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duh
duh
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Let $f:G\to \mathbb{C}$ be a finitely supported functions and let $m_f$ denote the associated multiplier on $C^*_r(G)$, the reduced group $C^*$-algebra: $$m_f(\alpha)(g)=f(g)\alpha(g)$$ for every $\alpha\in C^*_r(G)$.

We can look at two natural norms of $m_f$:

  • the completely bounded norm $\Vert m_f\Vert_{cb}$ of $m_f$ as a multiplier on $C^*_r(G)$
  • the multiplier norm $\Vert m_f\Vert_m$ of $m_f$ viewed as a bounded linear operator on $C^*_r(G)$.

In general, $$\Vert m_f\Vert_m\le \Vert m_f\Vert_{cb}.$$

Is there anything else known about the relation of these two norms? For instance, are there examples of $f$ with $\Vert m_f\Vert_{cb}=1$ and $\Vert m_f \Vert\to 0$$\Vert m_f \Vert_m\to 0$? Or should they be equal in some cases? I would be grateful for either an answer or direction where to look for this in the literature.

Let $f:G\to \mathbb{C}$ be a finitely supported functions and let $m_f$ denote the associated multiplier on $C^*_r(G)$, the reduced group $C^*$-algebra: $$m_f(\alpha)(g)=f(g)\alpha(g)$$ for every $\alpha\in C^*_r(G)$.

We can look at two natural norms of $m_f$:

  • the completely bounded norm $\Vert m_f\Vert_{cb}$ of $m_f$ as a multiplier on $C^*_r(G)$
  • the multiplier norm $\Vert m_f\Vert_m$ of $m_f$ viewed as a bounded linear operator on $C^*_r(G)$.

In general, $$\Vert m_f\Vert_m\le \Vert m_f\Vert_{cb}.$$

Is there anything else known about the relation of these two norms? For instance, are there examples of $f$ with $\Vert m_f\Vert_{cb}=1$ and $\Vert m_f \Vert\to 0$? Or should they be equal in some cases? I would be grateful for either an answer or direction where to look for this in the literature.

Let $f:G\to \mathbb{C}$ be a finitely supported functions and let $m_f$ denote the associated multiplier on $C^*_r(G)$, the reduced group $C^*$-algebra: $$m_f(\alpha)(g)=f(g)\alpha(g)$$ for every $\alpha\in C^*_r(G)$.

We can look at two natural norms of $m_f$:

  • the completely bounded norm $\Vert m_f\Vert_{cb}$ of $m_f$ as a multiplier on $C^*_r(G)$
  • the multiplier norm $\Vert m_f\Vert_m$ of $m_f$ viewed as a bounded linear operator on $C^*_r(G)$.

In general, $$\Vert m_f\Vert_m\le \Vert m_f\Vert_{cb}.$$

Is there anything else known about the relation of these two norms? For instance, are there examples of $f$ with $\Vert m_f\Vert_{cb}=1$ and $\Vert m_f \Vert_m\to 0$? Or should they be equal in some cases? I would be grateful for either an answer or direction where to look for this in the literature.

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duh
duh
  • 165
  • 5

Let $f:G\to \mathbb{C}$ be a finitely supported functions and let $m_f$ denote the associated multiplier on $C^*_r(G)$, the reduced group $C^*$-algebra: $$m_f(\alpha)(g)=f(g)\alpha(g)$$ for every $\alpha\in C^*_r(G)$.

We can look at two natural norms of $f$$m_f$:

  • the completely bounded norm $\Vert f\Vert_{cb}$$\Vert m_f\Vert_{cb}$ of $f$$m_f$ as a multiplier on $C^*_r(G)$
  • the multiplier norm $\Vert f\Vert_m$$\Vert m_f\Vert_m$ of $f$$m_f$ viewed as a bounded linear operator on $C^*_r(G)$.

In general, $$\Vert f\Vert_m\le \Vert f\Vert_{cb}.$$$$\Vert m_f\Vert_m\le \Vert m_f\Vert_{cb}.$$

Is there anything else known about the relation of these two norms? For instance, are there examples of $f$ with $\Vert f\Vert_{cb}=1$$\Vert m_f\Vert_{cb}=1$ and $\Vert f \Vert\to 0$$\Vert m_f \Vert\to 0$? Or should they be equal in some cases? I would be grateful for either an answer or direction where to look for this in the literature.

Let $f:G\to \mathbb{C}$ be a finitely supported functions and let $m_f$ denote the associated multiplier on $C^*_r(G)$, the reduced group $C^*$-algebra: $$m_f(\alpha)(g)=f(g)\alpha(g)$$ for every $\alpha\in C^*_r(G)$.

We can look at two natural norms of $f$:

  • the completely bounded norm $\Vert f\Vert_{cb}$ of $f$ as a multiplier on $C^*_r(G)$
  • the multiplier norm $\Vert f\Vert_m$ of $f$ viewed as a bounded linear operator on $C^*_r(G)$.

In general, $$\Vert f\Vert_m\le \Vert f\Vert_{cb}.$$

Is there anything else known about the relation of these two norms? For instance, are there examples with $\Vert f\Vert_{cb}=1$ and $\Vert f \Vert\to 0$? Or should they be equal in some cases? I would be grateful for either an answer or direction where to look for this in the literature.

Let $f:G\to \mathbb{C}$ be a finitely supported functions and let $m_f$ denote the associated multiplier on $C^*_r(G)$, the reduced group $C^*$-algebra: $$m_f(\alpha)(g)=f(g)\alpha(g)$$ for every $\alpha\in C^*_r(G)$.

We can look at two natural norms of $m_f$:

  • the completely bounded norm $\Vert m_f\Vert_{cb}$ of $m_f$ as a multiplier on $C^*_r(G)$
  • the multiplier norm $\Vert m_f\Vert_m$ of $m_f$ viewed as a bounded linear operator on $C^*_r(G)$.

In general, $$\Vert m_f\Vert_m\le \Vert m_f\Vert_{cb}.$$

Is there anything else known about the relation of these two norms? For instance, are there examples of $f$ with $\Vert m_f\Vert_{cb}=1$ and $\Vert m_f \Vert\to 0$? Or should they be equal in some cases? I would be grateful for either an answer or direction where to look for this in the literature.

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duh
duh
  • 165
  • 5

Multiplier norm vs cb norm

Let $f:G\to \mathbb{C}$ be a finitely supported functions and let $m_f$ denote the associated multiplier on $C^*_r(G)$, the reduced group $C^*$-algebra: $$m_f(\alpha)(g)=f(g)\alpha(g)$$ for every $\alpha\in C^*_r(G)$.

We can look at two natural norms of $f$:

  • the completely bounded norm $\Vert f\Vert_{cb}$ of $f$ as a multiplier on $C^*_r(G)$
  • the multiplier norm $\Vert f\Vert_m$ of $f$ viewed as a bounded linear operator on $C^*_r(G)$.

In general, $$\Vert f\Vert_m\le \Vert f\Vert_{cb}.$$

Is there anything else known about the relation of these two norms? For instance, are there examples with $\Vert f\Vert_{cb}=1$ and $\Vert f \Vert\to 0$? Or should they be equal in some cases? I would be grateful for either an answer or direction where to look for this in the literature.