PhyslibAlpha.AlgebraicFramework.CStarAlgebra.Stinespring.Dilation

Given a $C^*$-algebra $A$ and a complex Hilbert space $H$, this definition represents the algebraic tensor product $A \otimes_{\mathbb{C}} H$. It serves as the underlying carrier space for the Stinespring dilation construction.

Let $A$ be a $C^*$-algebra and $H$ be a complex Hilbert space. Given a completely positive map $J: A \to \mathcal{B}(H)$, an element $a \in A$, and a vector $h \in H$, this definition is the linear functional $f_{a,h} : A \otimes_{\mathbb{C}} H \to \mathbb{C}$ uniquely determined by its action on elementary tensors: $$f_{a,h}(b \otimes k) = \langle h, J(a^* b) k \rangle$$ where $b \in A$, $k \in H$, and $\langle \cdot, \cdot \rangle$ denotes the inner product on $H$. This functional serves as the fundamental building block for constructing the inner product in Stinespring's dilation theorem.

Let $A$ be a $C^*$-algebra and $H$ be a complex Hilbert space. For a completely positive map $J: A \to \mathcal{B}(H)$, elements $a, b \in A$, and vectors $h, k \in H$, the linear functional $f_{a,h}$ (the `baseFunctional`) evaluated at the elementary tensor $b \otimes k \in A \otimes_{\mathbb{C}} H$ satisfies: $$f_{a,h}(b \otimes k) = \langle h, J(a^* b) k \rangle$$ where $\langle \cdot, \cdot \rangle$ denotes the inner product on $H$ and $a^*$ is the adjoint of $a$.

Let $A$ be a $C^*$-algebra, $H$ be a complex Hilbert space, and $J: A \to \mathcal{B}(H)$ be a completely positive map. This definition is the map $S: A \times H \to (A \otimes_{\mathbb{C}} H \to \mathbb{C})$ which is conjugate-linear in both $a \in A$ and $h \in H$. For any fixed $a \in A$ and $h \in H$, the resulting linear functional $S(a, h)$ acts on the algebraic tensor product $A \otimes_{\mathbb{C}} H$ such that for any elementary tensor $b \otimes k \in A \otimes_{\mathbb{C}} H$: $$(S(a, h))(b \otimes k) = \langle h, J(a^* b) k \rangle$$ where $\langle \cdot, \cdot \rangle$ is the inner product on $H$ (conventionally conjugate-linear in the first argument). This map packages the `baseFunctional` into a sesquilinear form over $A$ and $H$.

Let $A$ be a $C^*$-algebra and $H$ be a complex Hilbert space. Given a completely positive map $J : A \to \mathcal{B}(H)$, this definition constructs a (possibly degenerate) sesquilinear form on the algebraic tensor product $A \otimes_{\mathbb{C}} H$. This form is represented as a map that sends an element $x \in A \otimes_{\mathbb{C}} H$ to a linear functional, such that the mapping is conjugate-linear. For elementary tensors $a \otimes h$ and $b \otimes k$ in $A \otimes_{\mathbb{C}} H$, the form satisfies: $$\langle a \otimes h, b \otimes k \rangle = \langle h, J(a^* b) k \rangle_H$$ where $\langle \cdot, \cdot \rangle_H$ denotes the inner product on $H$ (conventionally conjugate-linear in the first argument) and $a^*$ is the adjoint of $a$.

Let $A$ be a $C^*$-algebra and $H$ be a complex Hilbert space. Given a completely positive map $J : A \to \mathcal{B}(H)$, the sesquilinear form $\langle \cdot, \cdot \rangle$ defined on the algebraic tensor product $A \otimes_{\mathbb{C}} H$ satisfies the following identity for any elementary tensors $a \otimes h$ and $b \otimes k$ (where $a, b \in A$ and $h, k \in H$): $$\langle a \otimes h, b \otimes k \rangle = \langle h, J(a^* b) k \rangle_H$$ where $a^*$ is the adjoint of $a$ and $\langle \cdot, \cdot \rangle_H$ denotes the inner product on $H$, which is conventionally conjugate-linear in the first argument.

Let $A$ be a $C^*$-algebra, $H$ be a complex Hilbert space, and $J : A \to \mathcal{B}(H)$ be a completely positive map. Let $\langle \cdot, \cdot \rangle_J$ be the sesquilinear form on the algebraic tensor product $A \otimes_{\mathbb{C}} H$ defined by $\langle a \otimes h, b \otimes k \rangle_J = \langle h, J(a^* b) k \rangle_H$. For any $a, b \in A$ and $h, k \in H$, the form satisfies conjugate symmetry on elementary tensors: $$\overline{\langle a \otimes h, b \otimes k \rangle_J} = \langle b \otimes k, a \otimes h \rangle_J$$ where the overline denotes complex conjugation.

Let $A$ be a $C^*$-algebra and $H$ be a complex Hilbert space. Given a completely positive map $J : A \to \mathcal{B}(H)$, let $\langle \cdot, \cdot \rangle_J$ denote the sesquilinear form induced by $J$ on the algebraic tensor product $A \otimes_{\mathbb{C}} H$ (defined on elementary tensors by $\langle a \otimes h, b \otimes k \rangle_J = \langle h, J(a^* b) k \rangle_H$). For any elements $x, y \in A \otimes_{\mathbb{C}} H$, the form satisfies conjugate symmetry: $$\overline{\langle x, y \rangle_J} = \langle y, x \rangle_J$$ where the overline denotes complex conjugation.

Let $A$ be a $C^*$-algebra and $H$ be a complex Hilbert space. Given a completely positive map $J : A \to \mathcal{B}(H)$, the sesquilinear form $\langle \cdot, \cdot \rangle_J$ defined on the algebraic tensor product $A \otimes_{\mathbb{C}} H$ (where $\langle a \otimes h, b \otimes k \rangle_J = \langle h, J(a^* b) k \rangle_H$) is positive semi-definite. That is, for any $x \in A \otimes_{\mathbb{C}} H$, it holds that $0 \le \langle x, x \rangle_J$.

Given a completely positive map $J: A \to \mathcal{B}(H)$ from a $C^*$-algebra $A$ to the space of bounded linear operators on a complex Hilbert space $H$, this definition constructs a pre-inner product space structure on the algebraic tensor product $A \otimes_{\mathbb{C}} H$. The structure is defined by the sesquilinear form $\langle \cdot, \cdot \rangle_J$ (specifically, the `tensorInner` map), which satisfies $$\langle a \otimes h, b \otimes k \rangle_J = \langle h, J(a^* b) k \rangle_H$$ for elementary tensors $a \otimes h$ and $b \otimes k$. This structure includes the definitions for the inner product, conjugate symmetry, and non-negativity required to treat $A \otimes_{\mathbb{C}} H$ as a pre-Hilbert space.

For an element $a$ of a $C^*$-algebra $A$, this is the $\mathbb{C}$-linear map on the algebraic tensor product $A \otimes_{\mathbb{C}} H$ (where $H$ is a complex Hilbert space) that acts as left multiplication by $a$ on the first factor. Specifically, it is defined on simple tensors by $a \cdot (b \otimes h) = (ab) \otimes h$ for $b \in A$ and $h \in H$.

Let $A$ be a $C^*$-algebra and $H$ be a complex Hilbert space. For any elements $a, b \in A$ and any vector $h \in H$, the left multiplication operator $\text{leftMul}_a$ acting on the simple tensor $b \otimes h$ in the algebraic tensor product $A \otimes_{\mathbb{C}} H$ satisfies: $$\text{leftMul}_a(b \otimes h) = (ab) \otimes h$$ where $ab$ denotes the product in $A$.

Let $A$ be a $C^*$-algebra and $H$ be a complex Hilbert space. Given a completely positive map $J : A \to \mathcal{B}(H)$, let $\langle \cdot, \cdot \rangle_J$ be the sesquilinear form induced on the algebraic tensor product $A \otimes_{\mathbb{C}} H$ defined on elementary tensors by $\langle a \otimes h, b \otimes k \rangle_J = \langle h, J(a^* b) k \rangle_H$. For any element $a \in A$ and any tensors $x, y \in A \otimes_{\mathbb{C}} H$, the left multiplication operator satisfies the following adjoint relation: $$\langle a \cdot x, y \rangle_J = \langle x, a^* \cdot y \rangle_J$$ where $a \cdot x$ denotes the action of $a$ on the first factor of the tensor product (i.e., $a \cdot (b \otimes h) = (ab) \otimes h$) and $a^*$ is the adjoint of $a$ in $A$.

Let $A$ be a $C^*$-algebra and $H$ be a complex Hilbert space. For any elements $a, b \in A$ and any element $x$ in the algebraic tensor product $A \otimes_{\mathbb{C}} H$, the left multiplication operator satisfies the identity $\text{leftMul}(ab)x = \text{leftMul}(a)(\text{leftMul}(b)x)$.

Let $A$ be a $C^*$-algebra and $H$ be a complex Hilbert space. For any elements $a, b \in A$ and any element $x$ in the algebraic tensor product $A \otimes_{\mathbb{C}} H$, the left multiplication of $x$ by the sum $a + b$ is equal to the sum of the left multiplication of $x$ by $a$ and the left multiplication of $x$ by $b$, which can be written as $(a + b) \cdot x = a \cdot x + b \cdot x$.

Let $A$ be a $C^*$-algebra and $H$ be a complex Hilbert space. For any scalar $r \in \mathbb{C}$, any element $a \in A$, and any element $x$ in the algebraic tensor product $A \otimes_{\mathbb{C}} H$, the action of left multiplication by $r \cdot a$ on $x$ is equal to $r$ times the action of left multiplication by $a$ on $x$. That is, $\text{leftMul}(r \cdot a)x = r \cdot (\text{leftMul}(a)x)$.

Let $A$ be a $C^*$-algebra and $H$ a complex Hilbert space. For any element $x$ in the algebraic tensor product $A \otimes_{\mathbb{C}} H$, the left multiplication by the identity element $1 \in A$ acts as the identity map, i.e., $1 \cdot x = x$.

Let $A$ be a $C^*$-algebra and $H$ be a complex Hilbert space. For any element $a \in A$ and any element $x$ in the algebraic tensor product $A \otimes_{\mathbb{C}} H$, let $L_b$ denote the linear map of left multiplication by $b \in A$ on the first factor of the tensor product. Then the following identity holds: $$L_{\|a\|^2 \cdot 1 - a^* a}(x) = \|a\|^2 x - L_{a^* a}(x)$$ where $\|a\|$ is the norm of $a$, $1$ is the identity element of $A$, and $a^*$ is the adjoint of $a$.

Let $A$ be a $C^*$-algebra and $H$ a complex Hilbert space. Let $J: A \to \mathcal{B}(H)$ be a completely positive map, and let $\langle \cdot, \cdot \rangle_J$ be the sesquilinear form on the algebraic tensor product $A \otimes_{\mathbb{C}} H$ defined by $\langle a \otimes h, b \otimes k \rangle_J = \langle h, J(a^* b) k \rangle_H$. For any self-adjoint element $q \in A$ (i.e., $q^* = q$) and any $x \in A \otimes_{\mathbb{C}} H$, the following identity holds: $$\langle q \cdot x, x \rangle_J = \langle x, q \cdot x \rangle_J$$ where $q \cdot x$ denotes the left multiplication by $q$ on the first factor of the tensor product.

Let $A$ be a $C^*$-algebra and $H$ be a complex Hilbert space. Given a completely positive map $J : A \to \mathcal{B}(H)$, let $\langle \cdot, \cdot \rangle_J$ be the sesquilinear form on the algebraic tensor product $A \otimes_{\mathbb{C}} H$ defined by the relation $\langle a \otimes h, b \otimes k \rangle_J = \langle h, J(a^* b) k \rangle_H$. For any positive element $q \in A$ (denoted $q \geq 0$) and any element $x$ in the tensor product $A \otimes_{\mathbb{C}} H$, the following inequality holds: $$0 \leq \langle L_q x, x \rangle_J$$ where $L_q$ is the linear map that performs left multiplication by $q$ on the first factor of the tensor product.

Let $A$ be a $C^*$-algebra and $H$ be a complex Hilbert space. Given a completely positive map $J : A \to \mathcal{B}(H)$, this definition equips the algebraic tensor product $A \otimes_{\mathbb{C}} H$ with the structure of a seminormed additive commutative group. The seminorm is induced by the pre-inner product $\langle \cdot, \cdot \rangle_J$ defined on elementary tensors by $\langle a \otimes h, b \otimes k \rangle_J = \langle h, J(a^* b) k \rangle_H$.

Let $A$ be a $C^*$-algebra and $H$ be a complex Hilbert space. Given a completely positive map $J : A \to \mathcal{B}(H)$, this definition equips the algebraic tensor product $A \otimes_{\mathbb{C}} H$ with an inner product space structure over $\mathbb{C}$. The inner product is induced by the sesquilinear form $\langle \cdot, \cdot \rangle_J$ defined on elementary tensors by $$\langle a \otimes h, b \otimes k \rangle_J = \langle h, J(a^* b) k \rangle_H$$ where $a, b \in A$ and $h, k \in H$. This structure is defined in terms of the pre-inner product core `tensorCore` and is compatible with the seminormed group structure `tensorSeminormed`.

Let $A$ be a $C^*$-algebra and $H$ be a complex Hilbert space. Let $J : A \to \mathcal{B}(H)$ be a completely positive map, where $\mathcal{B}(H)$ denotes the space of bounded linear operators on $H$. Let $A \otimes_{\mathbb{C}} H$ be the algebraic tensor product equipped with the seminorm $\|\cdot\|$ induced by the $J$-dependent inner product, which is defined on simple tensors by $\langle a \otimes h, b \otimes k \rangle = \langle h, J(a^* b) k \rangle_H$. Then, for any $a \in A$ and any $x \in A \otimes_{\mathbb{C}} H$, the left multiplication action $a \cdot x$ (defined by $a \cdot (b \otimes h) = (ab) \otimes h$) satisfies the inequality: $$\|a \cdot x\| \le \|a\| \|x\|$$ where $\|a\|$ is the norm in the $C^*$-algebra $A$.

Let $A$ be a $C^*$-algebra and $H$ a complex Hilbert space. For any completely positive map $J : A \to \mathcal{B}(H)$, let $A \otimes_{\mathbb{C}} H$ be the algebraic tensor product equipped with the seminorm $\|\cdot\|_J$ induced by the pre-inner product $\langle a \otimes h, b \otimes k \rangle_J = \langle h, J(a^* b) k \rangle_H$. Then the addition operation $+ : (A \otimes_{\mathbb{C}} H) \times (A \otimes_{\mathbb{C}} H) \to A \otimes_{\mathbb{C}} H$ is uniformly continuous with respect to the pseudo-metric induced by this seminorm.

Let $A$ be a $C^*$-algebra and $H$ be a complex Hilbert space. Let $J: A \to \mathcal{B}(H)$ be a completely positive map. Consider the algebraic tensor product $A \otimes_{\mathbb{C}} H$ equipped with the seminormed additive commutative group structure induced by $J$. Then the negation map $x \mapsto -x$ is uniformly continuous on $A \otimes_{\mathbb{C}} H$ with respect to the induced pseudo-metric.

Let $A$ be a $C^*$-algebra, $H$ be a complex Hilbert space, and $J : A \to \mathcal{B}(H)$ be a completely positive map from $A$ to the space of bounded linear operators on $H$. Let $A \otimes_{\mathbb{C}} H$ be the algebraic tensor product equipped with the seminorm $\|\cdot\|_J$ induced by the $J$-dependent pre-inner product, which is defined on simple tensors by $\langle a \otimes h, b \otimes k \rangle_J = \langle h, J(a^* b) k \rangle_H$. For a given element $a \in A$, this definition is the continuous linear map $L_a : (A \otimes_{\mathbb{C}} H) \to (A \otimes_{\mathbb{C}} H)$ that acts as left multiplication by $a$ on the first factor (i.e., $L_a(b \otimes h) = (ab) \otimes h$). Its operator norm is bounded by the $C^*$-norm of $a$ ($\|L_a\| \le \|a\|$).

Let $A$ be a $C^*$-algebra and $H$ be a complex Hilbert space. Let $J : A \to \mathcal{B}(H)$ be a completely positive map from $A$ to the space of bounded linear operators on $H$. Let $A \otimes_{\mathbb{C}} H$ be the algebraic tensor product equipped with the seminorm induced by the $J$-dependent pre-inner product, defined on simple tensors by $\langle a \otimes h, b \otimes k \rangle_J = \langle h, J(a^* b) k \rangle_H$. For any $a \in A$, let $L_a$ be the continuous linear map on this seminormed space that performs left multiplication by $a$ on the first factor. Then for any $x \in A \otimes_{\mathbb{C}} H$, the action of $L_a$ on $x$ coincides with the algebraic left multiplication of $a$ on $x$.

Let $A$ be a $C^*$-algebra and $H$ be a complex Hilbert space. Given a completely positive map $J : A \to \mathcal{B}(H)$, let $K_J$ be the Hilbert space completion of the algebraic tensor product $A \otimes_{\mathbb{C}} H$ with respect to the $J$-induced semi-inner product defined by $\langle a \otimes h, b \otimes k \rangle_J = \langle h, J(a^* b) k \rangle_H$. For an element $a \in A$, this definition is the unique continuous linear map on $K_J$ that extends the left multiplication operator $L_a(b \otimes h) = (ab) \otimes h$ from the dense subspace $A \otimes_{\mathbb{C}} H$.

Let $A$ be a $C^*$-algebra, $H$ be a complex Hilbert space, and $J : A \to \mathcal{B}(H)$ be a completely positive map. Let $A \otimes_{\mathbb{C}} H$ be the algebraic tensor product equipped with the semi-inner product induced by $J$, defined on elementary tensors by $\langle a \otimes h, b \otimes k \rangle_J = \langle h, J(a^* b) k \rangle_H$. Let $K_J$ be the Hilbert space completion of $A \otimes_{\mathbb{C}} H$. For any $a \in A$, let $L_a$ be the continuous linear map on $A \otimes_{\mathbb{C}} H$ that acts as left multiplication by $a$ on the first factor, and let $\bar{L}_a$ be its unique continuous extension to $K_J$. Then for any $x \in A \otimes_{\mathbb{C}} H$, we have $\bar{L}_a(x) = L_a(x)$, where $x$ on the left-hand side is viewed as an element of the completion $K_J$.

Let $A$ be a $C^*$-algebra and $H$ be a complex Hilbert space. Given a completely positive map $J : A \to \mathcal{B}(H)$, let $K_J$ be the Hilbert space completion of the algebraic tensor product $A \otimes_{\mathbb{C}} H$ with respect to the semi-inner product defined by $\langle a \otimes h, b \otimes k \rangle_J = \langle h, J(a^* b) k \rangle_H$. For any $a \in A$, let $\pi(a)$ denote the unique continuous linear operator on $K_J$ extending the left multiplication $b \otimes h \mapsto (ab) \otimes h$. Then for any $a, b \in A$, the operator corresponding to the product $ab$ satisfies $$\pi(ab) = \pi(a) \circ \pi(b)$$ where $\circ$ denotes the composition of operators.

Given a $C^*$-algebra $A$, a complex Hilbert space $H$, and a completely positive map $J : A \to \mathcal{B}(H)$, the type `Stinespring.Canonical.Pre J` is a type synonym for the algebraic tensor product $A \otimes_{\mathbb{C}} H$. It is used to equip the tensor product with a specific semi-inner product induced by $J$, defined on elementary tensors as $\langle a \otimes h, b \otimes k \rangle = \langle h, J(a^* b) k \rangle_H$.

Given a $C^*$-algebra $A$, a complex Hilbert space $H$, and a completely positive map $J : A \to \mathcal{B}(H)$, the type `Stinespring.Canonical.Pre J` (which is the algebraic tensor product $A \otimes_{\mathbb{C}} H$) inherits the structure of an additive commutative group.

Given a $C^*$-algebra $A$, a complex Hilbert space $H$, and a completely positive map $J : A \to \mathcal{B}(H)$, the type `Stinespring.Canonical.Pre J` (which is a type synonym for the algebraic tensor product $A \otimes_{\mathbb{C}} H$) inherits the structure of a module (vector space) over the complex numbers $\mathbb{C}$.

Given a $C^*$-algebra $A$, a complex Hilbert space $H$, and a completely positive map $J : A \to \mathcal{B}(H)$, the type `Pre J` (a type synonym for the algebraic tensor product $A \otimes_{\mathbb{C}} H$) is equipped with the structure of a seminormed additive commutative group. This seminorm is induced by the $J$-dependent inner product $\langle a \otimes h, b \otimes k \rangle_J = \langle h, J(a^* b) k \rangle_H$.

Given a $C^*$-algebra $A$, a complex Hilbert space $H$, and a completely positive map $J: A \to \mathcal{B}(H)$, the space `Pre J` (which is a type synonym for the algebraic tensor product $A \otimes_{\mathbb{C}} H$) is equipped with a complex inner product space structure. The inner product is defined on elementary tensors by the relation $\langle a \otimes h, b \otimes k \rangle = \langle h, J(a^* b) k \rangle_H$, where $\mathcal{B}(H)$ denotes the space of bounded linear operators on $H$.

Let $A$ be a $C^*$-algebra, $H$ be a complex Hilbert space, and $J: A \to \mathcal{B}(H)$ be a completely positive map. Let $\text{Pre } J$ denote the algebraic tensor product $A \otimes_{\mathbb{C}} H$ equipped with the seminorm induced by the semi-inner product $\langle a \otimes h, b \otimes k \rangle_J = \langle h, J(a^* b) k \rangle_H$. Then $\text{Pre } J$ is a uniform additive group, meaning that its addition and subtraction operations are uniformly continuous with respect to the uniformity induced by its seminorm.

Let $A$ be a $C^*$-algebra, $H$ be a complex Hilbert space, and $\mathcal{B}(H)$ denote the algebra of bounded linear operators on $H$. Given a completely positive map $J: A \to \mathcal{B}(H)$, let $\text{Pre}(J)$ be the algebraic tensor product $A \otimes_{\mathbb{C}} H$ equipped with the seminormed structure induced by the $J$-dependent semi-inner product defined on elementary tensors by $\langle a \otimes h, b \otimes k \rangle_J = \langle h, J(a^* b) k \rangle_H$. Then the scalar multiplication by complex numbers $\mathbb{C}$ on $\text{Pre}(J)$ is a bounded operation.

Let $A$ be a $C^*$-algebra and $H$ be a complex Hilbert space. Given a completely positive map $J : A \to \mathcal{B}(H)$, let $\text{Pre } J$ denote the algebraic tensor product $A \otimes_{\mathbb{C}} H$ equipped with the seminormed structure induced by the semi-inner product $\langle a \otimes h, b \otimes k \rangle = \langle h, J(a^* b) k \rangle_H$. For any constant scalar $c \in \mathbb{C}$, the map $x \mapsto c \cdot x$ on $\text{Pre } J$ is uniformly continuous.

Let $A$ be a $C^*$-algebra, $H$ be a complex Hilbert space, and $J : A \to \mathcal{B}(H)$ be a completely positive map. Let $\text{Pre } J$ denote the pre-Hilbert space formed by the algebraic tensor product $A \otimes_{\mathbb{C}} H$ equipped with the semi-inner product $\langle a \otimes h, b \otimes k \rangle = \langle h, J(a^* b) k \rangle_H$. The space $K(J)$ is defined as the completion of $\text{Pre } J$ with respect to the induced seminorm. This space serves as the canonical Hilbert space for the Stinespring dilation of $J$.

Given a $C^*$-algebra $A$, a complex Hilbert space $H$, and a completely positive map $J: A \to \mathcal{B}(H)$, let $\text{Pre } J$ be the pre-Hilbert space consisting of the algebraic tensor product $A \otimes_{\mathbb{C}} H$ equipped with the semi-inner product $\langle a \otimes h, b \otimes k \rangle = \langle h, J(a^* b) k \rangle_H$. For an element $a \in A$, this definition is the $\mathbb{C}$-linear operator on $\text{Pre } J$ that acts as left multiplication on the first factor, mapping elementary tensors $b \otimes h$ to $(ab) \otimes h$.

Let $A$ be a $C^*$-algebra and $H$ a complex Hilbert space. Given a completely positive map $J : A \to \mathcal{B}(H)$, let $\text{Pre } J$ denote the algebraic tensor product $A \otimes_{\mathbb{C}} H$ equipped with the semi-inner product $\langle a \otimes h, b \otimes k \rangle = \langle h, J(a^* b) k \rangle$. For any $a \in A$ and $x \in \text{Pre } J$, the action of the left multiplication operator $\text{preLeftMul}_J(a)$ on $x$ is equal to the action of the generic algebraic left multiplication operator $\text{leftMul}_a$ on $x$.

Let $A$ be a $C^*$-algebra and $H$ be a complex Hilbert space. Given a completely positive map $J : A \to \mathcal{B}(H)$, let $\text{Pre } J$ denote the algebraic tensor product $A \otimes_{\mathbb{C}} H$ equipped with the seminorm induced by the semi-inner product $\langle a \otimes h, b \otimes k \rangle = \langle h, J(a^* b) k \rangle_H$. For any $a \in A$, this definition is the continuous linear operator on $\text{Pre } J$ that acts as left multiplication on the first factor, mapping elementary tensors $b \otimes h$ to $(ab) \otimes h$. Its continuity is established by the operator norm bound $\|a \cdot x\| \le \|a\| \|x\|$.