Let $(A,\alpha, G)$ be a $C^*$-dynamical system, where $G$ is a discrete group. Let $\Gamma$ be a subgroup of $G$, then we can form two universal crossed products $A\rtimes_\alpha \Gamma$ and $A\rtimes_\alpha G$. Question 1: Is there a the canonical inclusion map $A\rtimes_\alpha \Gamma \subseteq to A\rtimes_\alpha G$ injective? Question 2: what What about reduced crossed products $A\rtimes_{\alpha,r} \Gamma \subseteq to A\rtimes_{\alpha,r} G$?

For the universal case, I guess it's wrong even though I can not find a counter examplecounterexample. But if we let $\alpha$ be the trivial action of $G$, then we only need to look at $A\otimes_{max} C^* (\Gamma) \subseteq A\otimes_{max} C^* (G) $.

For the reduced case, I guess it's should be the case. Just follows from the facts that the left regular representaion of $G$, restricted to $\Gamma$, is a multiple of the left regular representation of $\Gamma$ and $A\rtimes_{\alpha,r} \Gamma = C^*(\pi(A), 1\otimes \lambda(G))$, where $\pi: A\subseteq B(H) \rightarrow B(H\otimes l^2(G))$ and $\lambda$ is the left regular representaion of $G$. Do I make any mistake?

Let $(A,\alpha, G)$ be a $C^*$-dynamical system, where $G$ is discrete group. Let $\Gamma$ be a subgroup of $G$, then we can form two universal crossed products $A\rtimes_\alpha \Gamma$ and $A\rtimes_\alpha G$. Question 1: Is there a canonical inclusion $A\rtimes_\alpha \Gamma \subseteq A\rtimes_\alpha G$? Question 2: what about reduced crossed products $A\rtimes_{\alpha,r} \Gamma \subseteq A\rtimes_{\alpha,r} G$?

For universal case I guess it's wrong even though I can not find a counter example. But if we let $\alpha$ be the trivial action of $G$, then we only need to look at $A\otimes_{max} C^* (\Gamma) \subseteq A\otimes_{max} C^* (G) $, which is the case if $C^* (\Gamma \subseteq C^* (G)$ hereditary subalgebra.only if?

For reduced case I guess it's should be the case. Just follows from the facts that the left regular representaion of $G$, restricted to $\Gamma$, is a multiple of the left regular representation of $\Gamma$ and $A\rtimes_{\alpha,r} \Gamma = C^*(\pi(A), 1\otimes \lambda(G))$, where $\pi: A\subseteq B(H) \rightarrow B(H\otimes l^2(G))$ and $\lambda$ is the left regular representaion of $G$. Do I make any mistake?

Let $(A,\alpha, G)$ be a $C*$-dynamical C^*$-dynamical system, where $G$ is discrete group. Let $\Gamma$ be a subgroup of $G$, then we can form two universal crossed product products $A\rtimes_\alpha \Gamma$ and $A\rtimes_\alpha G$. Question 1: Is there a canonical inclusion $A\rtimes_\alpha \Gamma \subseteq A\rtimes_\alpha G$? Question 2: what about reduced crossed product products $A\rtimes_{\alpha,r} \Gamma \subseteq A\rtimes_{\alpha,r} G$?

For universal case I guess it's wrong even though I can not find a counter example. But if we let $\alpha$ be the trivial action of $G$, then we only need to look at $A\otimes_{max} C*(\Gamma)\subseteq C^* (\Gamma) \subseteq A\otimes_{max} C*(G)$C^* (G) $, which is the case if $C^* (\Gamma \subseteq C^* (G)$ hereditary subalgebra. only if?

For reduced case I guess it's should be the case. Just follows from the fact facts that the left regular representaion of $G$, restricted to $\Gamma$, is a multiple of the left regular representation of $\Gamma$ and $A\rtimes_{\alpha,r} \Gamma = C*(\pi(A)C^*(\pi(A), 1\otimes \lambda(G))$, where $\pi: A\subseteq B(H) \rightarrow B(H\otimes l^2(G))$ and $\lambda$ is the left regular representaion of $G$ G$. Do I make any mistake?

Let $(A,\alpha, G)$ be a $C*$-dynamical system, where $G$ is discrete group. Let $\Gamma$ be a subgroup of $G$, then we can form two universal crossed product $A\rtimes_\alpha \Gamma$ and $A\rtimes_\alpha G$. Question 1: Is there a canonical inclusion $A\rtimes_\alpha \Gamma \subseteq A\rtimes_\alpha G$? Question 2: what about reduced crossed product $A\rtimes_{\alpha,r} \Gamma \subseteq A\rtimes_{\alpha,r} G$?

For universal case I guess it's wrong even though I can not find a counter example. But if we let $\alpha$ be the trivial action of $G$, then we only need to look at $A\otimes_{max} C*(\Gamma)\subseteq A\otimes_{max} C*(G)$.

For reduced case I guess it's should be the case. Just follows from the fact that $A\rtimes_{\alpha,r} \Gamma = C*(\pi(A), 1\otimes \lambda(G))$, where $\pi: A\subseteq B(H) \rightarrow B(H\otimes l^2(G))$ and $\lambda$ is the left regular representaion of $G$