On projections of free semialgebraic sets

Theorem 2.1

The question whether a formula (without free variables) holds for at least one size of matrices is undecidable.

Proof

It is shown in [2], basically going back already to [10], that there exists a recursively enumerable sequence of finitely presented groups (G_i)_i∈ℕ such that the set

is not recursive (=decidable). The statement that a finitely presented group has a nontrivial unitary representation can easily be expressed as a formula in our sense. In fact, if the group is generated by elements g₁, . . . , g_m subject to relations 1 = r₁(x₁, . . . , x_m) = ⋅⋅ ⋅ = r_s(x₁, . . . , x_m), the first naive formula would simply be

But since we quantify only over Hermitian matrices, we have to replace the conditions “∃U_j unitary" by

and to replace each U_j in the rest of the formula by A_j + iB_j. Nontriviality of the representation is achieved by asking that A_j + iB_j ≠ 1 for all least one index j.

So we obtain a recursively enumerable sequence of formulas (φ_i)_i∈ℕ for which there is no algorithm that decides for each φ_i, whether it holds for at least one size of matrices.

Definition 3.1

Let S be a subspace of Md(C)h. We call S definite, if there exists a definite element in S. We call S indefinite, if every element in S\{0} is indefinite.

Theorem 3.2

Let p∈Md(C〈x1,…,xn〉)h and let B1,…,Bm∈Md(C)h such that the vector space span_ℝ{B₁, . . . , B_m} is an indefinite subspace of 𝕄_d(ℂ)_h. Then for any Hilbert space ℋ and for any choice T₁, . . . , T_n ∈ 𝔹(ℋ)_h, the following are equivalent:

1. There exist S ₁, . . . , S_m ∈ 𝔹(ℋ)_h such that p(T₁, . . . , T_n) + B₁ ⊗ S₁ + ⋅⋅ ⋅ + B_m ⊗ S_m ⩾ 0.

2. For all r ∈ ℕ with r ≤ dim(ℋ) and for all W_j ∈ 𝕄_d,r(ℂ) with∑jWj∗BiWj=0for all i = 1, . . . , m, we have

Before we prove the theorem, let us comment on some of the conditions. Firstly, we assume that S := span_ℝ{B₁, . . . , B_m} is a definite subspace. Then clearly Condition (i) is fulfilled for any choice of T_i, in which case the whole question is not very interesting. But of course S could be neither definite nor indefinite (it could for example contain a semidefinite but no definite element). We will deal with this case below. Secondly, if ℋ is finite-dimensional, we can clearly restrict to the single case r = dim(ℋ) in (ii). If ℋ is infinite-dimensional, we have to use all r ∈ ℕ. Thirdly, the length of the appearing sums can be bounded in terms of r. This is a straightforward application of Carathéodory’s Theorem for convex cones.

Finally, note that the description from(ii) consists of inequalities only, but infinitely many. However, these inequalities are still parametrised well in terms of the input data. We call a set of the form appearing in (ii) a semi-algebraically parametrised free semi-algebraic set. We do not yet dare to make a precise definition of this term, but we sincerely hope that the above theorem and Theorem 3.5 will lead to a class of sets which are closed under certain projections. This then might guide further research on how to set up a model theoretic framework which satisfies a suitable form of quantifier elimination in form of a projection theorem.

Proof of Theorem 3.2

By conjugating all coefficients with W_j and summing up, it is obvious that (i) implies (ii). To prove the converse, first note that both Conditions (i) and (ii) are equivalent to the corresponding conditions where p is replaced by p_A and B_i is replaced by A^∗B_iA for an invertible matrix A ∈ 𝕄_d(ℂ). We can choose A ∈ 𝕄_d(ℂ)_h such that A² ∈ S^⊥ and A is invertible (we use the easy observation that since S is indefinite, S^⊥ is definite).

Thus, we can assume that tr(B_i) = 0 for i = 1, . . . , m. Further note that both conditions are equivalent to the conditions where a coefficient P_ω of p is replaced by any element in P_ω + S. Thus, we can assume that P_ω ∈ S^⊥ for every word ω. Finally note that we can assume that B₁, . . . , B_m are orthonormal.

Now consider

Observe that idd∈V follows from tr(B_i) = 0, and that V is closed under ∗ since each B_i is Hermitian. Thus V is an operator system in the C^∗-algebra 𝕄_d(ℂ); see for example [6] for details on operator systems. Now consider the following ∗-linear map

We claim that φ is r-positive, meaning that φ ⊗ id_{𝕄r (ℂ)} maps positive matrices to positive operators. So let (W_ij)_i,j ∈ 𝕄_r(V) be positive semidefinite. ∗So there are vectors w_1k , . . . , w_rk ∈ ℂ^d such that Wij=∑kwikwjk∗for all i, j. We now compute

where W_k is the matrix having w¯1k,…,w¯rkas its columns. From assumption (ii) we see that this operator is positive semidefinite, provided that∑kWk∗BiWk=0for all i. But this follows easily from the fact that W_jk ∈ V, i.e. 〈Bit,Wjk〉=0 for all i, j, k.

So as an r-positive map from an operator space to 𝔹(ℋ) (for all r ≤ dim ℋ), φ admits a completely positive extension ψ: 𝕄_d(ℂ) → 𝔹(ℋ), by Arveson’s Extension Theorem (see [6], Theorem 6.1 and Theorem 7.5). For any U ∈ 𝕄_d(ℂ) we have

For the second equality we have used that U−∑i〈U,Bit〉Bitlies in V since the Bitare orthonormal, and for the third that 〈P_ω, B_i〉 = 0 for all coefficients P_ω of p and all i.

Let E _kj ∈ 𝕄_d(ℂ) be the matrix with 1 in the (k, j)-entry, and zeros elsewhere. Then the Choi-matrix E = (E_jk)_j,k ∈ 𝕄_d(𝕄_d(ℂ)) is positive semidefinite, and thus

This implies (i).

Proposition 3.3

Let p ∈ 𝕄_d(ℂ〈x₁, . . . , x_n〉)_h and let B₁, . . . , B_m ∈ 𝕄_d(ℂ)_h such that span{B₁, . . . , B_m} is an indefinite subspace of 𝕄_d(ℂ)_h. Then for any Hilbert space ℋ and for any choice T₁, . . . , T_n ∈ 𝔹(ℋ)_h, the following are equivalent:

1. There exist S ₁, . . . , S_m ∈ 𝔹(ℋ)_h such that p(T₁, . . . , T_n) + B₁ ⊗ S₁ + ⋅⋅ ⋅ + B_m ⊗ S_m > 0.

2. There is some ε > 0 such that for all r ≤ dim(ℋ) and all V_j ∈ 𝕄_d,r(ℂ) with∑jVj∗BiVj=0for all i = 1, . . . , m and∑jVj∗Vj=idrwe have

Let us also comment on Condition (ii) here. In case that ℋ is finite-dimensional, there are only finitely many choices of r to check. For any such r, the set of all possible tuples of V_j fulfilling the condition is compact (again using that the length of the sum can be bounded). Thus it is easy to see that the statement ”There is some ε > 0 . . .” can simply be replaced by strict positivity: Σ_j p_Vj (T₁, . . . , T_n) > 0.

Proof of Proposition 3.3

(i)⇒(ii): By assumption (i) there are S₁, . . . , S_m and an ε > 0 such that

For all r ≤ dim(ℋ) and all V_j ∈ 𝕄_d,r(ℂ) with ∑jVj∗BiVj=0 for all i = 1, . . . , m and ∑jVj∗Vj=idr we immediately get

For (ii)⇒(i) define q := p − ε ⋅ id_d ⊗ 1 ∈ 𝕄_d(ℂ〈x₁, . . . , x_n〉). We want to apply Theorem 3.2 to q. So let W_j ∈ 𝕄_d,r(ℂ) such that ∑jWj∗BiWj=0 for all i = 1, . . . , m and let k be the rank of the matrix ∑jWj∗WjThen there is an invertible matrix U ∈ 𝕄_r(ℂ) with

Define

and V_j := W_jUV for all j. Then we have

for all i = 1, . . . , m and

It follows that the V_j fulfill the assumptions from (ii), and therefore

Now we claim that W_jUP = W_jU for all j. To prove this, let x ∈ ℂ^r, x₁ = Px and x₂ = x − x₁. Then we have W_jUPx = W_jUx₁ = W_jUx − W_jUx2. Hence, it suffices to show that W_jUx₂ = 0. We compute

Since every summand on the left hand side is nonnegative, we indeed get W_jUx₂ = 0 and therefore W_jUP = W_jU for all j. With this in hand we can further compute

Since we have chosen U to be invertible, we also get Σ_j q_Wj (T₁, . . . , T_n)⩾ 0. We have thus checked Condition (ii) from Theorem 3.2, and can conclude that there are S₁, . . . , S_m ∈ 𝔹(ℋ)_h such that

This is equivalent to

the desired result.

Lemma 3.4

Let p ∈ 𝕄_d(ℂ〈x₁, . . . , x_n〉)_h and let B ∈ 𝕄_d(ℂ)_h be semidefinite. Let the columns of W ∈ 𝕄_d,r(ℂ) form a basis of ker(B). Then for any Hilbert space ℋ and for any choice T₁, . . . , T_n ∈ 𝔹(ℋ)_h, the following are equivalent:

1. There exists S ∈ 𝔹(H)_h such that p(T₁, . . . , T_n) + B ⊗ S > 0.

2. p_W(T₁, . . . , T_n) > 0.

Proof. Again (i)⇒(ii) is obvious, and with S := λ ⋅ id the direction (ii)⇒(i) is an easy exercise.

Theorem 3.5

Let p ∈ 𝕄_d(ℂ〈x₁, . . . , x_n〉)_h and let B₁, . . . , B_m ∈ 𝕄_d(ℂ)_h. Then for any Hilbert space ℋ and for any choice T₁, . . . , T_n ∈ 𝔹(ℋ)_h, the following are equivalent:

1. There exist S ₁, . . . , S_m ∈ 𝔹(ℋ)_h such that p(T₁, . . . , T_n) + B₁ ⊗ S₁ + ⋅⋅ ⋅ + B_m ⊗ S_m > 0.

2. There is some ε > 0 such that for all r ≤ max{dim(ℋ), d} and for all V_j ∈ 𝕄_d,r(ℂ) with∑jVj∗BiVj=0for all i = 1, . . . , m and∑jVj∗Vj=idrwe have

Proof. The proof that (i) implies (ii) is the same as in Proposition 3.3.

For the converse, set S = span(B₁, . . . , B_m) and let k = dim S. We will prove the implication from (ii) to (i) by induction over k. For k = 0 this is obvious, by choosing V = id_d. Now assume k > 0. If S is definite, then we can find λ₁, . . . , λ_m ∈ ℝ such that λ₁B₁ + ⋅⋅ ⋅ + λ_mB_m > 0. If we set S_i = λ ⋅ λ_i id_ℋ for λ ∈ ℝ large enough, then we get (i). If S is indefinite, then (i) follows from Proposition 3.3.

Now assume that S is neither definite nor indefinite, and let B be a nonzero positive semidefinite element of S. Further let V ∈ 𝕄_d,r(ℂ) be a matrix such that the columns of V form an orthonormal basis of ker(B). Set q := p_V ∈ 𝕄_r(ℂ〈x₁, . . . , x_n〉)_h. Now for all t ≤ max{dim(ℋ), r} and all V_j ∈ 𝕄_r,t(ℂ) with∑jVj∗(V∗BiV)Vj=0 for all i = 1, . . . , m and ∑jVj∗Vj=It we have

where the inequality follows from assumption (ii). Since B is in the kernel of the surjective linear map

we have dim(V∗SV) < dim S = k. By induction hypothesis we find S₁, . . . , S_m ∈ 𝔹(ℋ)_h such that

By Lemma 3.4 there is an R ∈ 𝔹(ℋ)_h such that p(T₁, . . . , T_n)+ B₁ ⊗ S₁ +⋅ ⋅ ⋅+ B_m ⊗ S_m + B ⊗ R > 0. Since B ∈ S, we find λ₁, . . . , λ_m ∈ ℝ such that B = λ₁B₁ + ⋅⋅ ⋅ + λ_mB_m. Now we have

This proves (i).

Remark 3.6

It is not clear whether we can get rid of the assumption on span{B₁, . . . , B_m} in Theorem 3.2 as well. There is one obvious way to proceed. If a tuple (T₁, . . . , T_n) fulfills (ii) in Theorem 3.2, one can try to approximate it by a tuple that even fulfills (ii) from Theorem 3.5, and thus obtain (i) from Theorem 3.5 for the approximation. So the set defined by (i) in Theorem 3.2 is at least dense in the one defined by (ii), in a suitable sense. One example is the following statement, which holds for free spectrahedrops.

Corollary 3.7

Let A ₁, . . . , A_n , B₁, . . . , B_m ∈ 𝕄_d(ℂ)_h be such that e₁A₁ + ⋅⋅ ⋅ + e_nA_n = id_d for some point e ∈ ℝⁿ. Let ℋ be a Hilbert space and assume that T₁, . . . , T_n ∈ 𝔹(ℋ)_h fulfill the following condition:

For all r ∈ ℕ with r ≤ max{dim(ℋ), d} and all W_j ∈ 𝕄_d,r(ℂ) with∑jWj∗BiWj=0for i = 1, . . . , m, we have Σ_j p_Wj (T₁, . . . , T_n) ⩾ 0.

Then for all ε > 0 there exist S₁, . . . , S_m ∈ 𝔹(H)_h such that

Proof. The tuple (T₁ + εe₁idH, . . . , T₁ + εe₁idH) clearly fulfills (ii) of Theorem 3.5. The statement follows from (i) in Theorem 3.5 for this tuple.