Definition 5.5 .

The lattice is said to be distributive if

x ∨ ( y ∧ z ) = ( x ∨ y ) ∧ ( x ∨ z ) . 𝑥 𝑦 𝑧 𝑥 𝑦 𝑥 𝑧 x\vee(y\wedge z)=(x\vee y)\wedge(x\vee z). italic_x ∨ ( italic_y ∧ italic_z ) = ( italic_x ∨ italic_y ) ∧ ( italic_x ∨ italic_z ) . (5.1)
Definition 2.4.3 .

A left Loday (or Leibniz) algebra [ 13 , 16 ] is a vector space equipped with a bilinear operation [ - , - ] [-,-] [ - , - ] such that the following version of the Jacobi identity is satisfied

[ a , [ b , c ] ] = [ [ a , b ] , c ] + [ b , [ a , c ] ] 𝑎 𝑏 𝑐 𝑎 𝑏 𝑐 𝑏 𝑎 𝑐 [a,[b,c]]=[[a,b],c]+[b,[a,c]] [ italic_a , [ italic_b , italic_c ] ] = [ [ italic_a , italic_b ] , italic_c ] + [ italic_b , [ italic_a , italic_c ] ]
Definition 2.3

Suppose E , F 𝐸 𝐹 E,F italic_E , italic_F are Hilbert modules, with inner product in a C * superscript 𝐶 C^{*} italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT -algebra A 𝐴 A italic_A . We define ℒ ⁢ ( E , F ) ℒ 𝐸 𝐹 {\mathcal{L}}(E,F) caligraphic_L ( italic_E , italic_F ) to be the set of all maps t : E → F : 𝑡 → 𝐸 𝐹 t:E\to F italic_t : italic_E → italic_F for which there exists a map t * : F → E : superscript 𝑡 → 𝐹 𝐸 t^{*}:F\to E italic_t start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT : italic_F → italic_E such that

[ t ⁢ x , y ] = [ x , t * ⁢ y ] 𝑡 𝑥 𝑦 𝑥 superscript 𝑡 𝑦 [tx,y]=[x,t^{*}y] [ italic_t italic_x , italic_y ] = [ italic_x , italic_t start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT italic_y ]

for all x ∈ E , y ∈ F formulae-sequence 𝑥 𝐸 𝑦 𝐹 x\in E,y\in F italic_x ∈ italic_E , italic_y ∈ italic_F . We call ℒ ⁢ ( E , F ) ℒ 𝐸 𝐹 {\mathcal{L}}(E,F) caligraphic_L ( italic_E , italic_F ) the set of adjointable operators from E 𝐸 E italic_E to F 𝐹 F italic_F . We abbreviate ℒ ⁢ ( E , E ) ℒ 𝐸 𝐸 {\mathcal{L}}(E,E) caligraphic_L ( italic_E , italic_E ) as ℒ ⁢ ( E ) ℒ 𝐸 {\mathcal{L}}(E) caligraphic_L ( italic_E ) . It can be shown that every element of ℒ ⁢ ( E , F ) ℒ 𝐸 𝐹 {\mathcal{L}}(E,F) caligraphic_L ( italic_E , italic_F ) is a bounded A 𝐴 A italic_A -linear map.