Tripotent in a Jordan triple system

Definition

Let $V$ be a Hermitian Jordan triple system with triple product $\{x,y,z\}$ . A tripotent is an element $e\in V$ satisfying

\{e,e,e\}=e.

It is the ternary analogue of a Jordan idempotent, but its definition requires neither a binary product nor a distinguished unit.

Operator example

For the normalized triple product on operators,

\{x,y,z\}=\frac12(xy^*z+zy^*x),

the tripotent equation becomes $ee^*e=e$ . Thus the tripotents are exactly the partial isometries. Projections are tripotents, but a tripotent need not be self-adjoint and therefore need not be a projection.

Peirce decomposition

Set $D(e,e)z=\{e,e,z\}$ . For the normalization above, $D(e,e)$ has possible eigenvalues $1,\tfrac12,0$ , giving the Peirce decomposition

V=V_2(e)\oplus V_1(e)\oplus V_0(e), \qquad V_j(e)=\{z:D(e,e)z=\tfrac j2z\}.

The indices $2,1,0$ are traditional and record twice the eigenvalue. The triple product obeys Peirce arithmetic:

\{V_i(e),V_j(e),V_k(e)\} \subseteq V_{i-j+k}(e),

with a summand interpreted as zero when its index is outside $\{0,1,2\}$ . The top space $V_2(e)$ is a unital complex Jordan algebra under

x\circ_e z=\{x,e,z\},

with unit $e$ .

Orthogonality and rank

Two tripotents $e,f$ are orthogonal when $D(e,e)f=0$ ; in positive Hermitian Jordan triples this relation is symmetric. Sums of pairwise orthogonal tripotents are again tripotents. A nonzero tripotent is minimal when $V_2(e)=\mathbb Ce$ , and it is complete when $V_0(e)=0$ . Maximal families of mutually orthogonal minimal tripotents play the role of Jordan frames and define the rank of a finite-dimensional positive triple system.

Normalization warning

Some sources use the doubled product $xy^*z+zy^*x$ . Under that convention the tripotent equation is $\{e,e,e\}=2e$ , and the Peirce eigenvalues are $2,1,0$ . Formulas must not be moved between the two conventions without this rescaling.

References

Ottmar Loos, Bounded Symmetric Domains and Jordan Pairs, University of California, Irvine, 1977, Parts I–II. Catalog record.
Harald Upmeier, Symmetric Banach Manifolds and Jordan C-Algebras*, North-Holland, 1985, Chapters 2–4. Publisher record.