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Daniele Tampieri
Daniele Tampieri
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I would like to show the following isomorphy but not sure how to go about this:

$\mathbb{K}\cong M_{n}(\mathbb{K})$

Also in Blackadar (Operator Algebras, page 171) he states that this isomorphism induces: $B\cong M_{2}(B)$, $M(B)\cong M_{2}(M(B))$ and $Q(B)\cong M_{2}(Q(B))$, which he calls standard isomorphisms.

The notation is $B$ is a $C^{*}$ algebra, $Q$ is
In the outer multiplier algebra of $B$ (hence $Q = M(B)/B$), $M$ iscontext or the multiplier algebrapresent question, $M_{n}(\mathbb{K})$ are nxn matrices with entries from $\mathbb{K}$ where $\mathbb{K}$ is the C star algebra of compact operators on a separable, infinite-dimensional Hilbert space.notation has the following meaning:

  • $B$ is a $C^\ast$-algebra,
  • $Q$ is the outer multiplier algebra of $B$ (hence $Q = M(B)/B$),
  • $M$ is the multiplier algebra,
  • $M_{n}(\mathbb{K})$ are $n\times n$ matrices with entries from $\mathbb{K}$, and finally
  • $\mathbb{K}$ is the $C^\star$-algebra of compact operators on a separable, infinite-dimensional Hilbert space.

Anyway, I don't know how to show this and cannot find a reference on how to do this. Would be very thankful for a reference, how to do it or even some hint.

I would like to show the following isomorphy but not sure how to go about this:

$\mathbb{K}\cong M_{n}(\mathbb{K})$

Also in Blackadar (Operator Algebras, page 171) he states that this isomorphism induces: $B\cong M_{2}(B)$, $M(B)\cong M_{2}(M(B))$ and $Q(B)\cong M_{2}(Q(B))$, which he calls standard isomorphisms.

The notation is $B$ is a $C^{*}$ algebra, $Q$ is the outer multiplier algebra of $B$ (hence $Q = M(B)/B$), $M$ is the multiplier algebra, $M_{n}(\mathbb{K})$ are nxn matrices with entries from $\mathbb{K}$ where $\mathbb{K}$ is the C star algebra of compact operators on a separable, infinite-dimensional Hilbert space.

Anyway, I don't know how to show this and cannot find a reference on how to do this. Would be very thankful for a reference, how to do it or even some hint.

I would like to show the following isomorphy but not sure how to go about this:

$\mathbb{K}\cong M_{n}(\mathbb{K})$

Also in Blackadar (Operator Algebras, page 171) he states that this isomorphism induces: $B\cong M_{2}(B)$, $M(B)\cong M_{2}(M(B))$ and $Q(B)\cong M_{2}(Q(B))$, which he calls standard isomorphisms.
In the context or the present question, the notation has the following meaning:

  • $B$ is a $C^\ast$-algebra,
  • $Q$ is the outer multiplier algebra of $B$ (hence $Q = M(B)/B$),
  • $M$ is the multiplier algebra,
  • $M_{n}(\mathbb{K})$ are $n\times n$ matrices with entries from $\mathbb{K}$, and finally
  • $\mathbb{K}$ is the $C^\star$-algebra of compact operators on a separable, infinite-dimensional Hilbert space.

Anyway, I don't know how to show this and cannot find a reference on how to do this. Would be very thankful for a reference, how to do it or even some hint.

added 109 characters in body
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craaaft
craaaft
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I would like to show the following isomorphy but not sure how to go about this:

$\mathbb{K}\cong M_{n}(\mathbb{K})$

Also in Blackadar (Operator Algebras, page 171) he states that this isomorphism induces: $B\cong M_{2}(B)$, $M(B)\cong M_{2}(M(B))$ and $Q(B)\cong M_{2}(Q(B))$, which he calls standard isomorphisms.

The notation is $B$ is a $C^{*}$ algebra, $Q$ is the outer multiplier algebra of $B$ (hence $Q = M(B)/B$), $M$ is the multiplier algebra and, $M_{n}(\mathbb{K})$ are nxn matrices with entries from $\mathbb{K}$ where $\mathbb{K}$ is the C star algebra of compact operators on a separable, infinite-dimensional Hilbert space.

Anyway, I don't know how to show this and cannot find a reference on how to do this. Would be very thankful for a reference, how to do it or even some hint.

I would like to show the following isomorphy but not sure how to go about this:

$\mathbb{K}\cong M_{n}(\mathbb{K})$

Also in Blackadar (Operator Algebras, page 171) he states that this isomorphism induces: $B\cong M_{2}(B)$, $M(B)\cong M_{2}(M(B))$ and $Q(B)\cong M_{2}(Q(B))$, which he calls standard isomorphisms.

The notation is $B$ is a $C^{*}$ algebra, $Q$ is the outer multiplier algebra of $B$ (hence $Q = M(B)/B$), $M$ is the multiplier algebra and $M_{n}(\mathbb{K})$ are nxn matrices with entries from $\mathbb{K}$.

Anyway, I don't know how to show this and cannot find a reference on how to do this. Would be very thankful for a reference, how to do it or even some hint.

I would like to show the following isomorphy but not sure how to go about this:

$\mathbb{K}\cong M_{n}(\mathbb{K})$

Also in Blackadar (Operator Algebras, page 171) he states that this isomorphism induces: $B\cong M_{2}(B)$, $M(B)\cong M_{2}(M(B))$ and $Q(B)\cong M_{2}(Q(B))$, which he calls standard isomorphisms.

The notation is $B$ is a $C^{*}$ algebra, $Q$ is the outer multiplier algebra of $B$ (hence $Q = M(B)/B$), $M$ is the multiplier algebra, $M_{n}(\mathbb{K})$ are nxn matrices with entries from $\mathbb{K}$ where $\mathbb{K}$ is the C star algebra of compact operators on a separable, infinite-dimensional Hilbert space.

Anyway, I don't know how to show this and cannot find a reference on how to do this. Would be very thankful for a reference, how to do it or even some hint.

added 213 characters in body
Source Link
craaaft
craaaft
  • 129
  • 2

I would like to show the following isomorphy but not sure how to go about this:

$\mathbb{K}\cong M_{n}(\mathbb{K})$

Also in Blackadar (Operator Algebras, page 171) he states that this isomorphism induces: $B\cong M_{2}(B)$, $M(B)\cong M_{2}(M(B))$ and $Q(B)\cong M_{2}(Q(B))$, which he calls standard isomorphisms.

The notation is $B$ is a $C^{*}$ algebra, $Q$ is the outer multiplier algebra of $B$ (hence $Q = M(B)/B$), $M$ is the multiplier algebra and $M_{n}(\mathbb{K})$ are nxn matrices with entries from $\mathbb{K}$.

Anyway, I don't know how to show this and cannot find a reference on how to do this. Would be very thankful for a reference, how to do it or even some hint.

I would like to show the following isomorphy but not sure how to go about this:

$\mathbb{K}\cong M_{n}(\mathbb{K})$

Also in Blackadar (Operator Algebras, page 171) he states that this isomorphism induces: $B\cong M_{2}(B)$, $M(B)\cong M_{2}(M(B))$ and $Q(B)\cong M_{2}(Q(B))$, which he calls standard isomorphisms.

Anyway, I don't know how to show this and cannot find a reference on how to do this. Would be very thankful for a reference, how to do it or even some hint.

I would like to show the following isomorphy but not sure how to go about this:

$\mathbb{K}\cong M_{n}(\mathbb{K})$

Also in Blackadar (Operator Algebras, page 171) he states that this isomorphism induces: $B\cong M_{2}(B)$, $M(B)\cong M_{2}(M(B))$ and $Q(B)\cong M_{2}(Q(B))$, which he calls standard isomorphisms.

The notation is $B$ is a $C^{*}$ algebra, $Q$ is the outer multiplier algebra of $B$ (hence $Q = M(B)/B$), $M$ is the multiplier algebra and $M_{n}(\mathbb{K})$ are nxn matrices with entries from $\mathbb{K}$.

Anyway, I don't know how to show this and cannot find a reference on how to do this. Would be very thankful for a reference, how to do it or even some hint.

Source Link
craaaft
craaaft
  • 129
  • 2