Grassmann manifold of a Hilbert Jordan algebra Takaaki NOMURA

A (real) Hilbert Jordan algebra (JH-algebra for short) is a (real) Jordan algebra as well as a (real) Hilbert space with an associative inner product ⟨ · | · ⟩. In my talk, I will state some properties of the manifold ℑ_{p} of rank p idempotents in a topologically simple JH-algebra V. The manifold structure of ℑ_{p} is given in such a way that the tangent space T_{a}(ℑ_{p}) at a ∈ ℑ_{p} is identified with the Peirce 1/2space V_{1/2}(a) and that the inclusion ℑ_{p} ↪ V is an embedding. The manifold ℑ_{p} is thus a Riemannian manifold. For every a ∈ ℑ_{p}, let E_{1/2}(a) : V → V_{1/2}(a) be the Peirce projection. Then we have the canonical connection ∇ defined by ∇_{X}Y (a) := E_{1/2}(a)(Y′(a)(X(a))), where X, Y are tangent vector fields on ℑ_{p} considered as V -valued functions such that the values at a are in V_{1/2}(a). For each x ∈ V_{1/2}(a), the curve ℝ ∋ t ↦ γ(t) := (exp 4t [ L(x), L(a) ])a (L(·) being the Jordan multiplication) is a geodesic in ℑ_{p} such that γ(0) = a, γ̇(0) = x. It has the expression

γ(t) = [cos(2tL(x^{2})^{1/2})] a + \frac{1}{2} \frac{sin(2tL(x^{2})^{1/2})}{L(x^{2})^{1/2}} x + \frac{1}{2} \frac{I − cos(2tL(x^{2})^{1/2})}{L(x^{2})} x^{2}.

The Riemannian distance dist(a, b) on ℑ_{p} equals ∥Arcsin(a − b)∥, where Arcsin is defined through an ‘operator calculus’ on V regarded as a JTS. The sectional curvature k_{a}(x, y) (x, y ∈ V_{1/2}(a), ∥x∥ = ∥y∥ = 1, ⟨ x | y ⟩ = 0) is expressed as

(1)  k_{a}(x, y) = 2 ⟨ (x □ x − P (x))y | y ⟩

by using the operators x □ x := {x, x, ·} and P (x) := {x, ·, x} defined through the JTS structure of V_{1/2}(a). From (1) we obtain the estimate

0 ≦ k_{a}(x, y) ≦ 4 min( ||| x |||_{1/2}^{2}, ||| y |||_{1/2}^{2}) ≦ 2,

where ||| · |||_{1/2} is the spectral norm of the Hilbert JTS V_{1/2}(a). Finally, k_{a}(x, y) = 0 if and only if x, y are contained in a flat subspace of the JTS V_{1/2}(a).

Department of Mathematics, Faculty of Science, Kyoto University, Sakyo 606-01, Kyoto, Japan (nomura@kusm.kyoto-u.ac.jp)