Definition 3.1

Let P 𝑃 P italic_P be a C * superscript 𝐶 C^{*} italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT -planar algebra. A P 𝑃 P italic_P -module V 𝑉 V italic_V will be called a Hilbert P 𝑃 P italic_P -module if each V k subscript 𝑉 𝑘 V_{k} italic_V start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT is a finite dimensional Hilbert space with inner product ⟨ , ⟩ fragments normal-⟨ normal-, normal-⟩ \langle,\rangle ⟨ , ⟩ satisfying

⟨ a ⁢ v , w ⟩ = ⟨ v , a * ⁢ w ⟩ 𝑎 𝑣 𝑤 𝑣 superscript 𝑎 𝑤 \langle av,w\rangle=\langle v,a^{*}w\rangle ⟨ italic_a italic_v , italic_w ⟩ = ⟨ italic_v , italic_a start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT italic_w ⟩

for all v , w 𝑣 𝑤 v,w italic_v , italic_w in V 𝑉 V italic_V and a 𝑎 a italic_a in A ⁢ P 𝐴 𝑃 AP italic_A italic_P (in the graded sense).

Definition 2.1

Let * * * be a star product on ( M , ω ) 𝑀 𝜔 (M,\omega) ( italic_M , italic_ω ) then a trace is a ℂ ⁢ [ [ ν ] ] ℂ delimited-[] delimited-[] 𝜈 \mathbb{C}[\![\nu]\!] blackboard_C [ [ italic_ν ] ] -linear map τ : N c [ [ ν ] ] → ℂ [ ν - 1 , ν ] ] fragments τ : subscript 𝑁 𝑐 fragments [ fragments [ ν ] ] → C fragments [ superscript 𝜈 1 , ν ] ] \tau\colon N_{c}[\![\nu]\!]\to\mathbb{C}[\nu^{-1},\nu]\!] italic_τ : italic_N start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT [ [ italic_ν ] ] → blackboard_C [ italic_ν start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT , italic_ν ] ] satisfying

τ ⁢ ( u * v ) = τ ⁢ ( v * u ) . 𝜏 𝑢 𝑣 𝜏 𝑣 𝑢 \tau(u*v)=\tau(v*u). italic_τ ( italic_u * italic_v ) = italic_τ ( italic_v * italic_u ) .
Definition 5 .

Let p ⁢ ( t ) = ∑ c j ⁢ t j 𝑝 𝑡 subscript 𝑐 𝑗 superscript 𝑡 𝑗 p(t)=\sum c_{j}t^{j} italic_p ( italic_t ) = ∑ italic_c start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_t start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT be a non-zero complex power series. We say that p ⁢ ( t ) 𝑝 𝑡 p(t) italic_p ( italic_t ) is almost real if and only if

p ⁢ ( t ) = α ⁢ h ⁢ ( t ) 𝑝 𝑡 𝛼 ℎ 𝑡 p(t)=\alpha h(t) italic_p ( italic_t ) = italic_α italic_h ( italic_t )

where α ∈ ℂ ∗ 𝛼 superscript ℂ ∗ \alpha\in\mathbb{C}^{\ast} italic_α ∈ blackboard_C start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT and where h ⁢ ( t ) ℎ 𝑡 h(t) italic_h ( italic_t ) is a non-zero power series with real coefficients.

Definition 9

Consider a group epimorphism ϕ : G ~ → G . normal-: italic-ϕ normal-→ normal-~ 𝐺 𝐺 \phi:{\widetilde{G}}\rightarrow G. italic_ϕ : ~ start_ARG italic_G end_ARG → italic_G . Suppose that a group G 𝐺 G italic_G acts on X 𝑋 X italic_X and G ~ normal-~ 𝐺 \widetilde{G} ~ start_ARG italic_G end_ARG acts on X ~ . normal-~ 𝑋 \widetilde{X}. ~ start_ARG italic_X end_ARG . By a covering in the category of spaces with group actions we mean a covering map p : X ~ → X normal-: 𝑝 normal-→ normal-~ 𝑋 𝑋 p:\widetilde{X}\rightarrow X italic_p : ~ start_ARG italic_X end_ARG → italic_X such that

p ⁢ ( g ~ ⁢ ( x ~ ) ) = ϕ ⁢ ( g ~ ) ⁢ ( p ⁢ x ~ ) 𝑝 ~ 𝑔 ~ 𝑥 italic-ϕ ~ 𝑔 𝑝 ~ 𝑥 p({\widetilde{g}}(\widetilde{x}))=\phi{({\widetilde{g}})}(p{\widetilde{x}}) italic_p ( ~ start_ARG italic_g end_ARG ( ~ start_ARG italic_x end_ARG ) ) = italic_ϕ ( ~ start_ARG italic_g end_ARG ) ( italic_p ~ start_ARG italic_x end_ARG )

for any x ~ ∈ X ~ ~ 𝑥 ~ 𝑋 \widetilde{x}\in\widetilde{X} ~ start_ARG italic_x end_ARG ∈ ~ start_ARG italic_X end_ARG and g ~ ∈ G ~ ~ 𝑔 ~ 𝐺 \widetilde{g}\in\widetilde{G} ~ start_ARG italic_g end_ARG ∈ ~ start_ARG italic_G end_ARG .