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3 Fixed numbering

Consider finite-dimensional (for-simplicity) $\star$-algebras, that is, unital associative algebras over the complex numbers equipped with an antilinear antiautomorphism $\star$.

A state on a $\star$-algebra $A$ is a linear mapping $\psi: A \to \mathbb{C}$ satisfying

(i) $\psi(1) = 1$ [normalization]

(iii) ii) $\psi(a^*) = \overline{\psi(a)}$ [reality]

(ii) iii) $\psi(a^*a) \geq 0$ [positivity]

Denote $s(A)$ the set of all states on $A$.

Consider the following 2 ways to construct a mapping between states on different algebras:

(1) Consider $A$, $B$ $\star$-algebras and $f: A \to B$ a $\star$-homomorphism. Then we have $f^{-1}: s(B) \to s(A)$ defined by $f^{-1}(\psi)(a) := \psi(f(a))$.

(2) Consider $A$, $B$ $\star$-algebras and $\phi$ a state on $B$. Then we have $t_\phi: s(A) \to s(A \otimes B)$ defined by $t_\phi(\psi)(a \otimes b) = \psi(a) \phi(b)$.

Now, suppose we compose any number of mappings of the above 2 kinds. We get for any $\star$-algebras $A$, $B$, a certain class of mappings $s(A) \to s(B)$. I.e., these are mappings obtained by composing mapping of the above 2 kinds while inserting any number of intermediate algebras. Denote this set of mappings $\mathrm{Mor}(s(A), s(B))$.

Given a $\star$-algebra $A$, the positive cone of $A$ is the set of all linear combinations of elements of the form $a^*a$ with positive coefficients. Denote it $p(A)$.

Consider $A$, $B$ $\star$-algebras and $L: A \to B$ a linear mapping (not necessarily a homomorphism!) preserving 1 and $\star$. Suppose $L$ is positive in the sense that it maps $p(A)$ to $p(B)$. Then $L$ induces $L^{-1}: s(B) \to s(A)$ defined by $L^{-1}(\psi)(a) = \psi(L(a))$.

The Question:

Is $L^{-1}$ guaranteed to be in $\mathrm{Mor}(s(A), s(B))$?

2 I've LaTeX'd the question

Consider finite-dimensional (for-simplicity) *-algebras, $\star$-algebras, that is, unital associative algebras over the complex numbers equipped with an antilinear antiautomorphism *$\star$.

A state on a *-algebra A $\star$-algebra $A$ is a linear mapping psi: $\psi: A -> C \to \mathbb{C}$ satisfying

(i) psi(1) $\psi(1) = 1 1$ [normalization]

(iii) psi(a*) $\psi(a^*) = psi(a)^bar \overline{\psi(a)}$ [reality]

(ii) psi(a*a) >= 0 $\psi(a^*a) \geq 0$ [positivity]

Denote s(A) $s(A)$ the set of all states on A$A$.

Consider the following 2 ways to construct a mapping between states on different algebras:

(1) Consider A, B *-algebras $A$, $B$ $\star$-algebras and f: $f: A -> B \to B$ a *-homomorphism. $\star$-homomorphism. Then we have f^-1: $f^{-1}: s(B) -> s(A) \to s(A)$ defined by f^-1(psi)(a) $f^{-1}(\psi)(a) := psi(f(a))\psi(f(a))$.

(2) Consider A, B *-algebras $A$, $B$ $\star$-algebras and phi $\phi$ a state on B. $B$. Then we have t_phi: $t_\phi: s(A) -> \to s(A (x) B) \otimes B)$ defined by t_phi(psi)(a (x) $t_\phi(\psi)(a \otimes b) = psi(a) phi(b)\psi(a) \phi(b)$.

Now, supposed suppose we compose any number of mappings of the above 2 kinds. We get for any *-algebras A, B, $\star$-algebras $A$, $B$, a certain class of mappings s(A) -> s(B). I.e$s(A) \to s(B)$. I.e., these are mappings obtained by composing mapping of the above 2 kinds while inserting any number of intermediate algebras. Denote this set of mappings Mor(s(A), s(B))$\mathrm{Mor}(s(A), s(B))$.

Given a *-algebra A, $\star$-algebra $A$, the positive cone of A $A$ is the set of all linear combinations of elements of the form a*a $a^*a$ with positive coefficients. Denote it p(A).$p(A)$.

Consider A, B *-algebras $A$, $B$ $\star$-algebras and L: $L: A -> B \to B$ a linear mapping (not necessarily a homomorphism!) preserving *. $\star$. Suppose L $L$ is positive in the sense that it maps p(A) $p(A)$ to p(B). $p(B)$. Then L $L$ induces L^-1: $L^{-1}: s(B) -> s(A) \to s(A)$ defined by L^-1(psi)(a) $L^{-1}(\psi)(a) = psi(L(a))\psi(L(a))$.

Is L^-1 $L^{-1}$ guaranteed to be in Mor(s(A), s(B))$\mathrm{Mor}(s(A), s(B))$?

1

# Mappings between states on *-algebras

Consider finite-dimensional (for-simplicity) *-algebras, that is, unital associative algebras over the complex numbers equipped with an antilinear antiautomorphism *

A state on a *-algebra A is a linear mapping psi: A -> C satisfying

(i) psi(1) = 1 [normalization]

(iii) psi(a*) = psi(a)^bar [reality]

(ii) psi(a*a) >= 0 [positivity]

Denote s(A) the set of all states on A

Consider the following 2 ways to construct mapping between states on different algebras:

(1) Consider A, B *-algebras and f: A -> B a *-homomorphism. Then we have f^-1: s(B) -> s(A) defined by f^-1(psi)(a) := psi(f(a))

(2) Consider A, B *-algebras and phi a state on B. Then we have t_phi: s(A) -> s(A (x) B) defined by t_phi(psi)(a (x) b) = psi(a) phi(b)

Now, supposed we compose any number of mappings of the above 2 kinds. We get for any *-algebras A, B, a certain class of mappings s(A) -> s(B). I.e. these are mappings obtained by composing mapping of the above 2 kinds while inserting any number of intermediate algebras. Denote this set of mappings Mor(s(A), s(B)).

Given a *-algebra A, the positive cone of A is the set of all linear combinations of elements of the form a*a with positive coefficients. Denote it p(A).

Consider A, B *-algebras and L: A -> B a linear mapping (not necessarily a homomorphism!) preserving *. Suppose L is positive in the sense that it maps p(A) to p(B). Then L induces L^-1: s(B) -> s(A) defined by L^-1(psi)(a) = psi(L(a))

The Question:

Is L^-1 guaranteed to be in Mor(s(A), s(B))?