Questions tagged [multiplier-algebra]

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Multiplier algebra of Fock space

For any vector space, one may form the tensor algebra with multiplication being the tensor product. For a Hilbert space $\mathcal{H}$, the analogous construction is the Fock space $$ \mathcal{F}(\...
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If $S\subseteq A^*$ is separating, does $S$ also separate $M(A)$?

Let $A$ be a non-unital $C^*$-algebra. Let $S\subseteq A^*$ be a set of continuous functionals that separates the points of $A$. Every element $\omega \in A^*$ extends uniquely to a strictly ...
  • 405
9 votes
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Is a $*$-automorphism $M(A) \to M(A)$ automatically strictly continuous?

Let $A$ be a (non-unital) $C^*$-algebra with multiplier $C^*$-algebra $M(A)$. Let $\phi: M(A) \to M(A)$ be a $*$-automorphism. Is it true that $\phi$ is automatically strictly continuous (on bounded ...
2 votes
1 answer
140 views

$(\iota \otimes f)(X) = 0$ for all $f \in B^*$ implies $X=0$

Let $A$ and $B$ be $C^*$-algebras. Given $f \in B^*$, we can form the right slice map $$\iota \otimes f: A \otimes B \to A: a \otimes b \mapsto af(b)$$ which extends uniquely to a bounded linear map $$...
  • 405
4 votes
1 answer
114 views

Direct sum of multiplier algebras

Consider a collection of $C^*$-algebras $\{A_i\}_{i \in I}$. We can form the direct sum $$\bigoplus_{i \in I}^{c_0} A_i:= \left\{(a_i)_{i \in I} \in \prod_{i\in I} A_i: \lim_{i \in I} \|a_i\| = 0\...
  • 405
2 votes
1 answer
176 views

Strict topology on the multiplier algebra

Let $A$ be a $C^*$-algebra. Let $M(A)$ be its multiplier $C^*$-algebras. The strict topology on $M(A)$ is given by $$x_\lambda \to x \iff \forall a\in A: (\|x_\lambda a-xa\| + \|ax_\lambda - ax\| \to ...
  • 405
1 vote
1 answer
101 views

The mapping $\iota \otimes S$ on the multiplier algebra

Let $A$ be a non-degenerate algebra with multiplier algebra $M(A)$. Let $S: A \to M(A)$ be an antimultiplicative linear map, i.e. $$S(ab) = S(b)S(a).$$ Consider the mapping $$\iota \otimes S: A \...
  • 405
1 vote
1 answer
105 views

About extensions between morphisms on the multiplier algebra

Let $A$ be a non-degenerate algebra and let $\Delta: A \to M(A \otimes A)$ be a non-degenerate morphism. We can extend the algebra morphism $$\iota \otimes \Delta: M(A \otimes A) \to M(A \otimes A \...
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1 vote
1 answer
132 views

Non-degeneracy of comultiplication (multiplier Hopf algebras)

Consider the following fragment from the paper "Multiplier Hopf-algebras" by Van Daele. Can someone explain how the coassociativity in definition 2.2 (ii) and the requirement $(\Delta \...
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1 answer
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Antipode on a multiplier Hopf-algebra

Probably an easy question, but here goes: I'm reading the paper Multiplier Hopf algebras by Van Daele. Let $(A, \Delta)$ be a multiplier Hopf algebra. Let $L(A), R(A), M(A)$ be the left, right and ...
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2 votes
1 answer
104 views

Inclusion $M(A) \otimes M(B)\subseteq M(A\otimes B)$ of multiplier algebras

Consider the following definitions given in Timmerman's book "An invitation to quantum groups and duality": m Further in the book, it is claimed that if $A$ and $B$ are non-degenerate ...
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2 votes
1 answer
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Definition of multiplier bialgebra

Consider the following fragments from "An invitation to quantum groups and duality" by Timmerman: Question: In remark 2.1.6 (ii), it is stated that the homomorphism $\Delta\otimes \text{id}:...
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