1 Introduction↩︎

Matrix and noncommutative convexity constitute an important area of modern noncommutative analysis and have found significant applications in mathematical physics. See, for example, the discussion and references cited in the introductions of [1]–[3]. Indeed there are many compelling expositions of these connections in recent literature, such as the work of the ‘Helton school’ (we mention just [4] since it is a survey), or as another example forthcoming work of Kennedy and Skoufranis [5]. In the first part of our paper, we provide an abstract characterization of matrix convex sets, as well as of compact matrix convex sets; a brief description of this characterization is given three paragraphs below. This of course in some sense gives a new characterization of operator systems in view of the well known duality between the latter and matrix convex sets [3], [6]. Before this we give a quick review of the classical real case of this, that is abstract characterizations of convex sets and compact convex sets, and we will extend this to the complex case. Our approach is via a universal Banach space (resp.operator space) \(X_K\) of an abstract compact convex set (resp.matrix convex set) \(K\). This turns out to be a concrete construction of the (unique) base norm space (resp.) with base \(K\), together with a natural TVS topology. Nc base norm spaces are an important class of operator spaces which include duals and preduals of unital \(C^*\)-algebras and von Neumann algebras, and operator systems, where the ‘base’ is exactly the noncommutative convex set of (matrix) states on these. In a very recent paper [2] the first author and Hay generalized the theory of base norm spaces to the complex case, and further to the noncommutative setting relevant to ‘quantum convexity’. In that paper we spell out the importance of this, e.g.the connection to GPT’s (Generalized Probability Theories) in mathematical physics (see e.g.[7]–[9]). Or, as we said there, Quantum Channels or CPTP maps (completely positive trace preserving), are just our base morphisms, that is maps between matrices (viewed as nc base spaces in our language) preserving the base. The adjoint (dual) of such maps are the UCP maps. Indeed mathematical physicists usually prefer the base formulation, as any operator algebraist not familiar with physics knows who has tried to read a quantum physics or QIT article and found themselves having to ‘translate’ the predual or base formulations into statements at the algebra level.

In the later parts of our paper we give many applications, mostly to base norm spaces (classical and noncommutative). For example we show how the weak* topology on the dual \(M^*\) of a von Neumann algebra can be constructed from the weak* topology on \(M^*_+\), or from the weak* topology on the state space. We also refine the ‘main regularity results’ of [10], concerning regularity of convex sets (classical and noncommutative). For example if \(V\) is an LCTVS and if \(K\) is a compact convex set which spans \(V\) and lies in a hyperplane in \(V\) not passing through 0 then \(V\) is canonically isomorphic to \(A(K)^*\), the dual of the space of continuous affine functions on \(K\), via a continuous isomorphism which is a homeomorphism on bounded sets. Similarly in the nc case: If \(V\) is a complex \(*\)-LCTVS, and if \(K\) is a selfadjoint compact matrix convex set such that \(K_1\) spans \(V_{\rm sa}\) and lies in a hyperplane not passing through 0, then \(\mathbb{A}(K,\mathbb{C})^* \cong V\) via a continuous selfadjoint isomorphism which is a homeomorphism on bounded sets. We also give several interesting characterizations of base norm spaces (classical and noncommutative). Again, any such characterization will correspond by the duality results in [2] to a new characterization of operator systems, or in the classical case, function systems. For example, (complex) nc dual base norm spaces ‘are’ the matrix ordered \(*\)-LCTVS’s \(V\) with closed matrix cones such that (at level 1) \(V_+\) is locally compact and spans \(V\). Or, such that \(V\) (at level 1) has a linear base which is \(\tau\)-compact (see Theorem 25). Here \(\tau\) need not be the expected weak* topology, but it may be switched with it, and in any case is the same on ‘bounded sets’. (The majority of our characterizations are not however stated in terms of matrix ordered spaces.)

Most of our paper is developed in the setting of Wittstock’s complex matrix convexity as in [6], for example. Since our paper is already sizable, we will defer to a sequel the real matrix convexity case, which is a bit more technical, as well as the case relevant to Davidson and Kennedy’s nc convexity [3].

The answer to the question “What is a(n abstract) compact convex set?” is: it is an abstract convex \(K\) set which is compact and topologically convex (this is defined at the start of Subsection 2.3). Moreover such a set ‘is’ a compact convex set in an LCTVS if and only it is locally convex (that is, \(K\) has a basis of convex neighborhoods). See Theorem 2. In Section 3 we will answer the question “What is a noncommutative compact convex set?” in the complex case, for example as the matrix sets \(K\) in the following result:

The reader can find the conditions (M1), (M2)\(^{\prime}\), and (M3)\(^{\prime}\), and the proof of the theorem, in Section 3, in the matrix convex case. Suffice it to say that (M1) asserts that the \(\xi^* v \xi\) expressions ‘act as they ought’, (M2)\(^{\prime}\) asserts that such expressions determine \(v\), and (M3)\(^{\prime}\) phrases ‘direct sums’ and ‘compressions’ in terms of such expressions. We postpone the proof of the ‘respectively’ case of this theorem, namely the abstract characterization of Davidson and Kennedy’s nc compact convex sets, to the sequel paper.

Turning to definitions and notation, we will begin with the phrases used above. We write \(\mathbb{F}\) for \(\mathbb{R}\) or \(\mathbb{C}\). The letter \(H\) is usually reserved for real or complex Hilbert spaces, and \(K\) for a convex or matrix convex set. We will be considering real and complex vector spaces, which may also be normed spaces, ordered vector spaces, or \(*\)-vector spaces. Where the usual \(i\)-\(j\)-matrix subscripting conflicts with the lettering for the imaginary number \(i\), we sometimes write \(\iota\) for the latter. An ordered vector space is a vector space with a proper positive cone. For an ordered vector space \(E\), the cone of positive elements will be denoted by \(E_+\). By a \(*\)-vector space we mean a vector space with an involution (a period 2 automorphism) \(*\). If \(\mathbb{F}= \mathbb{C}\) we assume that the involution is conjugate linear. We write \(E_{\rm sa} = \{ x \in E : x = x^* \}\) for the elements in a set \(E\) which are selfadjoint. We assume that the positive cone of an ordered \(*\)-vector space \(E\) is contained in \(E_{\rm sa}\). We recall that the positive cone is called generating if \(E_+\) spans \(E\); e.g.if \(E_{\rm sa} = E_+ - E_+\). A real (resp.complex) Hausdorff LCTVS with a continuous involution will be called a \(*\)-LCTVS. Similarly for TVS’s. We will say that an ordered \(*\)-vector space \(X\) has a linear base \(K\), if \(K\) is a base for \(X_+\) (that is, every nonzero \(x \in X_+\) may be written uniquely as \(ck\) for \(c > 0\) and \(k \in K\)), and \(K\) spans \(X\). For basics on bases and base norm spaces see e.g.the prequel [2], which we expect the reader to be somewhat familiar with, at least for the many parts of our paper concerning base norm spaces and their nc variant. We write \(M_n(\mathbb{F})\) for the \(n \times n\) matrices, or sometimes simply \(M_n\) when the context is clear. We write \(M_n(E)\) for the \(n \times n\) matrices with entries from a vector space \(E\). We recall that \(*\)-vector space \(X\) is matrix ordered if there is a proper cone \(M_n(X)_+ \subset M_n(X)_{\rm sa}\) for all \(n \in \mathbb{N}\), such that \(\alpha^* M_n(X)_+ \alpha \subseteq M_m(X)_+\) for \(\alpha \in M_{n,m}(\mathbb{F})\). It follows that \(M_n(X)_+ \oplus M_m(X)_+ \subset M_{n+m}(X)_+\). A sequence \((C_n)\) of cones which satisfy these properties is also referred to as a matrix cone.

We assume that the reader is familiar with basic convexity theory. We denote the convex hull of a set \(A\) by \({\rm co} (A)\) and the closed convex hull by \(\overline{{\rm co}} (A)\). We should warn the reader about a potential source of subtle confusion in the literature. Namely sometimes the naked phrase “compact convex set” in the literature really means “compact convex set in an LCTVS”, as opposed to “compact convex set in a TVS”. This is sometimes not said explicitly, for example in some statements of Kadison’s theorem in the literature concerning the duality of the categories of ‘compact convex sets’ and ‘function systems’. They are not the same, not every compact convex set \(K\) in a TVS is affinely homeomorphic to a compact convex set in a LCTVS. Indeed this was an open problem until Roberts’ 1977 needle-point space counterexample, see e.g.[11]. For such \(K\), \(A(K)\) does not separate points of \(K\), which can be a source of deep trouble. Another implication of this for us is that not every base norm space with compact base \(K\) is a dual base norm space (Proposition 15). To be a dual base norm space one must add the topological condition that \(K\) is locally convex.

Operator systems are far-reaching noncommutative generalization of function systems which plays a central role in the theory of operator algebras and noncommutative functional analysis, generally. They may be characterized abstractly as the archimedean matrix order unit spaces, namely a matrix ordered \(*\)-vector space in the sense above which possesses a ‘matrix order unit’ (see [12], [13] for the latter definition). For \(n \in \mathbb{N}\) we define the amplification of a linear map \(T: V \to W\) by \[T^{(n)}: M_n(V) \to M_n(W)\] \[[x_{ij}] \mapsto [\varphi(x_{ij})] .\] The natural morphisms between matrix ordered spaces (resp.operator systems) are the completely positive (resp. unital completely positive (ucp)) maps. These are linear maps \(T: V \to W\) with every amplification positive (which implies that it is selfadjoint, that is \(*\)-preserving, that is \(T(x^*) = T(x)^*\) for \(x \in V\)). A (concrete) operator space is a linear space \(E\) of operators on a Hilbert space \(H\), together with norms on \(M_n(E)\) inherited from \(B(H^{(n)})\). These may be characterized as the vector spaces \(E\) with a norm on \(M_n(E)\) for all \(n \in \mathbb{N}\) satisfying certain conditions which we will not repeat here. The completely bounded norm is \(\| T \|_{\rm cb} = \sup_n \, \| T^{(n)} \|\), and \(T\) is completely contractive (resp.completely isometric) if \(\| T \|_{\rm cb} \leq 1\) (resp.each \(T^{(n)}\) is an isometry). A state (resp.matrix state) on an operator system \(V\) is a (selfadjoint) completely positive unital scalar valued (\(M_n\)-valued) map. Then \(({\rm ucp}(V, M_n))\) is the matrix state space.

For general background on complex operator systems and spaces, and in particular on the definitions etc.in this section, we refer the reader to e.g.[14]–[16]. For real operator systems and spaces see [13], [17]. The theory of complex \(C^*\)- and von Neumann algebra theory may be found in e.g.[18]. The connection between complex operator systems and matrix convex sets may be found in [6] (see also e.g. [3], [19]–[21] and references therein). A matrix set or nc set in a vector space \(E\) is a sequence \((X_n)\), with \(X_n \subseteq M_n(E)\). We call \(X_n\) the \(n\)th level of \(X\). A set with levels is sometimes called a graded set. A (real or complex) matrix convex set in a (real or complex) vector space \(E\) is a nc set \(K = (K_n)\) in \(E\) satisfying 1)\(x \in K_m\) and \(y \in K_n\) implies \(x \oplus y \in K_{m+n}\), and 2)\(a \in M_{n,m}(\mathbb{F})\) with \(a^* \, a = I_n\) and \(x \in K_n\) implies \(a^*xa \in K_m\). Here \(n, m \in \mathbb{N}\). A matrix affine combination is a finite sum \(\sum_i\, \gamma_i^* v_i \gamma_i\) for \(\gamma_i \in M_{n_i,n}\) with \(\sum_i\, \gamma_i^* \gamma_i = I_n\) and \(v_i\) at level \(n_i\). So matrix convex sets are the nc sets which are closed under matrix affine combinations. If \(E\) is a topological vector space, then we say \(K\) is closed (compact) if each \(K_n\) is closed (compact). The matrix state space \(({\rm UCP}(\mathcal{S}, M_n))\) of an operator system \(\mathcal{S}\) is the generic example of a compact matrix convex set [6]. We write \(A(K)\) or \(A(K,\mathbb{F})\) for the continuous affine scalar functions on a compact convex set \(K\), which are unital selfadjoint subspaces of \(C(K, \mathbb{F})\), the continuous functions on \(K\) with values in the field \(\mathbb{F}\). We write \(\mathbb{A}(K)\) or \(\mathbb{A}_{\mathbb{F}}(K)\) for the noncommutative version from [6], the matrix affine continuous nc functions into the scalars. We recall that a nc function is matrix affine if it preserves matrix affine combinations. Such \(\mathbb{A}(K)\) turn out to be the generic example of (complete) operator systems [6]. These results give dualities between the categories of operator systems and weak* compact matrix convex sets in an LCTVS (see [6], [3]), which generalize the ‘Kadison duality’ between function systems and compact convex sets in an LCTVS.

In classical functional analysis, base norm spaces appear as the objects that are dual to archimedean order unit spaces, or equivalently, by Kadison’s theorem, to the class of function systems. Both are ordered vector spaces whose order structure induces a norm. Whereas archimedean order unit spaces carry a norm which is induced by an order unit, the norm on a base norm space is induced by a base for the cone of positive elements. Similarly in the nc case: nc base norm spaces are the objects that are dual to operator systems. As we said earlier we expect the reader to be somewhat familiar with basics of base norm spaces and their nc variant in [2]. To avoid clutter in the Introduction, we will review the basic definitions and facts about base norm spaces and nc base norm spaces at the start of Subsection 2.4 and Section 5 respectively.

2 What is a convex set? Compact convex set? Matrix convex set?↩︎

2.1 What is a convex set?↩︎

Already in 1939 M. H. Stone had the idea of abstractly characterizing convex sets [22]. In 1954 Hausner gave a much cleaner characterization and proof [23]. Since then many mathematicians and mathematical physicists have needed and have rediscovered this abstract characterization of convex sets, often much less elegantly than in [23].

The abstract ‘convexity operation’ or ‘mixture’ \(K \times K \times [0,1] \to K\) is usually written as \(xty\) for \(x, y \in K, t \in [0,1]\). A map is abstract affine if \(f(xty) = f(x)tf(y)\) for \(x, y \in K, t \in [0,1]\). Mixtures play a role in the theory of utilities associated in part with von Neumann and Morgenstern (e.g.see the volume containing [23] for more details), and have also been of interest in certain areas of mathematical physics, e.g.see references in [2]. There are four Stone-Hausner axioms on this ‘convexity operation’ or ‘mixture’. Three are the obvious ones: 1)\(xtx = x\) (self-combination), 2)\(xty = y (1-t)x\) (commutativity), 3) If \(t \in (0,1]\) and \(xty = zty\) then \(x = z\) (cancellation law). The final axiom is an ‘associative law’, namely 4) \((xty)sz = xp(yrz)\) for the obvious scalars \(p,r\). If these four axioms hold then the Stone-Hausner theorem asserts that \(K\) is abstract affinely isomorphic to a convex subset of a vector space. Hausner’s proof is to show that 1)any abstract convex set \(K\) (Hausner calls these mixture spaces) is ‘affinely’ embedded in an ‘abstract cone’ \(\mathcal{C}\), and 2)any abstract cone may be ‘affinely’ embedded in a vector space \(X_K\) as a concrete cone. Indeed for 1), \(\mathcal{C}\) is \(K \times (0,\infty)\) with one point adjoined, namely the ‘zero element’ of the cone. The rough or guiding idea for 2)is to note that a cone is a semigroup under addition in such a way that the enveloping group (the Grothendieck group of the semigroup), is a vector space. (We remark that the axioms in the characterization of convex sets in [24] for example are different but include the strong requirement that affine maps separate points, which makes the proof rather trivial, although interesting.)

2.2 The universal vector space \(X_K\) of a convex set↩︎

If \(K\) is a convex set let \(\mathcal{C}\) be its ‘generated cone’ above, namely \(K \times (0,\infty)\) with one point adjoined, namely the ‘zero element’ of the cone. This has base \(K\), and an obvious well defined addition \(sk + t k' = (s+t) [\frac{s}{s+t} k + \frac{t}{s+t} k']\), for \(s,t \geq 0\) and \(k, k' \in K\). Let \(X_K\) be the Grothendieck group of \(\mathcal{C}\). Then \((X_K,\mathcal{C})\) is an ordered vector space and \(X_K = \mathcal{C}- \mathcal{C}\). If we desire a complex vector space generated by \(K\) then one takes the universal complex vector space of \(K\) to be the complexification \((X_K)_c\) of \(X_K\). This is a \(*\)-vector space with the obvious involution (making \(X_K\) the selfadjoint part). The base function of \(X_K\) (or its complexification) is the extension in Lemma 1 below of the constant 1 function on \(K\). It takes \(x \in \mathcal{C}\setminus \{ 0 \}\) to the unique scalar \(c\) such that \(x/c \in K\).

A hyperplane in a vector space \(V\) will be a set of the form \(\{x \in V : f(x) = 1\}\) for a linear functional \(f\) on \(V\). There is a well known trick to replace a convex set \(K\) in a vector space \(V\) which is in no hyperplane not passing through 0, by an affinely isomorphic convex set which is in such a hyperplane. Namely, replace \(K\) by \(K \times \{ 1 \}\) in \(V \oplus \mathbb{F}\).

We have the following universal property:

Remark. If \(f : K \to V\) is one-to-one we cannot assert that an extension \(X_K \to V\) is one-to-one, unless \(f\) maps into a hyperplane not passing through 0. For example, consider the map \(K \subset l^1_2(\mathbb{R}) \to \mathbb{R}\) taking \((t,1-t) \mapsto 2t-1\) for \(t \in [0,1]\). Here \(K = \{ (t,1-t) : t \in [0,1] \}\).The extension is not even one-to-one on the cone \(\mathcal{C}= \mathbb{R}_+ K\) generated by \(K\). In this example \(X_K \cong l^1_2(\mathbb{R})\).

Not only is \(X_K\) universal, but by (3) in the Lemma there is a copy of it in every vector space (resp.\(*\)-vector space) \(V\) in which \(K\) lies in a hyperplane not passing through 0 in \(V\) (resp.in \(V_{\rm sa}\)). Thus we will often identify a convex set with its affinely isomorphic image in the vector space \(X_K\) (or its complexification).

2.3 What is a compact convex set?↩︎

Now that we understand abstract convex sets, one may ask for a topological variant of the Stone-Hausner theorem, namely an abstract characterization of compact convex sets in a TVS. Or, in an LCTVS. In view of the Stone-Hausner theorem henceforth we have no need of abstract convex sets or abstract cones, but simply consider a convex set \(K\) which is also a compact Hausdorff topological space.

We say that a convex set is topologically convex if it has a Hausdorff topology with respect to which the convexity operation \(K \times K \times [0,1] \to K\) is continuous.

These authors of course built on earlier work of Klee, Keimel, and many others. Below we will treat these results within the framework of base norm spaces (see Lemma 3 and 6). This answers the question “What is a compact convex set?” as: it is an abstract convex set which is compact and topologically convex. Moreover such a set ‘is’ a compact convex set in an LCTVS if and only if it is locally convex.

If \(K\) is a compact topologically convex set then the cone \(\mathcal{C}\) above generated by \(K\) (see the first lines in Subsection 2.2), has a canonical Hausdorff topology with respect to which it is easy to see that it is locally compact and \(\sigma\)-compact, and the cone operations (addition and multiplication by a nonnegative scalar) are continuous. See [25]. Indeed [25] says that \(K\) is affinely homeomorphic to the compact base of a locally compact cone (with continuous addition and scalar product) in a Hausdorff TVS. We give a few more details which are occasionally useful. Indeed if \(L= (0,1] \times K\) with product topology, set \(\mathcal{C}_0\) to be the 1-point compactification of \(L\), with the one point identified with \(0\). By the uniqueness of the one-point compactification, \(\mathcal{C}_0\) is topologically identifiable with \(\{ x = t k \in \mathcal{C}: 0 \leq t \leq 1, k \in K \}\), with the topology with basis the sets \(I \times B\) for \(I = (a,b)\) for \(0< a < b\), and \(B\) chosen from a basis for \(K\), together with the sets \([0,\epsilon) \times K\) for \(\epsilon > 0\). Set \(\mathcal{C}_n = [n,n+1] \times K\) for \(n \in \mathbb{N}\), a compact set. We give \(\mathcal{C}\) the attachment space topology formed by gluing \(Y = \mathcal{C}_0\) to \(Z = [1,\infty) \times K\) along \(A = \{ 1 \} \times K\). Then \(\mathcal{C}\) is Hausdorff, normal, locally compact, and \(\sigma\)-compact (consider the compact \((\mathcal{C}_n)_{n \geq 0}\) above), and \(K\) may be identified with \(A\) as a compact subset and base of \(\mathcal{C}\). Note that \(Z \setminus A = (1,\infty) \times K\) (resp.\(\mathcal{C}_0\)) may be identified as an open (resp.compact) subset of \(\mathcal{C}\), and the latter two sets partition \(\mathcal{C}\). Indeed \(\mathcal{C}\setminus 0\) is homeomorphic to \((0,\infty) \times K\), the latter with product topology.

We will call \(\mathcal{C}\) above together with this topology the Lawson cone of \(K\). We give it this name even though its construction is not attributable to Lawson, to honor Lawson’s beautiful work (partly with coauthors).

The mechanics in the last proof are often used in what follows.

2.4 The universal Banach space \(X_K\) of a compact convex set↩︎

We will now need to get into base norm spaces, so we give some more definitions here (see [2] for a survey in the classical case, and many more details.) A real base norm space is an ordered normed space \(X\) with closed cone \(X_+\), with a convex base \(K\) in Ball\((X) \cap X_+\), such that Ball\((X) \subseteq t \, {\rm co} (K \cup (-K) )\) for all \(t > 1\). A real dual base norm space is a real base norm space with a Banach space predual such that the base \(K\) is weak* closed (and hence weak* compact). It follows that Ball\((X) = {\rm co} ( K \cup -K )\), and that the base function (that is, the unique functional on \(X\) which is \(1\) on \(K\)) is weak* continuous. The dual Banach space of a real unital function space is the generic real dual base norm space. Equivalently, the dual base norm spaces are exactly (up to appropriate isomorphism) the spaces \(A(K)^*\) for a compact convex set \(K\) in an LCTVS. The dual base of \(A(K)^*\) is \(\delta(K)\), where \(\delta : K \to A(K)^*\) is the canonical map. The generic real base norm space “is" the predual of a dual real function system, with the base corresponding to the normal state space. Similar results hold for complex base norm spaces and their duality with complex function systems [2]. This is based on the complexification of a real base norm space using the dual Taylor norm which we describe there. Indeed the theory of complex base norm spaces is a simple add-on to the classical theory of real base norm spaces. Essentially everything in the complex theory follows quickly from the real theory, together with the fact above that the complex base norm is completely determined, via the dual Taylor norm, by the classical ‘base norm’ on the selfadjoint part. Or in other words, the complex case is just a standard complexification of the real case. A base morphism is a positive linear map \(u : X \to Y\) between base norm spaces mapping base into base. This is equivalent (assuming \(u\) positive) to \(f_Y \circ u = f_X\), where \(f_X\) and \(f_Y\) are the base functions. A bijective base morphism between base norm spaces is an isometric order isomorphism by Lemma 5. Similarly a bijective nc base morphism between nc base norm spaces is a completely isometric order isomorphism.

If \(K\) is a convex set then there is a canonical seminorm \(\| \cdot \|_K\) that one can put on its universal space \(X_K\) from the earlier subsection, namely the Minkowski functional of co\((K \cup (-K))\). If this is a norm then \(X_K\) is a pre-base normed space in the sense of [2]. Indeed it is a norm under the conditions of the next result. As in the last subsection, in the following \(K\) is a compact topologically convex set, for example, a compact convex set in a TVS. Then the Lawson cone \(\mathcal{C}\) above is Hausdorff, locally compact, and \(\sigma\)-compact, and has \(K\) as a base. Suppose that \(K\) has a basis \(\mathcal{B}\) of open (or open and convex) sets. Then sets of form \(I \cdot B\), for \(I = (a,b)\) for \(0< a < b\), and \(B \in \mathcal{B}\), is a basis for \(\mathcal{C}\setminus 0\), while at \(0\) we take \([0,\epsilon) \cdot K\) to be the local basis. The Grothendieck group \(X_K\) of \(\mathcal{C}\) is as we said earlier, a real ordered vector space with positive cone \(\mathcal{C}\), and base \(K\) for \(\mathcal{C}\). Of course \(X_K = \mathcal{C}- \mathcal{C}\).

If we desire a complex space generated by \(K\) then one takes the universal complex normed space of \(K\) to be the dual Taylor complexification of \(X_K\), This may be identified with the projective tensor product \(X_K \hat{\otimes}_{\mathbb{R}} l^2_2(\mathbb{R})\). We write this as \((X_K)_c\) or sometimes simply as \(X_K\) when there is no confusion. This is a complex base norm space with norm closed base \(K\), if \(K\) is compact topologically convex [2], and is also a \(*\)-TVS with the ‘product topology’. whose selfadjoint part is \(X_K\).

The idea of a universal Banach space \(X_K\) of a convex set, and many of the ideas in the present subsection of our paper, have appeared in various forms or disguises in the literature over the decades. See e.g.[24] for a representative example. We have the following universal property of \(X_K\):

Thus if \(K\) is a compact convex set in a real TVS (resp.normed space) \(V\) then there is a linear (resp.and continuous) map \(X_K \to V\) onto \(\mathbb{R}_+ K - \mathbb{R}_+ K\). If \(K\) lies in a hyperplane not containing \(0\) then as we saw in Lemma 1 (3), this map is an isomorphism, so that \(V\) contains a base norm space with base \(K\). Thus not only is \(X_K\) universal, but there is a copy of it in every TVS (resp.\(*\)-vector space) \(V\) in which \(K\) lies in a hyperplane not passing through 0 in \(V\) (resp.in \(V_{\rm sa}\)). Thus we will often identify a compact convex set with its affinely homeomorphic (by the last assertion of Lemma 3) image in the base norm space and TVS \(X_K\) (or its complexification).

Lemma 3 embeds a compact topologically convex set in a (canonical) TVS. It is natural to ask when it may be embedded in an LCTVS, or when \(X_K\) is an LCTVS. This was answered in [25], [27]. The following works in both the real and complex case:

Remark. See the last assertion of [25] for the variant of this where \(K\) is locally compact.

Remarks. )The proof of the last result also uses the idea in the proof of [25].

2)The complex case of the last theorem from the real case, since a real LCTVS \(X\) is real linearly homeomorphically embedded in a complex LCTVS, namely \(X \oplus iX\) with the product topology.

Remark. We have geometric Hahn-Banach and separation theorems for a locally compact locally convex (topological) cone \(\mathcal{C}\). For example, if \(x \neq y\) in \(\mathcal{C}\) then there is a continuous linear functional \(f\) on \(X\) separating them, where \(X\) is an LCTVS in which \(\mathcal{C}\) is embedded by Theorem 3. Then \(f\) is affine on \(\mathcal{C}\), and separates \(x\) and \(y\).

Remarks. 1)In Corollary 5 we see in fact that \(\mathcal{C}\) is topologically embedded in \(X\) as a closed subset. See also Section 3.12 in [28].

2)If a topological cone \(\mathcal{C}\) is locally compact and locally convex then it is automatic that it has a compact base with \(\mathcal{C}\setminus \{ 0 \} \cong (0,\infty) \times K\) homeomorphically (and in this case \(X\) in the last result is a dual base norm space). To see this note that by Corollary 4 we can assume that \(\mathcal{C}\) is a locally compact cone in an LCTVS \(E\). However it is well known that in this case \(\mathcal{C}\) is closed in \(E\) and has a compact base [28]. Indeed the proof of this shows that the base function \(\varphi\) is continuous. Since \(K\) is locally convex \(X_K\) is a dual base norm space, and we have a continuous one-to-one map from \(X_K \to E\) taking \(\mathbb{R}_+ K\) onto \(\mathcal{C}\), which is a homeomorphism on \(K\). Suppose that \(c_t k_t \to c k\) in \(\mathcal{C}\), for \(c_t \geq 0, k_t \in K\). Applying \(\varphi\) we see that \(c_t \to c\). If \(c \neq 0\) then \(k_t \to k\). So \(c_t k_t \to ck\) in \(X\). So we have a homeomorphism between \(\mathcal{C}\) and \((0,\infty) \times K\) with product topology. As in the proof of Corollary 5 we have that \(\mathcal{C}\) is exactly the Lawson cone of \(K\), and \(X\)is a dual base norm space by Lemma 6.

3 What is a matrix convex set? A matrix compact convex set?↩︎

3.1 What is a matrix convex set?↩︎

In Subsection 2.1 we discussed the Stone-Hausner abstract characterization of convex sets, and in Subsection 2.3 we discussed the abstract characterization of compact convex sets. Now that we understand these, one may ask for an abstract characterization of matrix (or nc) convex sets \(K = (K_n)\), and of matrix (or nc) compact convex sets. One may write down a characterization similar to Stone-Hausner’s mixture conditions, that is in terms of multilinear maps in variables from \(K_n\) and matrix algebras, satisfying certain ‘mixture conditions’ resembling the four Stone-Hausner axioms. However this can be quite clumsy looking if not done carefully. Moreover in view of the Stone-Hausner theorem we have no need of abstract convex sets. Instead one may take a perspective from the theory of noncommutative functions (see e.g.[29]), namely that nc sets and nc functions are defined in terms of a particular fixed vector space. We will define a (complex) abstract matrix convex set \(K = (K_n)\) in terms of a canonical vector space \(Z\) which is determined by the abstract convex set \(K_1\). More particularly, \(Z = Z_{K_1}\) is the complexification of the real universal vector space \(X_{K_1}\) from Subsection 2.2. We say that \(K\) is a (complex) abstract matrix convex set if it is a complex matrix convex set in the usual sense (see [6], or our introduction) in the vector space \(Z = Z_{K_1}\).

If one is in a situation where further abstraction is convenient, we supply one such suitable framework, in terms of a kind of ‘quadratic form’. Let \(K\) be a sequence of sets \((K_n)\), such that \(K_1\) is an abstract convex set. Suppose that we have actions \(\mathbb{C}^n \times K_n \to K_1\), written as \(\xi^* v \xi\), for \(n \in \mathbb{N}, v \in K_n, \xi \in \mathbb{C}^n, \| \xi \| = 1\). We call these compressions to \(K_1\), and sometimes write it as \(v_\xi\). We may then talk about the (matrix) entries of an element \(k \in K_n\). These are a canonical linear combination in \(Z\) coming from the polarization identity \[\label{pn} \begin{align} k_{ij} &= \frac{1}{4} \sum_{k=0}^3 \, \iota^k (\eta + \iota^k \xi)^* k (\eta + \iota^k \xi), \qquad \eta = e_j, \xi = e_i , \\ k_{ii} &= e_i^* k e_i. \end{align}\tag{1}\] (Here \((e_i)\) is the canonical basis of \(\mathbb{C}^n\).) Then \([k_{ij}] \in M_n(Z)\). Let \(\tilde{K_n}\) be the set of such matrices in \(M_n(Z)\), for \(k \in K_n, n \in \mathbb{N}\). We require that

That is, the compressions to \(K_1\) act as they ought. We also require that (M2) the entries of \(k \in K_n\) determine \(k\). That is, \(k \to [k_{ij}]\) is one-to-one on each \(K_n\) for all \(n \in \mathbb{N}\). Assuming (M1), notice that (M2) is equivalent to

That is, if \(\xi^* k \xi = \xi^* k' \xi\), for \(k, k' \in K_n, \xi \in \mathbb{C}^n, \| \xi \| = 1\), then \(k = k'\). We will check this below.

We say that \(K\) is a (complex) abstract matrix convex set if (M1) and (M2) hold, as well as (M3)\(\tilde{K} = (\tilde{K_n})\) is a complex matrix convex set in the usual sense (see [6]), in the vector space \(Z\). Alternatively to (M3), one may write down the ‘matrix convexity operations’ for \(K\) in terms of the compressions to \(K_1\), and insist that \(K\) is closed under these. Namely, we have general compressions \[(\gamma^* k \gamma)_\xi = (\gamma \xi)^* \, k \, (\gamma \xi) = k_{\gamma \xi}, \qquad \gamma \in M_{m,n}, \gamma^* \gamma = I_n, k \in K_m, \xi \in \mathbb{C}^n, \| \xi \| = 1,\] and direct sum \[(k_1 \oplus k_2)_\xi = \xi_1^* k_1 \xi_1 + \xi_2^* k_1 \xi_2 , \qquad k_1 \in K_n, k_2 \in K_m , \xi \in \mathbb{C}^{n+m}, \| \xi \| = 1,\] where the \(\xi_k\) are the projection of \(\xi\) onto \(\mathbb{C}^{n} \oplus \vec{0}\) and \(\vec{0} \oplus \mathbb{C}^{m}\) in \(\mathbb{C}^{n+m}\), and \(\xi_j^* k_j \xi_j = \| \xi_j \|^2 \, (k_j)_{\xi_j'}\) for \(j = 1,2\), where we write \(\eta'\) for the unit vector normalization of a vector \(\eta \in \mathbb{C}^n\). Thus the formulation of the alternative to (M3) is:

An abstract nc affine map is a nc function which preserves such direct sums \(\oplus\) and general compressions.

Thus we now have an abstract definition of complex matrix convex sets, namely \(K = (K_n)\) satisfying the conditions in the last theorem. We will adopt a similar but more complicated approach to real matrix convex sets later. Momentarily we will define abstract compact matrix convex sets \(K = (K_n)\) in terms of a canonical TVS structure on \(Z\) which is determined by the abstract compact convex set \(K_1\).

A nc hyperplane in \(E\) is the sequence \((H_n)\) of sets \(H_n = \{x \in M_n(E): f^{(n)}(x) = I_n\}\), where \(f\) is a fixed linear functional on \(E\), and \(I_n\) is the \(n \times n\) identity matrix. The hyperplane trick mentioned above Lemma 1 works for a matrix convex set \(K\) in \(V\) to get a nc affinely isomorphic convex set which is in a nc hyperplane not passing through 0: consider \((K_n \times \{ I_n \})\) in \(V \oplus \mathbb{F}\).

Thus we will often identify a matrix convex set with its matrix affinely isomorphic image \(\tilde{K}\) in the vector space \(Z\).

3.2 What is a matrix compact convex set?↩︎

We say that an (abstract) matrix convex set \(K\) is matrix topologically convex if there is a Hausdorff topology \(\tau_n\) on \(K_n\) for all \(n\) with respect to which the ‘matrix convexity operations’ are continuous (so \(\sum_{k=1}^m \gamma_k^* v_k \gamma_k\) is continuous in the \(\gamma_k, v_k\)). This implies (using e.g.(1 )) that the entries \(k_{ij}\) are a continuous function of \(k \in K_n\) whenever \(K\) is a matrix convex set in a TVS (with the topology on \(K_n\) being the relative product topology). We will see below that \(K\) is matrix topologically convex if and only if \(K_1\) is topologically convex and the \(K_1\)-compressions \(\xi^* k \xi\) are continuous in \(k \in K_n\) for all \(n \in \mathbb{N}\) and fixed \(\xi \in \mathbb{C}^n, \| \xi \| = 1\). Or one may replace the last condition by saying that the entries \(k_{ij}\) are a continuous function of \(k \in K_n\) for all \(n \in \mathbb{N}\). See also e.g.Proposition 9 for another example of this trick.

Now suppose that the matrix topologically convex set \(K\) in the last paragraph is matrix compact, so that \(K_n\) is compact for all finite \(n\). In this case we simply say that \(K\) is an abstract matrix compactly convex set. The reader should take note of this somewhat subtle usage in the rest of our paper; we really mean by this an ‘abstract compact matrix topologically convex set’. Since \(K_1\) is compact and topologically convex there is a canonical TVS topology \(\mathcal{T}\) on the real base norm space \(X_{K_1}\) and on its complexification \(Z\) described in Lemma 3 and the remark after it. Hence there is also a canonical TVS topology \(\sigma_n\) on \(M_n(Z)\) for each \(n \in \mathbb{N}\) (the ‘product topology’ of \(n^2\) copies of \(\mathcal{T}\)).

The first statement of the next theorem yields an abstract characterization of (complex) compact matrix convex sets in a Hausdorff TVS, as the abstract matrix compactly convex sets. It follows from the later Corollary 19 that that TVS may be taken to be an LCTVS if and only if in addition \(K_1\) is locally convex.

The following is a useful criterion for when a nc map between matrix convex convex sets is nc affine or nc continuous:

3.3 The universal operator space \(X_K\) of a complex matrix convex convex set↩︎

If \(K\) is a selfadjoint complex matrix convex set in a complex \(*\)-vector space \(V\), and if \(K_1\) is contained in a hyperplane not passing through 0, let \[C_n = \{ \gamma^* k \gamma : k \in K_m , \gamma \in M_{m,n} : m \geq n \} \subset M_n(V)_{\rm sa}, \qquad \mathcal{C}= (C_n).\] This is the matrix cone generated by \(K\). It is well defined independently of the superspace \(V\) up to ‘nc affine zero-preserving isomorphism’. Indeed any nc affine embedding \(\epsilon : K \to W\) into another complex \(*\)-vector space extends to a linear isomorphism taking \(\mathcal{C}\) bijectively onto the matrix cone generated by \(\epsilon(K)\), by an argument in the proof of Proposition 7.

We will review basic definitions of nc base norm spaces at the start of Section 5. For now we merely state:

We will prove this in Section 5, together with many results which are the variants for the nc/matrix setting of results from Sections above, or from Section 4 below. For example, we will have the expected universal property of \(X_K\): Any nc bounded (resp.positive) matrix affine map \(f : K \to V_{\rm sa}\) into a complex \(*\)-operator space \(V\), has a unique complex linear completely bounded (resp.completely positive) extension \(\tilde{f} : X_K \to V\). Indeed since by Theorem 10 the universal operator spaces \(X_K\) are just the nc base norm spaces with nc base \(K\), their study may be viewed in some sense as essentially a part of the theory of nc base norm spaces.

4 Applications to base norm spaces↩︎

Remark. The ‘continuously’ in the statement of the last Corollorary is not necessarily ‘bicontinuously’, as can be seen by considering a topology on \(B(H)^*\) say which agrees on \(K = {\rm Ball}(B(H)^*)\) with the weak* topology.

In some sense Corollary 11 improves on, or refines, a ‘classical regularity result’ from [10]. It can be stated as saying the following:

Remarks. 1)Thus one may define a dual base norm space to be an LCTVS with a compact convex subset as in Corollary 12, or as in Corollary 13. Thus e.g.dual base norm spaces ‘are’ just the ordered LCTVS’s with a locally compact generating cone.

2)In \(\mathbb{R}^n\) it is known that a closed cone is proper if and only if it has a base. Indeed in this case the closed cone is obviously locally compact so by the fact in the last proof it has a compact base. Also it is well known that in this setting any base for the cone is compact.

3)Generalizing Remark 2), we saw in Theorem 3 that proper locally compact cones in an LCTVS, are up to affine homeomorphism the same as the abstract locally compact locally convex cones (Here ‘locally compact cone’ assumes that the cone operations are continuous.)

We give a nice application to the weak* topology on a real dual base norm space \(E\). The following result applies in particular to the selfadjoint part of the dual of a von Neumann algebra or the dual of an operator system \(\mathcal{S}\), or on \(C(K)^*\) for compact \(K\) (or more generally to the selfadjoint part of any complex dual base norm space). Namely, it shows how to retrieve the weak* topology on \(E\) from the weak* topology on \(E_+\), or from the weak* topology on \(K\) (by applying employing the Lawson cone construction from \(K\)). Of course once one has the weak* topology on the selfadjoint part of the dual of a complex operator system \(\mathcal{S}\), one has it on all of \(\mathcal{S}^*\) (by considering the product topology).

Remark. The category of complex base norm spaces is equivalent to the category of real base norm spaces. Indeed the first half of the proof of [2] shows that any complex base norm space \(F\) is the dual Taylor complexification of its selfadjoint part \(F_{\rm sa}\). Moreover any complex base morphism between complex base norm spaces is selfadjoint, hence restricts to a real base morphism between the selfadjoint parts. Conversely a real base morphism \(F_{\rm sa} \to G_{\rm sa}\) extends to a complex base morphism \(F \to G\) by Lemma 4.

5 Matrix convex sets and nc bases↩︎

In [2] we introduced and studied nc base norm spaces and nc dual base norm spaces. We refer the reader to [2] for various definitions of a nc base norm space, and we shall meet more equivalent definitions below. It is a matrix ordered \(*\)-vector space and operator space with closed matrix cones, and the base now is a closed matrix convex set \(K \subset ({\rm Ball}(M_n(X))_+)\), and every \(x \in M_n(X)_+\) is of form \(cxc\) for \(c \in (M_n)_+\) and \(x \in K_n\). For every \(t > 1\) and \(x \in {\rm Ball}(M_n(X))_{\rm sa}\) we may write \(x = c_1 x_1 c_1 - c_2 x_2 c_2\) for positive matrices \(c_i\), with \(\| c_1^2 + c_2^2 \| \leq t\) and \(x_i \in K_n\). This expression is a noncommutative version of the classical condition mentioned at the start of Section 2.4 that Ball\((X) \subseteq t \, {\rm co} (K \cup (-K) )\). It also essentially defines the nc base norm on \(M_n(X)_{\rm sa}\) in terms of an infimum of such \(\| c_1^2 + c_2^2 \|\) (see equation (2) in [2]). We then get the norm of a general \(x \in M_n(X)\) by the formula \(\| x \| = \| \tilde{x} \|\) where \(\tilde{x}\) is the selfadjoint matrix with rows \([0 \; x]\) and \([x^* \; 0]\). The nc base function is (the unique) selfadjoint functional \(f\) on \(X\) that is \(1\) on \(K_1\); indeed \(K_n = \{ x \in M_n(X)_+ : f^{(n)}(x) = I_n \}\) for each \(n\). A nc dual base norm space is a nc base norm space with an operator space predual such that the nc base \(K\) is weak* closed (and hence weak* compact). The dual operator space of an operator system is the generic nc dual base norm space (see below). Equivalently, the dual nc base norm spaces are exactly (up to appropriate isomorphism) the spaces \(\mathbb{A}(K)^*\) for a compact convex set \(K\). The dual base of \(\mathbb{A}(K)^*\) is \(\delta(K)\), where \(\delta : K \to \mathbb{A}(K)^*\) is the canonical map. The generic nc base norm space “is” the predual of a dual operator system, with the base corresponding to the normal matrix state space. A nc base morphism is a completely positive linear map \(u : X \to Y\) between nc base norm spaces mapping nc base into nc base. Again this is equivalent (assuming \(u\) completely positive) to \(f_Y \circ u = f_X\), where \(f_X\) and \(f_Y\) are the nc base functions. Indeed any map taking nc base into nc base is completely positive, selfadjoint, and completely contractive.

To see that the dual of an operator system \(\mathcal{S}\) is a dual nc base norm system with base the nc state space \(K = ({\rm UCP}(\mathcal{S}, M_n))\), first observe that \(K\) is the generic compact matrix convex set, and it is weak* compact at each level [3], [6]. In the notation above \(X = \mathcal{S}^*\), so that \(M_n(X) = M_n(\mathcal{S}^*) \cong CB(V,M_n)\), with \(M_n(\mathcal{S}^*)_+ \cong cp(V,M_n)\), the completely positive maps. Every such completely positive map may be written as \(c u(\cdot) c\) for \(c \in (M_n)_+\) and matrix state \(u \in {\rm UCP}(\mathcal{S}, M_n)\). The selfadjoint part of \(M_n(\mathcal{S}^*)\) correspond to the selfadjoint completely bounded maps from \(\mathcal{S}\) into \(M_n\), which equals cp\((\mathcal{S},M_n) - {\rm cp}(\mathcal{S},M_n)\). Any completely contractive map from \(\mathcal{S}\) into \(M_n\) may be written as a difference of two completely positive maps whose sum has norm \(\leq 1\) (this follows e.g.from [14]). In this case the nc base function is ’evaluation at \(1 \in \mathcal{S}\).

A base norm space or nc base norm space \(X\) with compact base \(K\) has a canonical TVS structure. Indeed \(X\) is isomorphic to the universal space \(X_{K_1}\) or its complexification \(Z\), so \(X\) inherits the TVS structure in e.g.Lemma 3 and the remark after it (we put the ‘product topology’ on the complexification. This observation also shows that our next result will immediately give a nc analogue of Lemmas 4 and 1, taking \(X\) to be the universal operator space \(X_K\) of \(K\), which we saw in Theorem 10 is a nc base norm space with nc base \(K\) if \(K\) is matrix compact.

Remark. We continue the discussion in the last lines of the proof concerning the fact stated in [2] that nc dual base norm spaces may be characterized as a nc base norm space which is an operator space dual with the nc base weak* compact. Indeed this fact follows easily from the last assertion of Theorem 18, or from the last part of its proof.

If \(K\) is an abstract real or complex matrix or nc convex set we define the nc bounded affine functions \(\mathbb{A}_{\rm b}(K)\) just as in [1], [3]. As in those sources, just as in the \(\mathbb{A}(K)\) case it is an operator system by the abstract characterization of operator systems. If \(f : K \to M_n\) is nc affine and selfadjoint then since \(-cI \leq f(k) \leq cI\) if and only if \(\| f(k) \| \leq c\), for \(k \in K_n\), it follows that the matrix norms on \(\mathbb{A}_{\rm b}(K)\) coincide with the matrix order unit norm on selfadjoint elements. Hence by the usual trick they coincide on all elements.

Remark. The last result generalizes the fact from [3] that \(\mathbb{A}_{\rm b}(K)\) is a dual operator system. Indeed in that setting, namely in the case that \(K\) is an abstract complex matrix compactly convex set with \(K_1\) locally convex, it is actually a bidual operator system. That is for such \(K\) we have \(\mathbb{A}_{\rm b}(K) \cong \mathbb{A}(K)^{**}\) completely isometrically and as operator systems.

Remark. Note that a complex matrix dual base norm space \(X\) with nc base \(K\) is also (with an equivalent norm) a complex dual base norm space with base \(K_1\). It is easy to see by an argument similar to one in the last proof, that the two associated weak* topologies are the same.

Again in some sense Corollary 23 allows a refinement on the main regularity results of [10]. Just as in the ‘commutative case’ in Corollary 12, this refinement of regularity can be stated as saying the following:

Thus one can define a complex nc dual base norm space e.g.to be a complex matrix ordered \(*\)-LCTVS \((V,\tau)\) such that \(V\) (at level 1) has a \(\tau\)-compact base, as in Theorem 25. As we saw above \(\tau\) may not be the desired weak* topology, but it may be switched with it, and in any case is the same on ‘bounded sets’.

Example. The following example is often useful in constructing potential counterexamples to results above, since it embeds any kind of badly behaved operator space \(X\) in a nc base normed space with a nc base that is usually tractable.

Let \(X\) be an operator space (resp.let \(W\) be a dual operator space), a subspace (resp.weak \(*\)-closed subspace) of \(B(H)\) say. Then its Paulsen system is a weak \(*\)-closed operator system in \(M = M_2(B(H)) \cong B(H^{(2)}\). In [2] we defined the base Paulsen system \({\mathfrak S}(X)\), which as a matrix ordered space is the same as \(\mathcal{S}(X)\), but is considered as a nc base norm space with base function the ‘normalized trace’ \(\tau\) on \(\mathcal{S}(X)\), namely the sum of the two scalars on the main diagonal of \(\mathcal{S}(X)\). This is strictly positive on \(\mathcal{S}(X)_+\) so that \(K_1 = \{ a \in \mathcal{S}(X)_+ : \tau(1) = 1 \}\) is a base for \(\mathcal{S}(X)_+\), indeed is a linear base for \(\mathcal{S}(X)\) in the sense of the introduction. Also in the dual operator space space case \(K_1\) is weak* compact, since it is weak* closed and bounded in \(M\). Thus by Theorem 25, it is a dual nc base norm space.

We showed in [2] that the nc base norms on the base Paulsen system \({\mathfrak S}(X)\) are equivalent (up to a constant) to the operator system norms. With some effort one can show the interesting fact that, just as is the case with the Paulsen system, the canonical map of an operator space \(X\) into the \(1\)-\(2\) corner of the base Paulsen system \({\mathfrak S}(X)\) is a complete isometry. So too are the canonical inclusions of the other three corners into \({\mathfrak S}(X)\). (Note that there is a multiplication by \(2\) in these inclusions because we are using the ‘normalized trace’ instead of the trace’, indeed obviously the norm of \(E_{11} I\) in \({\mathfrak S}(X)\) is 1/2.) Thus every operator space \(X\) ‘is a corner’ of a nc base norm space, its base Paulsen system.

Similarly the projections onto the spaces sitting in the four corners of \({\mathfrak S}(X)\) are contractions on \({\mathfrak S}(X)\) (multiplied by 2). Also, the variant of ‘the Paulsen lemma’ for the base Paulsen system holds. We will present this elsewhere.

We end with some remarks on functorial properties of base morphisms. A one-to-one base morphism \(f : X \to Y\) between base norm spaces need not be an isometry, however its range is a base norm space and \(f\) is an isometry (and order embedding) into that space, by Lemma 5. Similarly in the nc case, using Lemma 8. Quotients of base norm spaces behave well:

6 Closing remarks and Acknowledgements.↩︎

The real case of the nc theory in the last section is much more technical, and requires the complexification theory from [1] (or its matrix convexity variant). Thus we defer this to a sequel, together with the case of our results relevant to Davidson and Kennedy’s nc convexity [3].

We thank D. M. Hay for some comments on a preliminary version of this paper, and spotting several typos. This project was partially supported by NSF grant DMS-2154903.

References↩︎