Elements in Finite-Dimensional Power-Associative Algebras or, more intuitively, Generalizing the Spectral Radius without Dealing with Spectra Moshe Goldberg Department of Mathematics Technion – Israel Institute of Technology Advances in Applied Mathematics: in Memoriam Saul Abarbanel Tel Aviv University 18–20 December, 2018 1
The Minimal Polynomial Revisited Let � denote a finite-dimensional algebra over an arbitrary field � . We do not require commutativity, associativity, or even a unit element. Throughout this talk, we shall assume, however, that � is power-associative . This means that while � is not necessarily associative, the subalgebra of � generated by any one element is associative, or equivalently, that powers of each element in � are uniquely defined. Definition. A minimal polynomial of an element a in a power-associative algebra over � is a monic polynomial of lowest positive degree with coefficients in � that annihilates a . With this familiar definition, we can state the following non-surprising result: Theorem. Let � be a finite-dimensional power-associative algebra over a field � . Then every element of � possesses a unique minimal polynomial. 2
× � n n × matrices over � with the usual matrix operations. Example. Let be the algebra of n n × ∈ � n n ≠ , and consider the set ≠ ≠ ≠ Fix an idempotent matrix M , M I × = ∈ � n n � { MAM , A } × � n n Then � is a subalgebra of which contains the matrix M as an element. In fact, M is the unit element in � , so the minimal polynomial of M in � must be = = = = − . − − − p t ( ) t 1 × � n n On the other hand, the unit element in is I , and it is easily seen that the minimal × � n n polynomial of M as an element in is = 2 − . q t ( ) t t It follows that the minimal polynomial of an element may depend not only on the element, but also on the underlying algebra . The above example is a special case of a more general phenomenon: Theorem [G, Trans. AMS, 2007]. Let � and � be finite-dimensional power-associative algebras over a field � , such that � is a subalgebra of � . Let a be an element of � (and therefore of � ), and let p and q be the minimal polynomials of a in the algebras � and in = = � , respectively. Then either p q or ( ) q t tp t ( ) . 3
The Radius of an Element in a FDPA Algebra From now on, we shall restrict attention to the case where the base field � of our algebra is either � or � . Further, we shall abbreviate the expression finite-dimensional power-associative by FDPA. Main Definition [G, Trans. AMS]. Let � be a FDPA algebra over � or � . Let a be an element of � , and let p be the minimal polynomial of a in � . Then, the radius of a in � is defined as = λ λ λ λ λ λ λ λ ∈ � λ λ λ λ r a ( ) max{ : , is a root of }. p Unlike the minimal polynomial of an element a in � (which may depend on � ), the radius r a is independent of our algebra in the following sense: ( ) Proposition. Let � and � be FDPA algebras over � or � , such that � is a subalgebra of � . Then the radii of a in the algebras � and � coincide. Proof. Let p and q be the minimal polynomials of a in the algebras � and � , respectively. = = By the last theorem, either p q or ( ) q t tp t ( ) . Hence, the non-zero roots of p and q are identical; so λ λ λ λ λ λ λ λ ∈ � λ λ λ λ = λ λ λ λ λ λ λ λ ∈ � λ λ λ λ max{ : , is a root of } p max{ : , is a root of } q , and we are done. � 4
The radius has been computed for elements in several well-known FDPA algebras. For example, if � is an arbitrary matrix algebra over � or � with the usual matrix operations, then the radius of a matrix A ∈ � is the classical spectral radius, ∈ � ρ ρ ρ ρ = λ λ λ λ λ λ λ λ λ λ λ λ ( ) A max{ : , is an eigenvalue of A }. The following theorem, which is the heart of the matter, tells us that our newly defined radius retains all the basic properties of the classical spectral radius not only for matrix algebras with the usual matrix operations, but for arbitrary FDPA algebras as well: Main Theorem [G, Trans. AMS]. Let � be a FDPA algebra over a field � , either � or � . Then : (a) The radius is nonnegative. (b) The radius is homogeneous, i.e., for all a ∈ � and α α α α ∈ � , α α α α = α α α α r ( a ) r a ( ) . (c) For all a ∈ � and k = 1 ,2,3,... , k = k r a ( ) r a ( ) . (d) The radius vanishes only on nilpotent elements of � . (e) The radius is a continuous function on � . 5
A Non-Associative Example: The Cayley–Dickson Algebras � � … over the reals, � The Cayley–Dickson algebras constitute a series of algebras, , , , 0 1 2 = 2 n � where dim . n The first five Cayley–Dickson algebras are the reals � , the complex numbers � , the quaternions � , the octonions � , and the sedenions � , with dimensions 1, 2, 4, 8, and 16, respectively. While � and � are both commutative and associative, � is no longer commutative, and � and � are not even associative. Despite the deteriorating associativity properties of the low-dimensional Cayley–Dickson algebras, we do have: Theorem [G & Laffey, Proc. AMS, 2015]. All Cayley–Dickson algebras are power- associative . We note that in recent years, the Cayley–Dickson algebras have gained renewed interest via several important applications in applied mathematics and physics. For example the use of quaternions in GPS technology and in 3D computer graphics, and the employment of octonions and higher Cayley–Dickson algebras in modern physics (e.g., in Quantum Field Theory, and in Born-Infeld models). 6
How to Obtain the Cayley–Dickson Algebras The Cayley–Dickson algebras can be obtained inductively from each other by the following Cayley–Dickson doubling process: � , and by defining * 0 = � We begin by setting a , the conjugate of a real number a , to equal ≥ 1 � , has been determined, we define � n to be the set of all a . Then, assuming that , n − n 1 ordered pairs = ∈ � � {( , ) : a b a b , } , − n n 1 such that addition and scalar multiplication are taken componentwise on the Cartesian × � � product ; conjugation is given by − − n 1 n 1 = = = = − − − − ( , )* a b ( *, a b ) ; and multiplication is given by = − + ( , )( , ) a b c d ( ac d b da * , bc *) . With this definition, each element ∈ � n = ∈ � … a is of the form a ( a , , a ) , a ; and it follows n 1 j 2 that the conjugate of a is given by = − − … a * ( a , a , , a ) , n 1 2 2 and the unit element in � n is n = … 1 (1 ,0, ,0) . 7
� is a subalgebra of � n . It follows that since � , the We point out that by construction, − n 1 3 n ≥ ≥ ≥ ≥ algebra of the octonions, is no longer associative, all the Cayley–Dickson algebras for 3 are not associative . However, as mention earlier, all the Cayley–Dickson algebras are power-Associative. = ∈ � … Theorem [G & Laffey, Proc. AMS, 2015]. The radius of an element a ( a , , a ) is n 1 n 2 the Euclidean norm of a , i.e., = 2 + + 2 ⋯ r a ( ) a a . n 1 2 Corollary . The Cayley–Dickson algebras are void of nonzero nilpotent elements. Proof. By the Main Theorem, the radius vanishes only on nilpotent elements. Since the radius on the Cayley–Dickson algebras happens to be a norm, and since a norm vanishes a = , the proof is complete. ■ only at 0 8
Subnorms We shall now turn to two applications of our radius. Both application are associated with the concept of subnorm which we define next. Definition. Let � be an algebra over a field � , either � or � . Then a real-valued function → � � f : is a subnorm if for all ∈ ∈ � α α α α � and a , > ≠ f a ( ) 0, a 0, α α α α = α α α α f ( a ) f a ( ). ∈ � ∈ α α α α We recall that a real-valued function N is a norm on � if for all pairs � and a b , , > ≠ N a ( ) 0, a 0, α α α α = α α α α N ( a ) N a ( ), + ≤ + N a ( b ) N a ( ) N b ( ). Hence, a norm is a subadditive subnorm . In passing, we recall that in a finite-dimensional setting, a norm is always a continuous ≥ � function . In contrast, a subnorm may fail to be continuous when dim 2 . In fact, there exist pathological subnorms on nontrivial algebras which are discontinuous everywhere . 9
Application 1. Extending the Gelfand Formula to FDPA Algebras Gelfand's Formula. Let � be an (associatve) Banach Algebra over � with norm N . Then k 1/ k = ρ ρ ∈ �� ρ ρ lim N a ( ) ( ), a a . →∞ k ρ ρ ρ ρ where ( ) a is the spcteral radius of a . This well-known formula can be extended to subnorms on FDPA algebras as follows: Theorem GF [G, Linear & Multilinear Algebra] . Let f be a continuous subnorm on a FDPA algebra � over � or � , and let r denote the radius on � . Then for every ∈ � , a k 1/ k = lim ( f a ) r a ( ) . →∞ k � . Then Example. Let f be a continuous subnorm on the Cayley–Dickson algebra n k 1/ k = ≡ 2 + + 2 = ∈ ⋯ … � . lim ( f a ) r a ( ) a a , a ( a , , a ) 1 n 1 n n 2 2 →∞ k 10
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