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YCor
YCor
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Let$\DeclareMathOperator\op{op}$Let $A$ be a $C^*$-algebra. We know that $A$ admits a natural operator space structure, namely the operator space structure induced by any faithful $*$-representation of $A$ into some $B(\mathcal H)$ for some Hilbert space $\mathcal H.$ Now consider the opposite $C^*$ algebra $A^{op}$$A^{\op}$ which is the same as $A$ but multiplication reversed. Now $A^{op}$$A^{\op}$ has a natural operator space structure as well.

  1. Is it true that the matrix norms are given by $\|[x_{ij}]_{i,j=1}^n\|_{M_n(A^{op})}=\|[x_{ji}]_{i,j=1}^n\|_{M_n(A)}$$\|[x_{ij}]_{i,j=1}^n\|_{M_n(A^{\op})}=\|[x_{ji}]_{i,j=1}^n\|_{M_n(A)}$?
  2. Suppose $A$ and $A^{op}$$A^{\op}$ is isomorphic as $C^*$-algebras then there exists some constant $C>0$ such that for all $n\geq 1$ and $[x_{ij}]_{i,j=1}^n\in M_n(A)$ we have $\|[x_{ij}]_{i,j=1}^n\|_{M_n(A)}\leq C \|[x_{ji}]_{i,j=1}^n\|_{M_n(A)}$?

Let $A$ be a $C^*$-algebra. We know that $A$ admits a natural operator space structure, namely the operator space structure induced by any faithful $*$-representation of $A$ into some $B(\mathcal H)$ for some Hilbert space $\mathcal H.$ Now consider the opposite $C^*$ algebra $A^{op}$ which is the same as $A$ but multiplication reversed. Now $A^{op}$ has a natural operator space structure as well.

  1. Is it true that the matrix norms are given by $\|[x_{ij}]_{i,j=1}^n\|_{M_n(A^{op})}=\|[x_{ji}]_{i,j=1}^n\|_{M_n(A)}$?
  2. Suppose $A$ and $A^{op}$ is isomorphic as $C^*$-algebras then there exists some constant $C>0$ such that for all $n\geq 1$ and $[x_{ij}]_{i,j=1}^n\in M_n(A)$ we have $\|[x_{ij}]_{i,j=1}^n\|_{M_n(A)}\leq C \|[x_{ji}]_{i,j=1}^n\|_{M_n(A)}$?

$\DeclareMathOperator\op{op}$Let $A$ be a $C^*$-algebra. We know that $A$ admits a natural operator space structure, namely the operator space structure induced by any faithful $*$-representation of $A$ into some $B(\mathcal H)$ for some Hilbert space $\mathcal H.$ Now consider the opposite $C^*$ algebra $A^{\op}$ which is the same as $A$ but multiplication reversed. Now $A^{\op}$ has a natural operator space structure as well.

  1. Is it true that the matrix norms are given by $\|[x_{ij}]_{i,j=1}^n\|_{M_n(A^{\op})}=\|[x_{ji}]_{i,j=1}^n\|_{M_n(A)}$?
  2. Suppose $A$ and $A^{\op}$ is isomorphic as $C^*$-algebras then there exists some constant $C>0$ such that for all $n\geq 1$ and $[x_{ij}]_{i,j=1}^n\in M_n(A)$ we have $\|[x_{ij}]_{i,j=1}^n\|_{M_n(A)}\leq C \|[x_{ji}]_{i,j=1}^n\|_{M_n(A)}$?
Fixed typo in title, minor copyediting
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gmvh
gmvh
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Oppositie Opposite $C^*$ algebras

Let $A$ be a $C^*$-algebra. We know that $A$ admits a natural operator space structure, namely the operator space structure induced by any faithful $*$-representation of $A$ into some $B(\mathcal H)$ for some Hilbert space $\mathcal H.$ Now consider the opposite $C^*$ algebra $A^{op}$ which is the same as $A$ but multiplication reversed. Now $A^{op}$ has a natural operator space structure as well.

  1. Is it true that the matricialmatrix norms are given by $\|[x_{ij}]_{i,j=1}^n\|_{M_n(A^{op})}=\|[x_{ji}]_{i,j=1}^n\|_{M_n(A)}$?
  2. Suppose $A$ and $A^{op}$ is isomorphic as $C^*$-algebras then there exists some constant $C>0$ such that for all $n\geq 1$ and $[x_{ij}]_{i,j=1}^n\in M_n(A)$ we have $\|[x_{ij}]_{i,j=1}^n\|_{M_n(A)}\leq C \|[x_{ji}]_{i,j=1}^n\|_{M_n(A)}$?

Oppositie $C^*$ algebras

Let $A$ be a $C^*$-algebra. We know that $A$ admits a natural operator space structure, namely the operator space structure induced by any faithful $*$-representation of $A$ into some $B(\mathcal H)$ for some Hilbert space $\mathcal H.$ Now consider the opposite $C^*$ algebra $A^{op}$ which is the same as $A$ but multiplication reversed. Now $A^{op}$ has a natural operator space structure as well.

  1. Is it true that the matricial norms are given by $\|[x_{ij}]_{i,j=1}^n\|_{M_n(A^{op})}=\|[x_{ji}]_{i,j=1}^n\|_{M_n(A)}$?
  2. Suppose $A$ and $A^{op}$ is isomorphic as $C^*$-algebras then there exists some constant $C>0$ such that for all $n\geq 1$ and $[x_{ij}]_{i,j=1}^n\in M_n(A)$ we have $\|[x_{ij}]_{i,j=1}^n\|_{M_n(A)}\leq C \|[x_{ji}]_{i,j=1}^n\|_{M_n(A)}$?

Opposite $C^*$ algebras

Let $A$ be a $C^*$-algebra. We know that $A$ admits a natural operator space structure, namely the operator space structure induced by any faithful $*$-representation of $A$ into some $B(\mathcal H)$ for some Hilbert space $\mathcal H.$ Now consider the opposite $C^*$ algebra $A^{op}$ which is the same as $A$ but multiplication reversed. Now $A^{op}$ has a natural operator space structure as well.

  1. Is it true that the matrix norms are given by $\|[x_{ij}]_{i,j=1}^n\|_{M_n(A^{op})}=\|[x_{ji}]_{i,j=1}^n\|_{M_n(A)}$?
  2. Suppose $A$ and $A^{op}$ is isomorphic as $C^*$-algebras then there exists some constant $C>0$ such that for all $n\geq 1$ and $[x_{ij}]_{i,j=1}^n\in M_n(A)$ we have $\|[x_{ij}]_{i,j=1}^n\|_{M_n(A)}\leq C \|[x_{ji}]_{i,j=1}^n\|_{M_n(A)}$?
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A beginner mathmatician
A beginner mathmatician
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Oppositie $C^*$ algebras

Let $A$ be a $C^*$-algebra. We know that $A$ admits a natural operator space structure, namely the operator space structure induced by any faithful $*$-representation of $A$ into some $B(\mathcal H)$ for some Hilbert space $\mathcal H.$ Now consider the opposite $C^*$ algebra $A^{op}$ which is the same as $A$ but multiplication reversed. Now $A^{op}$ has a natural operator space structure as well.

  1. Is it true that the matricial norms are given by $\|[x_{ij}]_{i,j=1}^n\|_{M_n(A^{op})}=\|[x_{ji}]_{i,j=1}^n\|_{M_n(A)}$?
  2. Suppose $A$ and $A^{op}$ is isomorphic as $C^*$-algebras then there exists some constant $C>0$ such that for all $n\geq 1$ and $[x_{ij}]_{i,j=1}^n\in M_n(A)$ we have $\|[x_{ij}]_{i,j=1}^n\|_{M_n(A)}\leq C \|[x_{ji}]_{i,j=1}^n\|_{M_n(A)}$?