Spectral theorem for Euclidean Jordan algebras

Statement

Let $J$ be a Euclidean Jordan algebra of rank $r$ . For every $x\in J$ , there is a Jordan frame $(c_1,\ldots,c_r)$ and real numbers $\lambda_1,\ldots,\lambda_r$ such that

x=\lambda_1c_1+\cdots+\lambda_rc_r.

The multiset of eigenvalues $\{\lambda_1,\ldots,\lambda_r\}$ , including multiplicities, is uniquely determined by $x$ . The diagonalizing frame need not be unique when an eigenvalue is repeated.

Spectral idempotents

If $\mu_1,\ldots,\mu_s$ are the distinct eigenvalues, collecting equal terms gives

x=\mu_1e_1+\cdots+\mu_se_s,

where the $e_j$ are nonzero pairwise orthogonal idempotents summing to the unit. These coarser spectral idempotents are uniquely determined by $x$ ; each is the sum of the primitive frame idempotents carrying the same eigenvalue.

Functional calculus

For a real function $f$ defined on the spectrum of $x$ , set

f(x)=\sum_{i=1}^r f(\lambda_i)c_i.

This is independent of the choice of diagonalizing frame. In particular, one defines

\operatorname{tr}_J(x)=\sum_i\lambda_i, \qquad \det_J(x)=\prod_i\lambda_i.

The element $x$ is invertible exactly when no $\lambda_i$ vanishes, and then $x^{-1}=\sum_i\lambda_i^{-1}c_i$ . It lies in the cone of squares exactly when every eigenvalue is nonnegative.

Matrix model

For $J=H_n(\mathbb R)$ with product $a\circ b=(ab+ba)/2$ , this is the ordinary spectral theorem for real symmetric matrices. The $c_i$ are rank-one orthogonal projections, and the Jordan trace and determinant agree with the usual matrix invariants. The Jordan theorem extends the same mechanism to spin factors, quaternionic Hermitian matrices, and the exceptional Albert algebra.

References

Jacques Faraut and Adam Korányi, Analysis on Symmetric Cones, Oxford University Press, 1994, Chapter III, Theorem 1.2. Publisher record.
Kevin McCrimmon, A Taste of Jordan Algebras, Springer, 2004, Chapter 13. Publisher record.