By Karakhanyan M.I., Khor'kova T.A.

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A hermitian Jordan triple is a complex vector space J with triple product {x, y, z} = Lx,y (z) which is symmetric and C-linear in the outer variables x, z and conjugate-linear in the middle variable y, satisfying the 5-linear axiom {x, y, {z, w, v}} = {{x, y, z}, w, v} − {z, {y, x, w}, v} + {z, w, {x, y, v}} for a Jordan triple system. A ﬁnite-dimensional hermitian Jordan triple is positive if the trace form tr(Lx,y ) is a positive deﬁnite Hermitian scalar product. 24 Colloquial Survey Every nonzero element has a unique spectral decomposition x = λk ek for nonzero orthogonal tripotents e3k = ek and distinct positive singular values 0 < λ1 < · · · < λr ∈ R (called the singular spectrum of x); the spectral norm is the maximal size x := maxi λi = λr of the singular values.

A. Albert showed that it is indeed an exceptional Jordan algebra of dimension 27 (we now call such 27-dimensional exceptional algebras Albert algebras, and denote H3 (K) by A). The Physical End of Exceptional Algebras These results were deeply disappointing to physicists: there was only one exceptional algebra in this list, for n = 3; for n > 3 the algebra Hn (K) is not a Jordan algebra at all, and for n = 2 it is isomorphic to JSpin9 , and for n = 1 it is just R+ . This lone exceptional algebra H3 (K) was too tiny to provide a home for quantum mechanics, and too isolated to give a clue as to the possible existence of inﬁnite-dimensional exceptional algebras.

We now indicate what Jordan algebras were doing in such a strange landscape. Inverses and Isotopy For the ﬁrst application we need to say a few words about inverses in unital Jordan algebras. An element x is invertible in J iﬀ the operator Ux is an invertible operator on J, in which case its inverse is the operator Ux−1 = Ux−1 of the inverse element x−1 := Ux−1 (x). In special Jordan algebras J ⊆ A+ an element x is invertible iﬀ it is invertible in A and its associative inverse x−1 lies in J, in which case x−1 is also the Jordan inverse.

### A Characteristic Property of the Algebra C(Q)B by Karakhanyan M.I., Khor'kova T.A.

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