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InfiniteLooper
InfiniteLooper
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For a Hilbert space $H$, the Riesz representation theorem states that $H$ is isomorphic to its dual $H^*$ via $x \mapsto \langle x, -\rangle$.

It is often stated in the literature that this does not work in full generality for a Hilbert module over a $C^*$ algebra. For example, attempts to define the adjoint of a morphism between Hilbert modules run into the lack of such a representation theorem. To avoid this, we insist that a morphism between Hilbert modules admit an adjoint.

Is there a simple counterexample to Riesz representability for Hilbert modules?

To be more precise, take $E$ an Hilbert module over $A$ and $\phi : E \to A$ being $A$ linear. The question is on the existence of an $x$ in $E$ such that $\phi = \langle x, - \rangle$.

For a Hilbert space $H$, the Riesz representation theorem states that $H$ is isomorphic to its dual $H^*$ via $x \mapsto \langle x, -\rangle$.

It is often stated in the literature that this does not work in full generality for a Hilbert module over a $C^*$ algebra. For example, attempts to define the adjoint of a morphism between Hilbert modules run into the lack of such a representation theorem. To avoid this, we insist that a morphism between Hilbert modules admit an adjoint.

Is there a simple counterexample to Riesz representability for Hilbert modules?

For a Hilbert space $H$, the Riesz representation theorem states that $H$ is isomorphic to its dual $H^*$ via $x \mapsto \langle x, -\rangle$.

It is often stated in the literature that this does not work in full generality for a Hilbert module over a $C^*$ algebra. For example, attempts to define the adjoint of a morphism between Hilbert modules run into the lack of such a representation theorem. To avoid this, we insist that a morphism between Hilbert modules admit an adjoint.

Is there a simple counterexample to Riesz representability for Hilbert modules?

To be more precise, take $E$ an Hilbert module over $A$ and $\phi : E \to A$ being $A$ linear. The question is on the existence of an $x$ in $E$ such that $\phi = \langle x, - \rangle$.

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David Handelman
David Handelman
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couterexemple for riesz Counterexample to Riesz representation failure for hilbertHilbert modules

For a Hilbert space $H$, the Riesz representation theorem states that $H$ is isomorphic to its dual $H^*$ via $x \mapsto \langle x, -\rangle$.

It is often stated in the litteratureliterature that this does not work in full generality for ana Hilbert module over a $C^*$ algebra. For exemple when one wantexample, attempts to define the adjoint of a morphism between hilbertHilbert modules, one can be stucked by run into the non existence oflack of such a representation theorem. At the end (the end of the beggining)To avoid this, we defineinsist that a morphism of hilbert module to be a function thatbetween Hilbert modules admit an adjoint.

Is there a simple counterexample to RiesRiesz representability for hilbert module Hilbert modules?

couterexemple for riesz representation failure for hilbert modules

For a Hilbert space $H$ the Riesz representation theorem states that $H$ is isomorphic to its dual $H^*$ via $x \mapsto \langle x, -\rangle$.

It is often stated in the litterature that this does not work in full generality for an Hilbert module over a $C^*$ algebra. For exemple when one want to define the adjoint of a morphism between hilbert modules, one can be stucked by the non existence of such a representation theorem. At the end (the end of the beggining), we define a morphism of hilbert module to be a function that admit an adjoint.

Is there a simple counterexample to Ries representability for hilbert module ?

Counterexample to Riesz representation for Hilbert modules

For a Hilbert space $H$, the Riesz representation theorem states that $H$ is isomorphic to its dual $H^*$ via $x \mapsto \langle x, -\rangle$.

It is often stated in the literature that this does not work in full generality for a Hilbert module over a $C^*$ algebra. For example, attempts to define the adjoint of a morphism between Hilbert modules run into the lack of such a representation theorem. To avoid this, we insist that a morphism between Hilbert modules admit an adjoint.

Is there a simple counterexample to Riesz representability for Hilbert modules?

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InfiniteLooper
InfiniteLooper
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couterexemple for riesz representation failure for hilbert modules

For a Hilbert space $H$ the Riesz representation theorem states that $H$ is isomorphic to its dual $H^*$ via $x \mapsto \langle x, -\rangle$.

It is often stated in the litterature that this does not work in full generality for an Hilbert module over a $C^*$ algebra. For exemple when one want to define the adjoint of a morphism between hilbert modules, one can be stucked by the non existence of such a representation theorem. At the end (the end of the beggining), we define a morphism of hilbert module to be a function that admit an adjoint.

Is there a simple counterexample to Ries representability for hilbert module ?