that such a direct generalization can not be obtained.

\[ex a trivial Ky Fan fails\] Let $\mathcal{U}$ be the C$^*$-algebra of $2\times 2$ matrices over the complex numbers. Given $\alpha, \beta, \delta \in \mathbb{R}$, with $\alpha \delta=\beta^2 \neq 0$ and $\alpha^2+2\beta^2+\delta^2=1$, we have that the matrix $A=\begin{pmatrix}  \alpha & \beta \\ \beta & \delta   \end{pmatrix}$ is a minimal projection in $\mathcal{U}$ with eigenvalues $\lambda_1(A)=1$ and $\lambda_2(A)=0$. For $u_1=\begin{pmatrix}  1 & 0 \\ 0 & 0  \end{pmatrix}$ and $u_2=\begin{pmatrix}  0 & 0 \\ 0 & 1  \end{pmatrix}$, which are orthogonal minimal tripotents (actually projections) in $\mathcal{U}$, we have that $$\lambda_1(A)+\lambda_2(A)=1 < \sqrt{\alpha^2+\beta^2}+\sqrt{\delta^2+\beta^2} = \|A\|_{u_1}+\|A\|_{u_2}.$$

The latter example shows that a selection among the families of mutually orthogonal minimal tripotents has to be done if we pretend to extend Ky Fan maximum principle to the Jordan setting.

We introduce now families of ($p$-Schatten) seminorms associated to mutually orthogonal minimal tripotents in a JB$^*$-triple. This kind of seminorms (in the case $p=2$) were previously introduced in [@PerRod01] with more generality.

Let $U$ be a JB$^*$-triple. Let $u_1,\ldots,u_n$ be mutually orthogonal minimal tripotents in $\mathcal{U}$. We define the following family of seminorms $$\| x \|_{p,u_1,\ldots,u_n} = (\sum_{i=1}^n \|x\|_{u_i}^p)^{\frac{1}{p}} \hspace{1cm} (1\leq p)$$

Clearly all these 