Higher K-groups for operator systems

We extend our previous definition of K-theoretic invariants for operator systems based on hermitian forms to higher K-theoretical invariants. We realize the need for a positive parameter <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="delta"> <mml:semantics> <mml:mi> δ </mml:mi> <mml:annotation encoding="application/x-tex">\delta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> as a measure for the spectral gap of the representatives for the K-theory classes. For each <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="delta"> <mml:semantics> <mml:mi> δ </mml:mi> <mml:annotation encoding="application/x-tex">\delta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and integer <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p greater-than-or-equal-to 0"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo> ≥ </mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">p \geq 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> this gives operator system invariants <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper V Subscript p Superscript delta Baseline left-parenthesis minus comma n right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">V</mml:mi> </mml:mrow> <mml:mi>p</mml:mi> <mml:mi> δ </mml:mi> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mo> − </mml:mo> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {V}_p^\delta (-,n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , indexed by the corresponding matrix size. The corresponding direct system of these invariants has a direct limit that possesses a semigroup structure, and we define the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K Subscript p Superscript delta"> <mml:semantics> <mml:msubsup> <mml:mi>K</mml:mi> <mml:mi>p</mml:mi> <mml:mi> δ </mml:mi> </mml:msubsup> <mml:annotation encoding="application/x-tex">K_p^\delta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -groups as the corresponding Grothendieck groups. This is an invariant of unital operator systems, and, more generally, an invariant up to Morita equivalence of operator systems. Moreover, there is a formal periodicity that reduces all these groups to either <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K 0 Superscript delta"> <mml:semantics> <mml:msubsup> <mml:mi>K</mml:mi> <mml:mn>0</mml:mn> <mml:mi> δ </mml:mi> </mml:msubsup> <mml:annotation encoding="application/x-tex">K_0^\delta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> or <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K 1 Superscript delta"> <mml:semantics> <mml:msubsup> <mml:mi>K</mml:mi> <mml:mn>1</mml:mn> <mml:mi> δ </mml:mi> </mml:msubsup> <mml:annotation encoding="application/x-tex">K_1^\delta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We illustrate our invariants by means of the spectral localizer.