Radii of Elements in Finite-Dimensional Power-Associative Algebras - - PowerPoint PPT Presentation

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Radii of Elements in Finite-Dimensional Power-Associative Algebras

• r, 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

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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.

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• Example. Let

n n ×

• be the algebra of n

n × matrices over with the usual matrix operations. Fix an idempotent matrix

n n

M

×

∈ , M I ≠ ≠ ≠ ≠ , and consider the set { , }

n n

MAM A

×

= ∈

• Then is a subalgebra of

n n ×

• 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 ( ) 1 p t t = − = − = − = − . On the other hand, the unit element in

n n ×

• is I, and it is easily seen that the minimal

polynomial of M as an element in

n n ×

• is

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( ) 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 =

• r ( )

( ) q t tp t = .

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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 ( ) max{ : , is a root of }. r a p = ∈ λ λ λ λ λ λ λ λ λ λ λ λ 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 =

• r ( )

( ) q t tp t = . Hence, the non-zero roots of p and q are identical; so max{ : , is a root of } max{ : , is a root of } p q ∈ = ∈

• λ

λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ , and we are done.

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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, ( ) max{ : , is an eigenvalue of }. A A = ∈ ρ λ λ λ ρ λ λ λ ρ λ λ λ ρ λ λ λ 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 ∈, ( ) ( ) r a r a = α α α α α α α α . (c) For all a∈ and 1 ,2,3,... k = , ( ) ( )

k k

r a r a = . (d) The radius vanishes only on nilpotent elements of . (e) The radius is a continuous function on .

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A Non-Associative Example: The Cayley–Dickson Algebras

The Cayley–Dickson algebras constitute a series of algebras,

1 2

, , ,

• … over the reals,

where =

• dim

2n

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

• f quaternions in GPS technology and in 3D computer graphics, and the employment of
• ctonions and higher Cayley–Dickson algebras in modern physics (e.g., in Quantum Field

Theory, and in Born-Infeld models).

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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: We begin by setting

0 =

• , and by defining *

a , the conjugate of a real number a, to equal

• a. Then, assuming that

−

• 1

n

, ≥ 1 n , has been determined, we define n to be the set of all

• rdered pairs

1

{( , ) : , }

n n

a b a b

−

= ∈

• ,

such that addition and scalar multiplication are taken componentwise on the Cartesian product

− −

×

• 1

1 n n

; conjugation is given by ( , )* ( *, ) 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 = …

1 2

( , , )

n

a a a , ∈

j

a ; and it follows that the conjugate of a is given by

1 2 2

* ( , , , )

n

a a a a = − − … , and the unit element in n is (1 ,0, ,0)

n =

1 … .

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We point out that by construction,

−

• 1

n

is a subalgebra of n. It follows that since

3

, the algebra of the octonions, is no longer associative, all the Cayley–Dickson algebras for 3 n ≥ ≥ ≥ ≥ 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

1 2

( , , )

n

n

a a a = ∈ … is the Euclidean norm of a, i.e.,

2 2 1 2

( ) .

n

r a a a = + + ⋯

• 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

• nly at

a = , the proof is complete. ■

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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 ∈

a and ∈ α α α α , ( ) 0, 0, ( ) ( ). f a a f a f a > ≠ = α α α α α α α α We recall that a real-valued function N is a norm on if for all pairs ∈ , a b and ∈ α α α α , ( ) 0, 0, ( ) ( ), ( ) ( ) ( ). N a a N a N a N a b N a N b α α α α α α α α > ≠ = + ≤ + 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.

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Application 1. Extending the Gelfand Formula to FDPA Algebras

Gelfand's Formula. Let be an (associatve) Banach Algebra over with norm N. Then

1/

lim ( ) ( ), .

k k k

N a a a ρ ρ ρ ρ

→∞

= ∈ where ( ) a ρ ρ ρ ρ is the spcteral radius of a. This well-known formula can be extended to subnorms on FDPA algebras as follows: TheoremGF [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 ,

1/

lim ( ) ( )

k k k

f a r a

→∞

= .

• Example. Let f be a continuous subnorm on the Cayley–Dickson algebra

n

. Then

1/ 2 2 1 1 2 2

lim ( ) ( ) , ( , , )

n n

k k n k

f a r a a a a a a

→∞

= ≡ + + = ∈ ⋯ … .

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Application 2: Stability of Subnorms

• Definition. Let be a FDPA algebra over or . Then a subnorm f on is stable if

there exists a positive constant σ σ σ σ such that for all a∈ and 1 , 2,3,... k = , ( ) ( )

k k

f a f a ≤ σ σ σ σ . The notion of stability plays an important role in several areas of mathematics. In particular, it provides the main tool for studying the behavior of finite-difference approximations to time-dependent partial differential equations. With the above definition we may now quote: TheoremST [G, TAMS]. If f is a continuous subnorm on a FDPA algebra over or , and if is void of nonzero nilpotent elements, then f is stable if and only if f r ≥ on . If f r ≥ on , we shall often say that f majorizes the radius on . By the Main Theorem, if the algebra is void of nonzero nilpotents, than it is easily seen that the radius is a continuous subnorm on . Hence, TheoremST implies:

• Corollary. Let , a FDPA algebra over or , be void of nonzero nilpotent elements.

Then the radius is the smallest stable continuous subnorm on .

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We recall, for example, that the Cayley–Dickson algebras are void of nonzero nilpotent

• elements. Hence, TheoremST and its corollary can be rephrased in this case as follows:
• Theorem. Let f be a continuous subnorm on

n

. Then: (a) f is stable if and only if f majorizes the Euclidean norm on

n

. (b) The Euclidean norm is the smallest continuous stable subnorm on

n

. Let us illustrate this thorem by observing that for each fixed p, 0 p < ≤ ∞ < ≤ ∞ < ≤ ∞ < ≤ ∞, the function = + = ∈ ⋯ …

1/ 1 1 2 2

( ) , ( , , )

n n

p p p n p

a a a a a a , is a continuous subnorm on

n

(a norm iff ≤ ≤ ∞ 1 p ). So | |p ⋅ is stable on

n

if and only if

2 on n p

a a ≥ , which holds precisely for 0 2 p < ≤ . On the other hand, our TheoremGF tells us that all continuous subnorms on a FDPA algebra

• ver or satisfy the Gelfand Formula.

So confronting the TheoremGF with the above example for 2 p > > > > (where | |p ⋅ is an unstable norm), we see that while continuity of a subnorm f is enough to force the Gelfand Formula, it is not enough to force stability, not even when f is a norm and the underlying algebra is void of nonzero nilpotents.