Hermitian Euclidean Jordan Algebra

Bruce C. Dieffenbach ()
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Bruce C. Dieffenbach: Independent author

Chapter 30 in Conjugate Duality in Economic Analysis, 2026, pp 227-232 from Springer

Abstract: Abstract The complexification of a Euclidean space X is its natural extension to a complex vector space having a complex-valued inner product. The complexification expresses a vector as the sum of its real and complex parts, $$\boldsymbol {x}=\boldsymbol {y}+ \boldsymbol {z}\mathrm {i},$$ in which y and z belong to X. The complex inner product in the complexification is $$\begin {aligned}\left \langle \boldsymbol {x}^{\ast },\boldsymbol {x}\right \rangle & =\left \langle \boldsymbol {y}^{\ast }+ \boldsymbol {z}^{\ast }\mathrm {i},\boldsymbol {y}+ \boldsymbol {z}\mathrm {i} \right \rangle :=\left ( \boldsymbol {y}^{\ast \top }-\boldsymbol {z}^{\ast \top }\mathrm {i} \right ) \left ( \boldsymbol {y}+ \boldsymbol {z}\mathrm {i}\right ) \!. \end {aligned}$$ The concept of the adjoint carries over to the complexification: the square linear transformation A+Bi is self-adjoint if and only if A is self-adjoint and B is skew. A simple Euclidean Jordan algebra, the Hermitian Euclidean Jordan algebra HX is the real vector space of self-adjoint linear transformations on the complexification. The Jordan product is $${\textit {{\sf { {X}}}}}^{\ast }\bullet {\textit {{\sf { {X}}}}}=\frac {1}{2}\left ( {\textit {{\sf { {X}}}}}^{\ast }{\textit {{\sf { {X}}}}}+ {\textit {{\sf { {X} {X}}}}}^{\ast }\right ) \!.$$ Using the spectral decomposition, one can verify that HX satisfies the axioms for a Euclidean Jordan algebra.

Date: 2026
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DOI: 10.1007/978-3-032-21396-9_30

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