Intersection and sum

Let $X$ and $Y$ be operator spaces such that $M_1(X)$ and $M_1(Y)$ are embedded in a Hausdorff topological vector space. $M_n(X \cap Y)$ is given a norm via $M_n(X \cap Y) := M_n(X) \cap M_n(Y)$. So we have:

$${ \left\Norm [x_{ij}] \right\Norm }_{M_n(X \cap Y)} = \max \left... ...\right\Norm }_{M_n(X)} , { \left\Norm [x_{ij}] \right\Norm }_{M_n(Y)} \right\}$$   .$$$$

The operator space $X \cap Y$ is called the intersection of $X$ and $Y$.
For operator spaces⁷⁵ $X$ and $Y$, by embedding $X \oplus Y$ in $(X^* \oplus_\infty Y^*)^*$ we obtain an operator space structure $X \oplus_1 Y$. We write $\diamondsuit := \left\{(x,-x)\right\} \subset X \oplus_1 Y$. The quotient operator space $\left(X \oplus_1 Y\right) / \diamondsuit$ is called the sum of $X$ and $Y$ and is denoted by $X+Y$. We have

$${ \left\Norm [x_{ij}] \right\Norm }_{M_n(X+Y)} = \inf_{[x_{ij}] ... ...{ \left\Norm [\left(x_{ij_X},x_{ij_Y}\right)] \right\Norm }_{M_n(X \oplus_1 Y)}$$   .

Footnotes

... spaces⁷⁵

Let $E$, $F$ be Banach spaces. Then we have their 1-direct sum $E \oplus_1 F$ with the norm

$${ \left\Norm (x_E,x_F) \right\Norm }_{} = { \left\Norm x_E \right\Norm }_{} + { \left\Norm x_F \right\Norm }_{}$$

and their sum $E + F$ with the quotient norm

$${ \left\Norm x \right\Norm }_{E+F} = \inf_{x = x_E + x_F} \left( ... ...eft\Norm x_E \right\Norm }_{E} + { \left\Norm x_F \right\Norm }_{F} \right) .$$