An ECA f π f italic_f is additive is there are binary operators β direct-sum \oplus β and β tensor-product \otimes β (not necessarily distinct), and a neutral element e π e italic_e such that for all x π₯ x italic_x , x β e = e β x = x direct-sum π₯ π direct-sum π π₯ π₯ {x\oplus e=e\oplus x=x} italic_x β italic_e = italic_e β italic_x = italic_x and with
Let ( N , h ) π β (N,h) ( italic_N , italic_h ) be a two-dimensional Lorentzian manifold. A map y : S β M normal-: π¦ normal-β π π y:S\rightarrow M italic_y : italic_S β italic_M is called wave map iff it satisfies the following differential equation:
where the trace is understood with respect to the metric h β h italic_h on N π N italic_N , that means in coordinates t , x π‘ π₯ t,x italic_t , italic_x which are orthogonal at a point p β N π π p\in N italic_p β italic_N :
at p π p italic_p .
Let π π {\mathfrak{K}} fraktur_K be a real Hilbert space. The algebra π β’ ( π ) π π {\mathfrak{A}}({\mathfrak{K}}) fraktur_A ( fraktur_K ) is the unital * * * -algebra with generators { Ο β’ ( ΞΎ ) : ΞΎ β π } conditional-set π π π π \{\omega(\xi):\xi\in{\mathfrak{K}}\} { italic_Ο ( italic_ΞΎ ) : italic_ΞΎ β fraktur_K } and relations
for all ΞΎ π \xi italic_ΞΎ , Ξ· β π π π \eta\in{\mathfrak{K}} italic_Ξ· β fraktur_K and Ξ» π \lambda italic_Ξ» , ΞΌ β π π π \mu\in\mathbf{R} italic_ΞΌ β bold_R .