A generalization of K-theory to operator systems
Walter D. van Suijlekom













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Institute for Mathematics, Astrophysics and Particle Physics, Radboud
University Nijmegen, Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands.

waltervs@math.ru.nl




A generalization of K-theory to operator systems
    Walter D. van Suijlekom
    September 4, 2024
================================================



§ ABSTRACT 
  We propose a generalization of K-theory to operator systems. Motivated by spectral truncations of noncommutative spaces described by C^*-algebras and inspired by the realization of the K-theory of a C^*-algebra as the Witt group of hermitian forms, we introduce new operator system invariants indexed by the corresponding matrix size. A direct system is constructed whose direct limit possesses a semigroup structure, and we define the K_0-group as the corresponding Grothendieck group. This is an invariant of unital operator systems, and, more generally, an invariant up to Morita equivalence of operator systems. For C^*-algebras it reduces to the usual definition. We illustrate our invariant by means of the spectral localizer.  
  




§ INTRODUCTION

The last few years has seen many new interactions between noncommutative geometry and operator theory. 
In particular, the development of noncommutative geometry to describe spectral truncations <cit.>, the corresponding spectral localizer <cit.> (cf. <cit.> and references therein), and spaces up to tolerance relations <cit.> has led to the development of many interesting new structures and applications in operator system theory. These include operator system duality <cit.>, noncommutative convex geometry <cit.>, quantum metric spaces <cit.> and the notion of Morita equivalence for operator systems <cit.>.

A key ingredient for many applications in noncommutative geometry based on C^*-algebras is K-theory. Given the above fruitful interactions based on the replacement of C^*-algebras by operator systems, one is naturally led to the question whether there is an analogue of K-theory for operator systems. We address this question in the present paper.

On our wish list for a candidate for K-theory we have put that




  * it should capture the spectral localizer <cit.> for spectral truncations as a pairing with K-theory;

  * it should capture compressions of the element Y describing quanta of geometry in <cit.>;

  * it should be invariant under Morita equivalence <cit.>;

  * it should be a refinement of K-theory for C^*-algebras, taking the operator system structure into account.
  
Some results in the direction of K-theory for operator systems already exist in the literature. For instance, projections in operator systems were defined in <cit.>. However, it is not difficult to see that spectral compressions PpP of a projection p in a C^*-algebra are in general not projections. As a matter of fact, in general they are not even ϵ-projections in the sense of <cit.> so that also quantitative K-theory may not be suitable for these kind of applications (instead, quantitative K-theory is more fitting for the examples coming from tolerance relations). The K-theory for absolute matrix order unit spaces of <cit.> need an additional structure (the absolute value map) on the operator system which we would like to avoid.

The aforementioned applications based on the spectral localizer suggests that one should look for invertible, or non-degenerate elements instead.
In fact, this is nicely aligned with the elegant description of C^*-algebraic K-theory K_0(A) as the Witt group of hermitian forms over A (see <cit.>). Indeed, these are given by invertible self-adjoint elements in M_n(A). It is precisely the matrix structure of the operator system that allows us to formulate such a notion of hermitian forms for operator systems as well. Based on this, we will develop  and analyze K-theoretic invariants for operator sysetems. 




This paper is structured as follows. After some brief background on operator systems and unital completely positive (ucp) maps, we introduce in Section <ref> a notion of hermitian forms in a unital operator system E. These are given by non-degenerate self-adjoint elements x ∈ M_n(E).
A natural notion of homotopy equivalence allows to introduce sets (E,n) of equivalence classes up to homotopy, which are labeled by the matrix size n. We will show that they are invariants for the operator system structure and compatible with the notion of direct sum. A semigroup of hermitian forms then appears when we introduce maps (E,n) →(E,n+1) and consider the direct limit (E,n). The K_0-group of the operator system is then defined to be the Grothendieck group of this semigroup. We show that it coincides with C^*-algebraic K-theory in case the unital operator system is a unital C^*-algebra, and that it is stably equivalent when taking matrix amplifications. It is not to be expected that this notion of K_0-theory behaves nicely with respect to ucp maps, as can be seen already in the simple case of a ucp map given by compression by a projection. Nevertheless, for some maps —including complete order isomorphisms— we have induced maps between the corresponding K_0-groups. 



In Section <ref> we consider the extension of K-theory to non-unital operator systems in the sense of <cit.>. This allows to formulate and prove the stability of K_0-theory, which as a consequence of <cit.> then shows invariance of K_0 under Morita equivalence. 
In Section <ref> we return to our initial motivation and formulate the spectral localizer <cit.> in terms of the K-theory groups for the spectral truncations. An illustrative example is given by spectral compressions of projections on the torus. 






 §.§ Acknowledgements

I would like to thank Alain Connes, Jens Kaad, Matthew Kennedy, Marc Rieffel, and Steffen Sagave for fruitful discussions and suggestions. I thank Malte Leimbach for a very careful proofreading of a draft of this paper. 

  



§ BACKGROUND ON OPERATOR SYSTEMS


We start by briefly recalling the theory of operator systems, referring to <cit.> for more details. 


A unital operator system (E,e) is a matrix-ordered *-vector space E, equipped with an Archimedean order unit e. A map ϕ: E → F between operator systems determines a family of maps ϕ^(n) : M_n(E) → M_n(F) given by ϕ^(n)([x_ij]) = [ϕ(x_ij)]. A map ϕ: E → F between unital operator systems is called completely positive if each ϕ^(n) is positive (n ≥ 1). We also abbreviate  completely positive by cp, and unital completely positive by ucp. 

A dilation of a ucp map ϕ: E → B()̋ of a unital operator system is a ucp map ψ: E → B(), where  is a Hilbert space containing $̋ such thatP_ψ̋(x)|_=̋ ϕ(x)for allx ∈E. 
The ucp mapϕis called  maximal if every dilation ofϕis obtained by attaching a direct summand.




A non-zero cp mapϕ:E →B()̋is said to be  pure if the only cp maps satisfying0 ≤ψ≤ϕare scalar multiples ofϕ.


We may viewEas a concrete operator system in theC^*-algebraC^*(E)it generates; in this case, we say that a ucp mapϕ: E →B()̋has the  unique extension property if it has a unique ucp extension toC^*(E)which is a*-representation. If, in addition, the*-representation is irreducible, it is called a  boundary representation<cit.>. The following result is well-known in the literature <cit.> but for completeness we include a proof. 

  Let ϕ: E → B()̋ be a ucp map. Then
  
  * ϕ is maximal if and only if it has the unique extension property.
    
  * ϕ is pure and maximal if and only if it is a boundary representation. 
    
(1) is <cit.>.
(2) If the extensionϕ̃: C^*(E) →B()̋ofϕis reducible, then there exists a non-trivial projectionP ∈ϕ̃(C^*(E))' . Then the mapψ:E →B()̋defined byψ(x) = P ϕ̃(x)is cp and0 ≤ψ≤ϕ. Butψ(1) = Pwhileϕ(1) = 𝕀_$̋ so that ψ is not a scalar multiple of ϕ; hence ϕ is not pure.

For the other implication, suppose that ϕ is a boundary representation and take a cp map ψ such that 0 ≤ψ≤ϕ. By Arveson's extension theorem, we have cp maps ψ̃ and ϕ-ψ from C^*(E) to B()̋. Since ψ̃+ ϕ-ψ extends ϕ it follows from the unique extension property of ϕ that ϕ-ψ =ϕ̃- ψ̃. Consequently, ψ̃≤ϕ̃ so that by <cit.> there exists an operator T ∈ϕ̃(C^*(E))' such that

    ψ̃(a) = T ϕ̃(a) ;     ∀ a∈ C^*(E).

Since ϕ̃ is irreducible, it follows that T = t ·𝕀_$̋ for somet ∈[0,1]so thatψ̃= t ϕ̃and, consequently, alsoψ= t ϕ. Henceϕis pure.      


§ K-THEORY FOR UNITAL OPERATOR SYSTEMS

Even though in an operator system we cannot speak about invertible elements, we may use the pure and maximal ucp maps to introduce the following notion of nondegeneracy. 


Let (E,e) be a unital operator system.  A self-adjoint element x=x^* ∈ M_n(E) is called a  hermitian form if it is non-degenerate in the sense that there exists g>0 such that for all pure and maximal ucp maps  ϕ: E → B()̋ we have


    |ϕ^(n)(x)| ≥ g ·𝕀_^̋⊕ n


The smallest real number g>0 such that (<ref>) holds is called the  gap of x.


We denote the set of hermitian forms contained inM_n(E)byH(E,n). 
The following result will be of crucial importance to us, as it makes it feasible to check the non-degeneracy condition in concrete cases:

A self-adjoint element x ∈ M_n(E) is a hermitian form if and only if _E^(n)(x) is an invertible element in the C^*-envelope C^*_(E). Moreover, x has gap g>0 if and only if |_E^(n)(x)| ≥ g · 1^⊕ n_C^*_(E).
  
In <cit.> theC^*-envelope ofEis constructed as the direct sum of all boundary representations(_̋σ, σ):_E : E →⊕_σ B( _̋σ ).Now ifx ∈Ethen_E(x)is invertible if and only if| _E(x)|≥g ·𝕀_⊕_σ(_̋σ), which holds if and only if|σ(x)| ≥g ·𝕀__̋σfor all boundary representationsσ. A similar statement holds ifx ∈M_n(E). Since by Proposition <ref> a ucp map is a boundary representation if and only if it is pure and maximal, the result follows.




A trivial example of a hermitian form is the order unitx = e(with gapg =1). Other examples that in fact motivated the above definition are


  *  (Hermitian forms and Witt groups)  The relation between the K-theory and the Witt group for rings <cit.> and (unital) C^*-algebras <cit.> stresses the role played by hermitian forms. In fact, any hermitian form on a finitely generated projective module of the form e A^n is described by a self-adjoint element x ∈ e M_n(A)e. The fact that the form is non-degenerate translates to invertibility of the hermitian element h = x+ (1-e) ∈ M_n(A), so that h^2 ≥ g^2 · 1_A for some g>0. But then 
  
    x^2      = (ehe)^2 = eh^2 e  ≥ g^2 · e,

  since h and e commute. We find that x is a hermitian form in the operator system (eM_n(A)e,e)  with gap g (note that since eM_n(A)e is a C^*-algebra, it coincides with its C^*-envelope). This should also explain the above terminology.

Note that in this case one may just as well consider the invertible element x+ (1-e) as a hermitian form in M_n(A).

  

  *  (Almost projections and quantitative K-theory) In <cit.> quantitative (aka controlled) K-theory K_0^ϵ,r(A) of a filtered C^*-algebra A = (A_r) was defined in terms of ϵ-r-projections,  i.e. elements p ∈ A_r such that p^2 - p < ϵ where ϵ < 1/4. But then x= 1_A -2p is a hermitian form in the unital operator system (A_r,1_A) with gap g = √(1-4 ϵ) >0 since
  
x^2 = 1_A - 4 (p-p^2) ≥ (1-4 ϵ )· 1_A

  *  (Projections in operator systems)
  In <cit.> projections in unital operator systems are abstractly defined. These then turn out to be precisely those elements p ∈ E such that _E(p) is a projection in the C^*-envelope. If we set x= e-2p we find that _E(x)^2 =e so that again such projections define hermitian forms (with gap 1). 

  

  *  (The even spectral localizer) The even spectral localizer <cit.> is defined as a spectral compression x = P H P of an invertible self-adjoint element H in a C^*-algebra A ⊆ B()̋ by a projection in $̋. If the spectrum ofHdoes not intersect with the interval[-g,g]then we compute for the operator system(PAP,P)that
  
    x^2 =  P H P P H P =P H^2P + P H[P,H] P  = P H^2P + P [P,H][P,H] P

  Hence if we setδ = [P,H]we find thatx^2 ≥ (g^2 -δ^2) Pin theC^*-extensionC^*(PAP) ⊆ B()̋. By the universal property of theC^*-envelope, there is a surjective*-homomorphismρ: C^*(PAP) → C^*_(PAP)which ensures that_E(x)^2 ≥ (g^2 -δ^2) Pso thatxis a hermitian form with gap√(g^2- δ^2).



  *  (Spectral truncations of quanta of geometry)
  As a special case of an invertible self-adjoint element we may also consider the operatorY ∈ Adescribing the so-called quanta of geometry in <cit.>. It satisfiesY^2 = 1so thatY has spectrum contained in{ -1,1}. IfPis a projection, we find that the compressionsPYPdefine hermitian forms with gap√(1-δ^2)providedδ =  [P,Y]<1. 


  Let us continue the general treatment of hermitian forms in operator systems, and derive the following rigidity result. 

Let x be a hermitian form with gap g>0. If y =y^* ∈ M_n(E) with  x - y≤ϵ for some ϵ  < g^2/2 x, then y is a hermitian form with gap √(g^2  - 2 ϵ x). 

The norm estimate implies that-2 ϵ x  e ≤_E (x)_E (y-x) + _E(y-x)_E( x) ≤ 2  ϵ xeHence by writingy = x + (y-x)and noting that_E (y-x)^2 ≥ 0we find that

    _E(y)^2    = _E (x)^2 + _E (x)_E (y-x) + _E(y-x)_E( x) + _E (y-x)^2
      
       ≥ ( g^2  - 2 ϵ x  )e.

We end this subsection by another standard operation of hermitian forms, which is their direct sum.

  Let x ∈ H(E,n) and x' ∈ H(E,n') be hermitian forms with respective gaps g,g'. Their  direct sum is the hermitian form given by
  
x ⊕ x' = [  x  0;  0 x' ]∈ H(E,n+n')

which has gap equal to min{ g,g'}. 
  

 §.§ Homotopy equivalence of hermitian forms

One of the defining properties of an operator system is its matrix-order structure. This motivates us to consider homotopies of hermitian forms inH(E,n)for a fixed matrix sizen ≥ 0, as follows:

  Let x,x' ∈ H(E,n) be hermitian forms. We say that x ∼_n x' if there exists a hermitian form x̃∈ H(C([0,1]) ⊗ E),n) such that
  x̃(0) = x ;     x̃(1) = x'
  
 
 
 
We denote the equivalence class of a hermitian form x ∈ H(E,n) by [x]_n, and the set of equivalence classes of hermitian forms in H(E,n) by (E,n), or, equivalently,
(E,n) = π_0 (H(E,n)).


This should be confronted with the usual notion of homotopy equivalence in, say, K-theory forC^*-algebras, wherenis allowed to vary (see also Section <ref> below). The following is immediate:

  Let E and F be unital operator systems. If ϕ:E → F is a ucp map for which there exists a *-homomorphism ϕ̃:C^*_(E) → C^*_(F) that makes the following diagram commute,
  
    E [r]^ϕ[d]__E    F  [d]__F
    
        C^*_(E) [r]^ϕ̃    C^*_(F)

  then there is an induced map ϕ^*:  (E,n) →(F,n) defined by
  ϕ^* ([x]_n) = [ϕ^(n)(x)]_n.
  



  If E and F are completely order isomorphic then (E,n) ≅(F,n) for all n ≥ 0.
  
  Consider E=. Then (,n) is the set of homotopy equivalence classes of invertible hermitian n × n matrices. Any such matrix x can be diagonalized with a unitary matrix, and since the unitary group U(n) is connected, there is a homotopy between x and the corresponding diagonal matrix. In turn, this diagonal matrix is homotopic (in the space of invertible matrices) to the corresponding signature matrix, which is unique up to ordering.  In other words, (,n) can be parametrized by the signature s of 
  the hermitian forms
  , yielding an isomorphism
  (,n) ≅{  -n , -n+2 …, n-2, n }
  Note that the direct sum of two hermitian forms translates to the addition of the corresponding signatures: (s,s')↦ s+s'.
  
  Anticipating the discussion in Section <ref> there are maps _nm : (,n) →(,m) for n ≤ m given by
  _nm ([x]_n) = [ x ⊕ e_m-n ]_m.
  
In terms of the signature, 
we find that _nm (s) = s + m-n. Moreover, there are commuting diagrams for any m ≥ n ≥ 1:
 
    (,n) [rd]_ρ_n[rr]^_nm      (,m) [ld]^ρ_m

 where ρ_n([x]) is defined to be the so-called  negative index of inertia of x,  i.e. the number of negative eigenvalues of the hermitian form x. When expressed in terms of the signature s of x, we have ρ_n(s) =1/2 (n-s) from which commutativity of the diagram follows at once. This suggests that  is the direct limit (,n); we will come back to this soon (in Section <ref>).

  
Let E and F be unital operator systems. Then for any n we have (E⊕ F,n) ≅(E,n) ×(F,n).

Let(x,y)∈ M_n(E) ⊕ M_n(F)be a hermitian form; this has gapgiff_E ⊕ F(x,y)^2 ≥ g^2 · (e_E, e_F) _E (x)^2 ≥ g^2 e_E  and _E (y)^2 ≥ g^2 e_F.Moreover,(x,y) ∼ (x',y')iffx ∼ x'andy ∼ y'. 
From this the stated isomorphism follows. 


The following result is a useful computational tool for(E,n). Recall the multiplierC^*-algebraA_Eof the operator systemE:A_E := { a ∈ C^*_(E): a E ⊂ E }Let u be a unitary in M_n(A_E), and let x ∈ M_n(E) be a hermitian form with gap g. Then u x u^* is a hermitian form with gap g, and u x u^* ⊕ e_n is equivalent to x ⊕ e_n as hermitian forms in H(E,2n).   
  
Clearly,_E(u x u^*)^2 = u _E (x)^2 u^* ≥ g^2 e. For the homotopy equivalence, recall that by Whitehead's Lemma there exists a continuous path of unitariesw_t ∈ M_2n(A_E)such thatw_0 = [ 1 0; 0 1 ];     
w_1 = [   u   0;   0 u^* ].But thenw_t ( x ⊕ e_n) w_t^*is a self-adjoint element inC[0,1]⊗ M_2n(E), also with gapg, establishing an equivalence betweenx ⊕ e_nandu x u^* ⊕ e_n, as desired. 

  Let x,y ∈ H(E,n) be hermitian forms both with gap g for which x - y < ϵ such that ϵ < 4 g^2. Then there is a homotopy x̃ of hermitian forms in C([0,1])⊗ E with gap √(g^2 - ϵ^2/4) such that x̃(0) = x and x̃(1)=y.
  
We claim that a homotopy of hermitian forms with the indicated gap is given byx̃(t) = t x + (1-t)y. It follows from x - y < ϵthat 

    (_E(x-y))^2 ≤ϵ^2 e,

which implies that_E(x) _E(y)+_E(y) _E(x) ≥_E(x)^2 + _E(y)^2 - ϵ^2 e.We will use this to find the gap ofx̃as follows:

    _C([0,1])⊗ E( x̃)^2    = t^2 _E(  x)^2  + (1-t)^2   _E(y)^2 + t (1-t) (_E(x) _E(y)+_E(y) _E(x))   
       ≥ (t^2 g^2 + (1-t)^2 g^2 +  t(1-t)(2 g^2 - ϵ^2 ))e
       ≥ (g^2 -  ϵ^2/4)e.

  Let x ∈ H(E,n) be a hermitian form with gap g and suppose that y =y ^*∈ M_n(E) is such that x - y < ϵ for some ϵ >0 such that g^2 > 2 ϵx  + ϵ^2/4.
  Then there is a homotopy between x and y given by hermitian forms with gap √(g^2- 2 ϵ x  - ϵ^2/4).

From Lemma <ref> it follows that the elementyis a hermitian form with gap√(g^2-2ϵ x )and the same applies tox. By Proposition <ref> it now follows that there is homotopy betweenxandyof hermitian forms with gap√(g^2-2ϵ x - ϵ^2/4).


 §.§ Semigroup of hermitian forms and K_0-group

In the above, we have stressed the role of the matrix sizen ≥ 0for the invariants(E,n). However, in order to prepare for a comparison withK-theory forC^*-algebras we will need the limit object(E) := (E,n)that we will now construct.

Given a unital operator system(E,e)we consider the direct system of sets((E,n), _nm)where form ≥ n_nm: (E,n)    →(E,m)
    
    [x]_n    ↦ [x ⊕ e_m-n]_m,

We denote the direct limit of the direct system (<ref>) by(E,n). A more explicit description is given as follows: forx ∈ H(E,n)andx' ∈ H(E,n')we writex ∼ x'if there exists ak ≥ n,n'such that x ⊕ e_k-n∼_k x'⊕ e_k-n'inH(E,k). We will write[x]_E, or simply[x], for the equivalence class corresponding tox ∈ H(E,n)and(E) := ⨿_n (E,n)/_∼for the corresponding set of equivalence classes. The following is then clear from the definition of the direct limit. 

  The set (E) is the direct limit (E,n) of the direct system (<ref>). Moreover, it is a semigroup when equipped with the direct sum [x]+ [x'] = [x ⊕ x'] and identity element 0 = [e]. 



We now arrive at our tentative definition of K-theory for unital operator systems.

  Let (E,e) be a unital operator system. We define the K-theory group K_0(E) of E to be the Grothendieck group of (E). 


Before addressing the properties of the associationK_0(E)to an operator system, let us first check the consistency of our definition with the notion ofK_0-groups when the unital operator system is in fact a unitalC^*-algebra. 

  For a unital C^*-algebra A the group K_0(A) is isomorphic to the C^*-algebraic K-theory group of A. 
    
Since a hermitian formxin aC^*-algebra is invertible we may define a map between the semigroups (where𝒫(A)denotes the semigroup of projections inM_∞(A)up to homotopy equivalence):

    Φ: (A)    →𝒫(A)
    
      [x]    ↦[p= 12 (1-x |x|^-1)  ]

This map is well-defined, since ifx ∼ x'then also the corresponding projectionsp,p'are homotopy equivalent. It also maps the neutral element0 = [e]to0=[0] ∈𝒫(A), whilst being compatible with direct sums. Let us check it is bijective.

For injectivity, let[x],[x'] ∈(A)be mapped to[p],[p']and assume that[p]=[p']. The homotopyp_tof projections (t ∈ [0,1]) that implements the equivalence betweenp_0 = p, p_1 = p'can be written asp_t = 1/2 (1-y_t)in terms of a self-adjoint elementy_tsatisfyingy_t^2 = 1, to wity_t = 1-2 p_t. This a continuous family of hermitian forms such thaty_0 = x|x|^-1andy_1 = x' |x'|^-1while alsox |x|^-1∼ xvia the homotopyx |x|^-t(t ∈ [0,1]) of hermitian forms, and the same applies tox' |x'|^-1∼ x'. Hencex ∼ x'which proves thatΦis injective.

Surjectivity follows by taking a[p] ∈𝒫(A)and defines a hermitian formx = 1-2p. ThenΦ([x]) = [p]as desired. 

  
The next result prepares for invariance ofK_0under stable isomorphism (in Section <ref> below).

  Let E be a unital operator system and let N be a natural number. Then (E) is isomorphic to (M_N(E)) (and so are the corresponding K_0-groups).

We will give an explicit proof in the caseN=2. For anyn>0we define a map^E,n : M_n(E)→ M_n(M_2(E))by

    ^E,n (x)= [ x_11    0 
     0     e                     x_12    0 
     0     0                                       ⋯                     x_1n    0 
     0     0; x_21    0 
     0     0                     x_22    0 
     0     e                                       ⋯                                       ⋮;                   ⋮                                       ⋮                                       ⋱                                       ⋮; x_n1    0 
     0     0                                       ⋯                                       ⋯                     x_nn    0 
     0     e ]

Letu_σbe the permutation matrix corresponding to the permutationσ = [    1    2    3    ⋯    n  n+1    ⋯   2n;    1    3    5    ⋯ 2n-1    2    ⋯   2n ]This shuffles the columns and rows in such a way thatu_σ·^E,n(x)· u_σ^* = [   x   0;   0 e_n ]Sinceu_σ∈ M_2n() ⊆ M_2n(A_E)it follows from Lemma <ref> that^E,n(x) ∼ x, relative toE. Similarly, it follows that^E,n+n'(x⊕ x') ∼^E,n(x) ⊕^E,n'(x'), while the map^E,nalso respects non-degeneracy and self-adjointness. Note that also^E,n(e_n) = e_2n, and, in fact,^E,nis unital completely positive. As such, it is contractive, hence continuous so that it maps homotopy equivalences to homotopy equivalences. We conclude from all of this that the induced map^E: (E) ↦(M_2(E)), [x] ↦ [^E,n(x)]is a well-defined morphism of semigroups.

Let us show that^Eis surjective: take[y] ∈(M_2(E),m). Then by the above it follows that^M_2(E),m(y) ∼ y, relative toM_2(E). Upon identifyingM_m(M_2(E))withM_2m(E)there is an elementỹ∈ H(E,2m)satisfying^E,2m(ỹ) ∼^M_2(E),m (y). But then^E([ỹ ]_E) = [y]_M_2(E)which shows surjectivity.

For injectivity  of^Esupposex ∈ H(E,n), x'∈ H(E,n')are such that^E,n(x) ∼^E,n'(x'),  i.e. that^E([x]) = ^E([x']). Since^E,n(x) ∼ xand^E,n'(x') ∼ x'by the above, it follows thatx ∼ x'. But then[x] = [x']which shows that^Eis injective. This completes the proof. 

More generally, we may consider maps

    ^E,n_NM : M_n(M_N(E)) → M_n(M_M(E)) 
     
      x ↦ v_σ^* ·[     x     0;     0 e_M-N ]· v_σ,

in terms of a suitable permutation matrix v_σ shuffling the rows and columns to identify M_n(M_M(E)) with M_M(M_n(E)). The corresponding maps _NM: (M_N(E)) →(M_M(E)) then yield a direct system, and the above Theorem generalizes to an isomorphism (M_N(E)) ≅(E). 


 §.§ K_0 and maps between operator systems

The behavior of hermitian forms with respect to ucp maps as obtained in Proposition <ref> translates into the following: 

Let E,F be unital operator systems and ϕ:E → F a restriction of a *-homomorphism between the corresponding C^*-envelopes (so that Equation (<ref>) is satisfied). Then the induced map ϕ^* : (E) →(F) of semigroups is well-defined and induces a map between the corresponding K_0-groups (also denoted ϕ^*).

We already know from Proposition <ref> that hermitian forms are mapped to hermitian forms. Clearly the mapϕbehaves well with respect to direct sums of hermitian forms, and unitality ofϕimplies that[e_E]is mapped to[e_F]. Finally, as in Proposition <ref> homotopies are mapped to homotopies.

Let E,F be unital operator systems. If E and F are completely order isomorphic, then K_0(E) ≅ K_0(F).
  
This follows from the previous Proposition in combination with the fact that a unital complete order isomorphismϕ: E → Fextends to a unital*-isomorphismϕ̃: C^*_(E) → C^*_(F)<cit.>. 



The following is then immediate. 

  *  For every unital operator sytem E, 𝕀_E^* = 𝕀_K_0(E);
  
  *  If E, F,G are unital operator systems, and if ϕ:E → F, ψ:F → G are restrictions of *-homomorphisms between the respective C^*-envelopes, then ψ^* ∘ϕ^*= ψ^* ∘ϕ^*;
  
  * K_0({0}) = { 0};

    

We also record the following behaviour ofK_0with respect to direct sums of operator systems, which is a direct consequence of Lemma <ref>.

  Let E,F be unital operator systems. Then K_0(E ⊕ F) ≅ K_0(E) × K_0(F). 
  

§ NON-UNITAL OPERATOR SYSTEMS AND STABILITY OF K-THEORY


 §.§ Non-unital operator systems

Recall <cit.> ( cf.<cit.>) that the partial unitization of a non-unital operator systemEis given by the*-vector spaceE^♯ = E ⊕with matrix order structure:(x,A) ≥ 0  iff  A ≥ 0   and ϕ(A_^-1/2 x A_^-1/2 ) ≥ -1for all >0and noncommutative statesϕ∈𝒮_n(E), and whereA_ = 𝕀_n + A. The matrix order unitse^♯_ninM_n(E^♯)are given by the identity matrices𝕀_n ∈ M_n(). This turns(E^♯, e^♯)into a unital operator system. Moreover, given a completely contractive completely positive mapϕ:E → Fwe have that the canonical extensionϕ^♯: E^♯→ F^♯is a ucp map (<cit.>) . 

We now extend the definition of K-theory to non-unital operator systems as follows: 

  Let E be a non-unital operator systems and E^♯ its unitization. We define the sets 
 
    (E,n)    := π_0 ( { (x,A) ∈ H(E^♯,n) : A ∼_n 𝕀_n })

 The Grothendieck group of the direct limit semigroup is denoted by K̃_0(E).

Let E and F be operator systems. If E and F are completely isometric, completely order isomorphic, then (E) ≅(F) as semigroups. Consequently, in this case K̃_0(E) ≅K̃_0(F). 

This follows from the fact that for a completely isometric, complete order isomomorphismϕ:E → Fthe induced mapϕ^♯ :E^♯→ F^♯is a complete order isomorphism which furthermore respects the propertyA ∼_n 𝕀_n. 



Let us also compare this with with our previous definition of K-theory in the case of  unital operator systems. 


  Let E be a unital operator system and let E^♯ be its partial unitization. Then for any n ≥ 1 there are isomorphisms (E,n) ≅(E,n). Consequently, in this case K̃_0(E) ≅ K_0(E).
  
In the unital case, there is a unital complete order isomorphism betweenE^♯andE ⊕given by(x, λ) ↦ (x+λ e, λ), where we have equippedE ⊕with the induced direct sum order structure <cit.>. By Lemma <ref> we have thatH(E ⊕,n) = H(E,n) × H(,n)from which it follows that{ (x,A) ∈ H(E^♯,n) : A ∼_n 𝕀_n }≅ H(E,n) ×{ A ∈ H(,n) : A ∼_n 𝕀_n }This isomorphism clearly respects direct sums and the unit, so that taking homotopy equivalence classes of this set of hermitian forms yields the statement. 


 §.§ Stability of K_0

One of the crucial features ofK-theory forC^*-algebras is that it is Morita invariant, or, equivalently, invariant under stable isomorphism. We will now establish thatK-theory for unital operator systems shares this property,  i.e. we will relate theK_0-groups ofEand𝒦⊗ E. 


Consider a direct system of Hilbert subspaces{ P_N }̋_N ≥ 0in$̋ with P_N ≅̋^N so that we realize = ()̋≅ M_N(). We then obtain the stabilization of E by the following series of completely contractive, completely positive maps defined for M ≥ N:
κ_NM : M_N(E) → M_M(E),      x ↦[     x     0;     0 0_M-N ],

The inductive limit of the sequence (M_N(E),κ_NM) is 𝒦⊗ E with connecting maps κ_N,∞ : M_N(E) →𝒦⊗ E.


  The maps 
  (κ_NM^♯)_*: (M_N(E),n) →(M_M(E),n) induced by κ_NM make the following diagram commute for any n:
(M_N(E),n) [d]_≅[rr]^- (κ_NM^♯)_*      (M_M(E),n)[d]_≅
 (M_N(E),n) [rr]^-_NM       (M_M(E),n)

where _NM is defined in Equation (<ref>) and the vertical isomorphisms are the ones from Proposition <ref>.

The maps κ_NM^♯: M_N(E)^♯→ M_M(E)^♯ and their amplifications are given explicitly by

(κ_NM^♯)^(n) : M_n(E^♯) → M_n( M_N(E)^♯) ,      (x,A) ↦( u_σ[        x        0;        0 0_n(M-N) ] u_σ^* , A ).

Here u_σ is a permutation matrix similar to the one appearing in the proof of Theorem <ref>: it  identifies M_M(M_n(E)) with M_n(M_M(E)) by a suitable shuffle of rows and columns. 
When we identify M_N(E)^♯ with the direct sum operator systems M_N(E) ⊕ ( cf. Proposition <ref>), then the map (κ_1N^♯)^(n) becomes

(x,A) ∈ M_n(M_N( E)) ⊕ M_n() ↦(u_σ( [       x       0;       0 A e_M-N ]) u_σ^*, A ) ∈ M_n( M_M(E)) ⊕ M_n() 

If we assume that A ∼_n _n we see that up to homotopy this map coincides with _NM, which completes the proof.

For a unital operator system E we have K̃_0(𝒦⊗ E) ≅ K_0(E). 
  
The above Lemma, in combination with Theorem <ref> and Proposition <ref>, yields that (M_N(E)) ≅(E). We will show that the universal map u: (M_N(E)) →(⊗ E) for the direct limit in the following diagram is an isomorphism:
(M_N(E)) [rd] @/^-2.0pc/[rdd]_(κ_N∞^♯)_*[rr]^(κ_NM^♯)_*      (M_M(E)) [ld] @/^2.0pc/[ldd]^(κ_M∞^♯)_*
   (M_N(E)) @–>[d]_u   
   (⊗ E)



For injectivity of u, take [(x,A)] ∈(M_N(E),n) and [(x',A')] ∈(M_N(E),n') for some N and n,n'. 

Assume that [(κ_N,∞^(n)(x),A)] = [(κ_N,∞^(n')(x),A)] as elements in (⊗ E). In other words, there exists a family (x̃,Ã) ∈ H ( C[0,1] ⊗ (𝒦⊗ E)^♯ ,k) for some k ≥ n,n' such that the following hold:

    x̃(0) = κ_N,∞ ^(n)(x) ⊕ 0_k-n ,     Ã(0)=A ⊕_k-n, 
    x̃(1) = κ_N,∞^(n) (x') ⊕ 0_k-n'    Ã(1) = A' ⊕_k-n'.

Consider the compressions R_M: 𝒦⊗ E  → M_M(E) given by y ↦ (P_M ⊗ e)y (P_M ⊗ e). Then for the above x ∈ M_n(M_N(E)) we have
κ_NM^(n)(x) = R_M^(n) (κ_N ∞^(n) (x)) ≡  (P_M ⊗ e_n) κ_N ∞^(k)(x) (P_M ⊗ e_n).

and similarly for x'. We may also compress the homotopy to give (R_M^♯)^(k)((x̃, Ã))  = (R_M^(k) (x̃),Ã ) with end points:

    (R_M^♯)^(k)( ( (x̃(0)),Ã (0)) = ( κ_NM^(n)(x) ⊕ 0_k-n,  A ⊕_k-n)
    
      (R_M^♯)^(k)( ( (x̃(1)),Ã (1)) = ( κ_NM^(n')(x') ⊕ 0_k-n',  A' ⊕_k-n')

A priori the family (R_M^♯)^(k)((x̃, Ã)) lies in M_k(C[0,1] ⊗  (M_M( E))^♯) so we need to show that it completely lies in ( M_M(E),n), at least for some sufficiently large M.
In other words, we need to show that the gap of (R_M^(k) (x̃),Ã)  is strictly positive, while Ã(t) ∼𝕀_k for all t. The latter fact is clear from the assumption that both A ∼_n and A' ∼_n'. For the former claim,  we identify C[0,1] ⊗ (M_M (E))^♯ with C[0,1] ⊗ (M_M (E) ⊕) which maps (R_M^(k) (x̃),Ã  ) to (R_M^(k) (x̃) + P_M ⊗Ã e,Ã) since P_M ⊗ e is the order unit of M_M(E). In order to check that the latter has strictly positive gap, we realize M_k((⊗ E)^♯)  as concrete operators in B( ⊗̋( '̋)^⊕ k) for some Hilbert space '̋. We compute


    (R_M^(k)(x̃ )+ P_M ⊗Ã  )^2    = ((P_M ⊗ e_k) (x̃ + 𝕀_⊗̋Ã )(P_M ⊗ e_k)   )^2
        =  ((P_M ⊗ e_k) (x̃ + 𝕀_⊗̋Ã )^2(P_M ⊗ e_k)   
            +  (P_M ⊗ e_k  )[P_M ⊗ e_k ,x̃ + P_M ⊗Ã ]^2 (P_M ⊗ e  )
       ≥ g^2(P_M ⊗ e_k)  - [P_M ⊗ e_k , x̃] ^2.

We have used that (x̃(t),Ã(t)) ∈ H( (⊗ E)^♯,k) has gap given by some g>0 for all t. 
Moreover, [P_M ⊗ e_k , x̃ ]→ 0 as N →∞ so that it follows that (R_M^(k)(x̃),Ã ) is a hermitian form. We thus find that [(R_M^(k)(x̃)(t),Ã(t) )] ∈( M_M(E),k) for all t, so that [(κ_NM^(n)(x),A)] = [(κ_NM^(n')(x'), A)] in (M_M(E)). Since we also know that κ_NM^♯ induces isomorphisms between (M_N(E)) and (M_M(E)) we conclude that [(x,A)] = [(x', A)] as elements in (M_N(E)).


For surjectivity, take an arbitrary [(x,A)] ∈( ⊗ E, n). We may approximate x by a finite rank operator (for instance using the compression R_N^(n)) so that for all ϵ>0 there exists a x_0∈ M_n(M_N(E)) for some N so that
 x - κ_N,∞^(n)(x_0)  < ϵ.

We then also have
 (x,A) -  (κ_N,∞^♯)^(n) (x_0,A) < ϵ
so that if (x,A) has gap g and we choose ϵ small enough so that g^2 - ϵ x -  ϵ^2/4 >0, then (x,A) and (κ_N,∞^♯)^(n) (x_0,A) are homotopy equivalent by Corollary <ref>. The element (x_0,A) is the sought-for element in (M_N(E)) ≅(E) that maps to [(x,A)].

Let E and F be Morita equivalent unital operator systems. Then K_0(E) ≅ K_0(F). 
  
In <cit.> it is shown that E and F are Morita equivalent whenever ⊗ E ≅⊗ F via a completely isometric, complete order isomorphism.


§ APPLICATION TO THE SPECTRAL LOCALIZER

In <cit.> the spectral localizer was introduced as a powerful tool for computing index pairings. They relate a certain Fredholm index to the signature of a finite-dimensional matrix —the so-called spectral localizer. We will put it in the context of our notion of K-theory, realizing the spectral localizer as a 
map (E,n) →. 
the spectral localizer, including its relation to spectral flow, we refer to the excellent textbook <cit.> and references therein.


In order to describe the index map on (E,n), we start with an operator system spectral triple <cit.>. 

  A (unital)  operator system spectral triple is given by a triple (E,,̋D) where E is a unital operator system realized concretely so that E ⊆ C^*_(E) ⊆ B()̋, 
  and a self-adjoint operator D: (D) →$̋ such that 
  
  *  the commutators [D,x] extend to bounded operators for all x ∈ for a dense *-subspace ⊆ E;
    
  *  the resolvent (i+D)^-1 is a compact operator.
  
  An operator system spectral triple is called  even if in addition to the above, there is a grading operatorγon$̋ (so that γ^* = γ, γ^2 = 1_$̋) which commutes with allx ∈ Eand anti-commutes withD. Otherwise, it is called  odd.

In the even case, we can decompose=̋_̋+ ⊕_̋-according to the eigenvalues ofγ, and decompose accordinglyD = [     0   D_0; D_0^*     0 ].Given an even operator system spectral triple, a parameterκ >0and a hermitian formx ∈ H(,n)we now define the  even spectral localizer<cit.> as

    L_κ (D,x)    = [             x   κ (D_0)^⊕ n; κ (D_0^*)^⊕ n           - x ].

  Let (E,,̋D) be an even finite-dimensional operator system spectral triple. 






  
  Let x ∈ H(,n) with gap g>0 and let κ_0 = g^2   [D,x] ^-1. Then 
  the index map defined by the signature of the spectral localizer, 
     _D([x]) = 1/2 (L_κ(D,x)),
  
       
is constant for all κ < κ_0 and invariant under homotopy equivalence. Consequently, it induces a map _D: (,n) →.

  
Without loss of generality we taken=1and compute very similar to <cit.> that

    L_κ(D,x)^2 = [ κ^2 D_0 D_0^* + x^ 2            κ [D_0,x];          κ [D_0,x]^*   κ^2 D_0^* D_0 +x^2 ]≥( g^2 -κ [D,x]) 1_M_2(B()̋).

HenceL_κ(D,x)is invertible (and thus has well-defined signature) providedκ< κ_0and, moreover, the signature is constant for allκ <κ_0as no eigenvalues will cross the origin. 


Also, one may easily check that(L_κ(D,x ⊕ x')) =  (L_κ(D,x ))+  (L_κ(D,x' )).In order to see that (L_κ(D,e )) = 0consider an eigensystem{v_λ =  (v_λ^+, v^-_λ)}_λin_̋+ ⊕_̋-such thatD_0 v^-_λ = λ v^+_λandγ v_λ^± = ± v_λ^±. We may write the matrix ofL_κin this eigensystem as⟨ v_λ, L_κ(D,x) v_λ'⟩  =δ_λλ'[  1  λ;  λ -1 ]∼δ_λλ'[  √(1+λ^2)         0;         0 -√(1+λ^2) ]and the signature of the latter matrix vanishes. 


Consider now a homotopyx̃inH(E,n)betweenx = x̃(0)andx' = x̃ (1). We defineκ_0' = g^2 / sup_t  [D,x̃(t)]and note that for allκ < κ_0'L_κ(D,x̃)^2 ≥ g^2 - κsup_t  [D, x̃(t)]by a computation similar to the one in Eq. (<ref>). So as long asκ < κ_0'we find that L_κ(D,x̃(t))is constant int. We combine this with the fact thatκ < κ_0' < κ_0to conclude that L_κ(D,x̃(0)) =  L_κ(D,x̃(1)). This completes the proof. 


We may now rephrase the main results of <cit.>. For an invertible self-adjoint elementx ∈ Aone considers the class[p=1/2 (1-x|x|^-1)]in the K-theory of theC^*-algebraA. Given a spectral triple(A,,̋D)one would like to compute the index pDp. As shown in  loc.cit. this index may be computed in terms of a spectrally truncated (operator system) spectral triple(P A P,P ,̋ P D P)for a spectral projectionP(ofD) of sufficiently high rank. Indeed, we then have

    (pDp)= _PDP  ([P x P ])

where the map on the right-hand side is the index map on(PAP,n)that appeared in Proposition <ref>. 


 §.§ Example: spectral localizer on the torus

Let us illustrate the spectral localizer for a spectrally truncated two-torus. First, recall from <cit.> the class of projections on the torus ( cf.<cit.>, much inspired by the so-called Powers–Rieffel projections on the noncommutative torus <cit.>:

    p = [      f g+hU^*;  g+ hU    1-f ]∈ M_2(C(^2)),

wheref,g,hare real-valued (periodic) functions of the first variablet_1, andUis a unitary depending only on the second variablet_2, sayU(t_2)=e^i m t_2. The projection propertyp^2=ptranslates into the two conditionsgh =  0,      g^2 + h^2 = f-f^2.A possible solution of these relations is given by0 ≤ f ≤ 1   such that  f(0)=1,    f(π) = 0 
,and theng = χ_[0,π]√(f-f^2)andh = χ_[π,2π]√(f-f^2), whereχ_Xis the indicator function for the setX(see Figure <ref>).

The Dirac operator on the two-torus is defined on the coreC^∞(^2) ⊗^2byD= [     0   D_0; D_0^*     0 ];      D_0 = i ∂_t_1 + ∂_t_2.The corresponding spectrum is{±√(n_1^2+n_2^2)}_n_1,n_2∈.

Let us now consider a spectral projection corresponding to a (discrete) ball of radiusρ,  i.e.P_ρ = χ(|D|≤ρ). We  consider the corresponding spectral compression ofY=1-2p,  i.e.P_ρ  Y P_ρ  = [          P_ρ -2 P_ρ  fP_ρ  -2P_ρ gP_ρ -2P_ρ h U P_ρ; -2P_ρ gP_ρ -2P_ρ h U^*P_ρ          -P_ρ  +2P_ρ fP_ρ ]∈ M_2(P_ρ  C^∞(^2)P_ρ )For suitableP_ρthese are hermitian forms in the truncated operator system, that is,P_ρ YP_ρ∈(P_ρ  C(^2)P_ρ ,2). 



The spectral localizer is now given by the following matrix:L_κ,ρ:=
L_κ(P_ρ D P_ρ,P_ρ Y P_ρ)  = [          P_ρ YP_ρ    κ P_ρ  D_0 P_ρ; κ P_ρ   D_0^* P_ρ         -P_ρ YP_ρ ]The main result of <cit.> ( cf. Eq. (<ref>)) now implies that for suitableρ,κthe index ofpD pcan be expressed in terms of the signature of the above spectral localizer.  More precisely, in terms of the above index map on(P_ρ C(^2)P_ρ , 2)we have:(pDp) = _P_ρ DP_ρ([P_ρ YP_ρ ])  ≡1/2Sig L_κ,ρIn Figure <ref> we illustrate the resulting signature by showing the negative and positive eigenvalues ofL_κ,ρ. We find already for lowρthat the signature of the spectral localizer is equal to (twice) the winding numberm, that is to say, the index ofpD p. 





§ OUTLOOK

We have proposed a generalization of K-theory forC^*-algebras to operator systems, which in the spirit of Witt was based on hermitian forms. Several aspects are still to be developed, and which we leave for future research. This includes a careful study of the functorial properties ofK_0and, in particular, howK_0behaves with respect to approximations results. For instance for approximations ofC^*-algebra by finite-dimensional algebras in <cit.> where the maps are approximately multiplicative, or approximately order-zero cpc maps. Or in the context of quantum metric spaces where it turns out that small Hausdorff distance between two such spaces —the state spaces of twoC^*-algebrasA,B— allows one to relate projections inAto projections inB<cit.>. 

Another open problem is to identify the higher K-groups, starting withK_1. Again, one needs to find a analogue of the assumption of being unitary. However, just demanding invertibility appears too weak; one can easily show that the set of all invertible elements in a finite-dimensional operator system is contractible —arguing much as in the case ofGL(n,). Nevertheless, from the odd spectral localizer <cit.> we know that some index-theoretic information is maintained after spectrally truncating a unitary. Again this fact will be leading in the development ofK_1, which we also leave for future research. This extends of course to finding an analogue of Bott periodicity. 

Dually, one may be interested also in defining K-homology for operator systems. Note here that the notion of a Fredholm module makes perfect sense for unital operator systems realized concretely in Hilbert space, but that this immediately gives rise to a Fredholm module for the pertinentC^*-extension. The task is thus to find the right notion of equivalence. 







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