# New PDF release: Amenable Banach algebras

By Pier, Jean-Paul

ISBN-10: 0470210664

ISBN-13: 9780470210666

ISBN-10: 0582014808

ISBN-13: 9780582014800

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Point" and " l i n e " are considered to be dual concepts and "incidence," as extended in this section, to be self-dual. The definition of each new concept will be accompanied by the definition of its dual concept, unless, of course, it is self-dual. " Two new concepts follow. 1. AXIOMS FOR A PROJECTIVE PLANE 31 Points P, Q, R (lines /, m, ri) are said to be collinear (concurrent) if there exists a line k (a point S) such that P, Q, Ro k (/, m,no S). In what follows, whenever applicable, the principle of duality will be applied without explicit statement to this effect.

B) Prove the theorem in general. (c) Use (^, if, o) of part (a) to interpret the comments following the theorem. 54 4. 3. THE IMBEDDING THEOREM Throughout this section ( ^ , if l5 o ^ is to be an arbitrary affine plane. We shall construct a projective plane (P, L, o) containing a deletion subgeometry (P',L', Θ) isomorphic to the given affine plane (^ l 5 5£x, o j . We shall say that the affine plane is imbedded, as a deletion subgeometry, into a projective plane. Further, we shall show that there is essentially one (in the sense of isomorphism) such projective plane.

The projective plane of the last theorem is unique up to isomorphism. PROOF. Let us continue in the notation of the last theorem. Let (J, Ji, Δ) be a projective plane and (J', Ji', Δ) the deletion subgeometry associated with m* e Jt. l5 o x ). ) Define, for Peâ, 71 tL(n0P) \L(À0l) if P A m * , if PAm*, where / e Jt' such that/ Δ Ρ ; and define, for / e ,M, /{V} UU if l*m*9 if l = m*. ) First, we must show that nP is well defined in the case P A m*. To this end, note that there exists / G M such that / Δ P and / φ m*, that is, / e Jt'.