Separation theorems

An important tool in the theory are the following separation theorems.

Let $\langle V,W\rangle$ be a (non degenerate) duality of complex vector spaces. Thus $V$ and $W$ have weak topologies, and the matrix levels have the corresponding product topology. For $v=[v_{i,j}]\in M_n(V)$ and $w=[w_{k,l}]\in M_m(W)$, $\langle v,w\rangle$ is defined by the joint amplifications of the duality:

$$\langle v,w\rangle=[\langle v_{i,j},w_{k,l}\rangle]_{(i,k),(j,l)}$$   .

Note that the matrices, ordered by the cone of the positive semidefinite matrices, are not totally ordered; $\not\leqslant$ does not imply $\geqslant$.

Theorem:⁶²Let $\langle V,W\rangle$ be a duality of complex vector spaces, $K$ a closed set of matrices over $V$ and $v_0\in M_n(V)\setminus K_n$ for some $n$.

a)

[WW99, Thm. 1.6] If $K$ is matrix convex, then there are $w\in M_n(W)$ and $\alpha\in (M_n)_{\mathrm{sa}}$ such that for all $m\in{\mathbb{N}}$ and $v\in K_m$

$$\mathrm{Re}\langle v,w\rangle\leqslant\mathrm{1\!\!\!\:l}_m\otimes\alpha$$   , but $$\mathrm{Re}\langle v_0,w\rangle\not\leqslant\mathrm{1\!\!\!\:l}_n\otimes\alpha$$   .

b)

[EW97b, Thm. 5.4] If $K$ is matrix convex and $0\in K_1$, then there is $w\in M_n(W)$ such that for all $m\in{\mathbb{N}}$ and $v\in K_m$

$$\mathrm{Re}\langle v,w\rangle\leqslant\mathrm{1\!\!\!\:l}_{nm}$$   , but $$\mathrm{Re}\langle v_0,w\rangle\not\leqslant\mathrm{1\!\!\!\:l}_{n^2}$$   .

c)

[Bet97, p. 57] If $K$ is a matrix cone, then there is $w\in M_n(W)$ sucht that for all $m\in{\mathbb{N}}$ and $v\in K_m$

$$\mathrm{Re}\langle v,w\rangle\leqslant 0$$   , but $$\mathrm{Re}\langle v_0,w\rangle\not\leqslant 0$$   .

d)

[EW97a, Thm. 4.1] If $K$ is absolutely matrix convex, then tere is $w\in M_n(W)$ such that for all $m\in{\mathbb{N}}$ and $v\in K_m$

$$\Vert\langle v,w\rangle\Vert\leqslant 1$$   , but $$\Vert\langle v_0,w\rangle\Vert >1$$   .

One can prove Ruan's theorem using the separation theorem for absolutely matrix convex sets, applied to the unit ball of a matricially normed space .

If $V$ is a complex involutive vector space, one can find selfadjoint separating functionals:

Theorem: ⁶³Let $\langle V,W\rangle$ be a duality of complex involutive vector spaces, $K$ a closed set of selfadjoint matrices over $V$ and $v_0\in M_n(V)\setminus K_n$ for some $n$.

b)

If $K$ is matrix convex and $0\in K_1$, then there is a $w\in M_n(W)_{\mathrm{sa}}$ such that for all $m\in{\mathbb{N}}$ and $v\in K_m$

$$\langle v,w\rangle\leqslant\mathrm{1\!\!\!\:l}_{nm}$$   , but $$\mathrm{Re}\langle v_0,w\rangle\not\leqslant\mathrm{1\!\!\!\:l}_{n^2}$$   .

Footnotes

...Theorem:⁶²

From this theorem one can easily get the following sharper version of the parts a), b) and d):

a)

If $K$ is matrix convex, then there are $w\in M_n(W)$, $\alpha\in (M_n)_{\mathrm{sa}}$ and $\varepsilon>0$ such that for all $m\in{\mathbb{N}}$ and $v\in K_m$

$$\mathrm{Re}\langle v,w\rangle\leqslant\mathrm{1\!\!\!\:l}_m\otimes(\alpha-\varepsilon\mathrm{1\!\!\!\:l}_n)$$   , but $$\mathrm{Re}\langle v_0,w\rangle\not\leqslant\mathrm{1\!\!\!\:l}_n\otimes\alpha$$   .

b)

If $K$ is matrix convex and $0\in K_1$, then there are $w\in M_n(W)$ and $\varepsilon>0$ such that for all $m\in{\mathbb{N}}$ and $v\in K_m$

$$\mathrm{Re}\langle v,w\rangle\leqslant(1-\varepsilon)\mathrm{1\!\!\!\:l}_{nm}$$   , but $$\mathrm{Re}\langle v_0,w\rangle\not\leqslant\mathrm{1\!\!\!\:l}_{n^2}$$   .

d)

If $K$ is absolutely matrix convex, then there are $w\in M_n(W)$ and $\varepsilon>0$ such that for all $m\in{\mathbb{N}}$ and $v\in K_m$

$$\Vert\langle v,w\rangle\Vert\leqslant 1-\varepsilon$$   , but $$\Vert\langle v_0,w\rangle\Vert >1$$   .

... ⁶³

This theorem can be obtained from the above separation theorem, part a) and b). Note that for selfadjoint $v$, the mapping $w\mapsto\langle v,w\rangle$ is selfadjoint.