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Motivated by the Araki-Lieb-Thirring inequality tr( A1/2BA1/2 )rp ≤ tr( Ar /2BrAr /2 )pfor  p  ≥ 0,  r  ≥ 1 (tr( A1/2BA1/2 )rp ≥ tr( Ar /2BrAr /2 )pfor  p  ≥ 0, 0 ≤  r  ≤ 1) for Hermitian positive semidefinite matrices and the Golden-Thompson inequality tr(exp ( A  +  B )) ≤ tr(exp ( A )exp ( B )) for Hermitian matrices, in this paper, we extend these inequalities to the setting of Euclidean Jordan algebras in the formfor  p  ≥ 0,  r  ≥ 1 (for  p  ≥ 0, 0 ≤  r  ≤ 1) for  a ,  b  ≥ 0 and tr(exp ( a  +  b )) ≤ tr(exp ( a )°exp ( b )) for all  a  and  b , where  Pxand  x °  y  denote, respectively, the quadratic representation and Jordan product of  x  and  y  in Euclidean Jordan algebras.

The Araki-Lieb-Thirring inequality and the Golden-Thompson inequality in Euclidean Jordan algebras