Work in progress

On non-commutative Minkowski spheres

L. L. STACHÓ and W. WERNER 29/08/2011, Constanta

L. L. STACHÓ and W. WERNER  ()  On non-commutative Minkowski spheres  29/08/2011, Constanta  1 / 14

MOTIVATION

M := {(t, x) ∈ IR × IR^{3} : t^{2} − ∥x∥^{2} = 1, t ≥ 1} M = [MINKOWSKI SPHERE in upper light cone] IR^{3} ≡ Mat(1, 3, IR), ∥a∥ = [aa^{T}]^{1/2}

ϕ(a) = [\begin{matrix}ϕ_{1}(a) \\ ϕ_{2}(a) \end{matrix}] (a ∈ IR^{3}, ∥a∥ ≠ 1)

ϕ_{1}(a) := \frac{1 + ∥a∥^{2}}{1 − ∥a∥^{2}} , ϕ_{2}(a) := 2(1 − ∥a∥^{2}a)^{−1}a .

Observation: ϕ : Ball(IR^{3}) ↔ M by MAPLE (simplify)

L. L. STACHÓ and W. WERNER  ()  On non-commutative Minkowski spheres  29/08/2011, Constanta  2 / 14

MOTIVATION

ϕ^{−1}(t, x, y, z) = \{ \begin{matrix}\frac{r}{1+\sqrt{1+r^{2}}}(x, y, z) \\ if (t, x, y, z) ≠ (1, 0, 0, 0) \\ (0, 0, 0) \\ if (t, x, y, z) = (1, 0, 0, 0) \end{matrix}

ALTERNATIVE VIEW

where r^{2} := x^{2} + y^{2} + z^{2} Mat(3, 3, IR) × IR^{3} instead of IR × IR^{3}

M := \{(t, x) ∈ \underbrace{Mat(3, 3, IR)_{+}}_{pos.self −adj.op}×IR^{3} : t^{2} − ∥x∥^{2} = 1, t ≥ I \}

ϕ(a) = [\begin{matrix}ϕ_{1}(a) \\ ϕ_{2}(a) \end{matrix}] (a ∈ IR^{3}, ∥a∥ ≠ 1)

ϕ_{1}(a) := (I + a^{T}a)(I − a^{T}a)^{−1} ,

ϕ_{2}(a) := 2a(I − a^{T}a)^{−1} .

Mat(3, 3, IR) × IR^{3} instead of IR × IR^{3}

M := \{(t, x) ∈ \underbrace{Mat(3, 3, IR)_{+}}_{pos.self −adj.op}×IR^{3} : t^{2} − ∥x∥^{2} = 1, t ≥ I \}

ϕ(a) = [\begin{matrix}ϕ_{1}(a) \\ ϕ_{2}(a) \end{matrix}] (a ∈ IR^{3}, ∥a∥ ≠ 1)

ϕ_{1}(a) := (I + a^{T}a)(I − a^{T}a)^{−1} ,

ϕ_{2}(a) := 2a(I − a^{T}a)^{−1} .

Mat(3, 3, IR) × IR^{3} instead of IR × IR^{3}

M := \{(t, x) ∈ \underbrace{Mat(3, 3, IR)_{+}}_{pos.self −adj.op}×IR^{3} : t^{2} − ∥x∥^{2} = 1, t ≥ I \}

ϕ(a) = [\begin{matrix}ϕ_{1}(a) \\ ϕ_{2}(a) \end{matrix}] (a ∈ IR^{3}, ∥a∥ ≠ 1)

ϕ_{1}(a) := (I + a^{T}a)(I − a^{T}a)^{−1} ,

ϕ_{2}(a) := 2a(I − a^{T}a)^{−1} .

ALTERNATIVE VIEW

Observation

ϕ_{1}(a)^{2}−ϕ_{2}(a)^{T}ϕ_{2}(a) =
= \underbrace{(I − a^{T}a)^{−1}} (I + a^{T}a)^{2} \underbrace{(I − a^{T}a)^{−1}} −

− \underbrace{(I − a^{T}a)^{−1}} (4a^{T}a) \underbrace{(I − a^{T}a)^{−1}} =

= (I − a^{T}a)^{−1}(I − a^{T}a)^{2}(I − a^{T}a)^{−1} =
= I .
L. L. STACHÓ and W. WERNER  ()  On non-commutative Minkowski spheres  29/08/2011, Constanta  5 / 14

NON-COMMUTATIVE SCALARS

Alternative case: ⟨a|a⟩ := a^{T}a ∈ Mat(3, 3, IR) Hilbert C^{∗}-modules E vector space, A C^{∗}-algebra with 1 (over IR, C)

\left\{{αx ∈ E LEFT \atop xα ∈ E RIGHT}\right\} A-module structure on E ⟨x|y⟩ ∈ A non-comm scalar prod. ⟨y|x⟩ = ⟨x|y⟩^{∗} , {α⟨x|y⟩ = ⟨αx|y⟩ ⟨x|y⟩α = ⟨xα|y⟩ Example. Mat(1, 3, IR) ∼ L(IR^{3}, IR) Motivation→left A = Mat1, 1, IR ≡ IR, Alternative→right A = Mat3, 3, IR, Mat(1, 3, IR) ↩ Mat(3, 3, IR).

L. L. STACHÓ and W. WERNER  ()  On non-commutative Minkowski spheres  29/08/2011, Constanta  6 / 14

TRO SETTING

Ternary Ring of Operators H Hilbert space T ⊂ L(H) subspace preserving xy^{∗}z TRO = Hilbert C^{∗}-module

gen. constants: A_{left} = {x^{∗}y : x, y ∈ T }^{C_{1}^{∗}span} AT ⊂ T A_{right} = {xy^{∗} : x, y ∈ T }^{C_{1}^{∗}span} T A ⊂ T

Remark. Enveloping TRO: L(H). IT SUFFICES ϕ(a) = \left[{ϕ_{1}(a) \atop ϕ_{2}(a)}\right] (a ∈ Ball(L(H))) ϕ_{1}(a) := (1 + a^{∗}a)(1 − a^{∗}a)^{−1}, ϕ_{2}(a) := 2a(1 − a^{∗}a)^{−1}

L. L. STACHÓ and W. WERNER  ()  On non-commutative Minkowski spheres  29/08/2011, Constanta  7 / 14

TRO SETTING

Proposition. ϕ : Ball(L(H)) ↔ M where M := \{[t, x] ∈ L(H)^{2} : \underbrace{t ≥ 0}_{pod.def}, t^{2} − x^{∗}x = 1\}.

moreover ϕ^{−1} [t, x] = x(1 + t)^{−1}

Corollary. If (E, A, ⟨.|.⟩) is a right [left] Hilbert C^{∗}-module, ϕ_{1}(a) := (1 + ⟨a|a⟩)(1 − ⟨a|a⟩)^{−1}, ϕ_{2}(a) := 2a(1 − ⟨a|a⟩)^{−1} [resp. 2(1 − ⟨a|a⟩)^{−1}a]

then [ϕ_{1}, ϕ_{2}] : Ball(E) ↔ {[t, x] ∈ A_{0} × E : t^{2} − ⟨x|x⟩ = 1}

L. L. STACHÓ and W. WERNER  ()  On non-commutative Minkowski spheres  29/08/2011, Constanta  8 / 14

GEOMETRIC ISSUE (motivation)

Example. H := C, ϕ(a) = [\frac{1+|a|^{2}}{1+|a|^{2}}, \frac{2a}{1+|a|^{2}}] ⟹

[ϕ′_{1}(a)u]^{2} − |ϕ′_{2}(a)u|^{2} = −4\frac{|u|^{2}}{[1 − |a|^{2}]^{2}} .

Riemann(-Poincaré-Bergman) metric on B := Ball(C) is given by

⟨u|u⟩_{a}^{B} := |g′_{a}(0)^{−1}u|^{2} = \frac{|u|^{2}}{[1 − |a|^{2}]^{2}} (a ∈ B, u ∈ C)

g_{a}(z) := (z + a)/(1 − \overline{a}z) Möbius transformation. Observation. Ψ := ϕ^{−1}, ⟹

t̃^{2} − |x̃|^{2} = −4⟨[Ψ′(t, x)](t̃, x̃)|[Ψ′(t, x)](t̃, x̃)⟩_{Ψ(t,x)}^{B}.

L. L. STACHÓ and W. WERNER  ()  On non-commutative Minkowski spheres  29/08/2011, Constanta  9 / 14

GEOMETRIC ISSUE (TRO setting)

ϕ : B :=Ball(L(H))↔M :={(t, x) : t ≥0, t^{2}−x^{∗}x =1}, Ψ := ϕ^{−1} g_{a}(z) := (1 − aa^{∗})^{−1/2}(z + a)(1 − a^{∗}z)^{−1}(1 − a^{∗}a)^{−1/2} Möbius trf.

⟨u|v⟩_{a}^{B} := [[g′_{a}(0)]^{−1}u]^{∗}[[g′_{a}(0)]^{−1}v] =
= (1 − a^{∗}a)^{−1/2}u^{∗}(1 − aa^{∗})^{−1}v(1 − a^{∗}a)^{−1/2}.

Theorem. For (\overline{t}, \overline{x}) ∈ T_{(t,x)}M we have \overline{t}t + t\overline{t} = \overline{x}^{∗}x + x^{∗}\overline{x} and ⟨(\overline{t}, \overline{x})|(\overline{t}, \overline{x})⟩_{(t,x)}^{M} := −4⟨[Ψ′(t, x)](\overline{t}, \overline{x})|Ψ′(t, x)](\overline{t}, \overline{x})⟩_{Ψ(t,x)}^{B} = (I +t)^{1/2}\{2[\overline{t}^{2}−\overline{x}^{∗}\overline{x}]−[\overline{x}^{∗}x(I +t)^{−1}x^{∗}\overline{x}−\overline{t}x^{∗}\overline{x}−\overline{x}^{∗}x\overline{t}+\overline{t}(t−I )\overline{t}]\}(I +t)^{1/2} Problem. Find ”smart” indefinite inner products ⟨.|.⟩_{(t,x)} on L(H)^{2}

extending ϕ^{#}⟨.|.⟩_{(t,x)} from T_{t,x}M

L. L. STACHÓ and W. WERNER  ()  On non-commutative Minkowski spheres  29/08/2011, Constanta  10 / 14

GEOMETRIC ISSUE (further motivation)

Remark. In the 1-dimensional case H = C, we have

1)  ϕ^{#}g_{b} \left[{t \atop x}\right] = ϕ ∘ g_{b} ∘ ϕ^{−1} \left[{t \atop x}\right] = [ \frac{1+|b|^{2}}{1−|b|^{2}}t + \frac{2b}{1−|b|^{2}}x \frac{2b}{1−|b|^{2}}t + \frac{1+|b|^{2}}{1−|b|^{2}}x ];

2) g_{tanh(a)} is the exponential of the vector field [a − \overline{a}u^{2}]\frac{∂}{∂u}. Hence ϕ^{#}g_{tanh(a)} \left[{t \atop x}\right] = \left[{cosh(2a)t + sinh(2a)x \atop sinh(2a)t + cosh(2a)x}\right] for \left[{t \atop x}\right] ∈ M .

In particular ϕ^{#}g_{b} is the restriction of a linear mapping of C^{2} to M.

L. L. STACHÓ and W. WERNER  ()  On non-commutative Minkowski spheres  29/08/2011, Constanta  11 / 14

GEOMETRIC ISSUE (TRO setting)

Notation. A_{a}(u) := [a − ua^{∗}u]\frac{∂}{∂u} (a ∈ L(H)) For fixed a ∈ L(H) and u ∈ BallL(H), u_{τ} := [exp(τA_{a})]u (τ ∈IR) satisfies \frac{d}{dτ}u_{τ} =A_{a}(u_{τ}), u_{0} =u. Well-known: b = [exp(A_{a})]0 ⇒ g_{b}(u) = [exp(A_{a})]u (∥u∥<1). Proposition. Let a ∈ L(H) and \left[{t \atop x}\right] ∈ M. Then

ϕ^{#}A_{a} \left[{t \atop x}\right] = \left[{a^{∗}x + x^{∗}a \atop a(1 + t) + x(1 + t)^{−1}x^{∗}a}\right] .

BEYOND TRO: Self-adjoint operators

A = A(H)={a∈H : a^{∗} =a} REAL JORDAN ALGEBRA in L(H)

ϕ^{#}A_{a} \left[{t \atop x}\right] = \left[{ax + xa \atop at + ta}\right] = [\begin{matrix}0 & L_{a} + R_{a} \\ L_{a} + R_{a} & 0 \end{matrix}] \left[{t \atop x}\right]

Theorem. a ∈ A ⇒ 1) g_{tanh(a)} : Ball(A) ↔ Ball(A),

2) ϕ : Ball(A)↔M ∩ A^{2} = \{[{t \atop x}] ∈ A_{+} × A : t^{2}−x^{2} =1\}

3) ϕ^{#}g_{tanh(a)}=ϕ ∘ [exp A_{a}] ∘ ϕ^{−1} \left[{t \atop x}\right]=\left[{cosh(L_{a}+R_{a})t+sinh(L_{a}+R_{a})x \atop sinh(L_{a}+R_{a})t+cosh(L_{a}+R_{a})x}\right]

cosh(L_{a} + R_{a})z = \frac{1}{2}[exp(a)]z[exp(a)] + \frac{1}{2}[exp(−a)]z[exp(−a)] , cosh(L_{a} + R_{a})z = \frac{1}{2}[exp(a)]z[exp(a)] − \frac{1}{2}[exp(−a)]z[exp(−a)] .

L. L. STACHÓ and W. WERNER  ()  On non-commutative Minkowski spheres  29/08/2011, Constanta  13 / 14

BEYOND TRO: JB*-triples

Observation. ϕ_{1}(a) = 1 + 2a^{∗}ϕ_{2}(a) in TRO case (E, ∥.∥, {…}) JB^{∗}-triple: {…} : E^{3} → E, D(a, b) : x ↦ {abx}

1) {xyz} symmetric bilinear in x, z, antilinear in y

2) Jordan-identity for {…}

3) ∥{aaa}∥ = ∥a∥^{3}

4) ∥ exp (ζD(a, a))∥ ≤ 1 if Re(ζ) ≤ 0 Canonical JB^{∗}-product on L(H): \{xyz\} := \frac{1}{2}xy^{∗}z + \frac{1}{2}zy^{∗}x ϕ_{2}(a) := 2[1 − D(a, a)]^{−1}a, ϕ_{1}(a) := 1 + \underbrace{D(ϕ_{2}(a), a)}_{=D(a,ϕ_{2}(a))} ∈ L(E)

Proposition. ϕ^{−1} \left[{t \atop x}\right] = (1 + t)^{−1}x for \left[{t \atop x}\right] ∈ range(ϕ)

L. L. STACHÓ and W. WERNER  ()  On non-commutative Minkowski spheres  29/08/2011, Constanta  14 / 14