Hilbert C*-module

Builds on Positive element Hilbert space

Definition

Let $A$ be a C*-algebra. Let $E$ be a complex vector space that is a right $A$ -module, with $\lambda (x a) = (\lambda x) a = x (\lambda a)$ for $\lambda \in \C$ , $x \in E$ and $a \in A$ . An $A$ -valued inner product on $E$ is a map $\ip{\cdot}{\cdot} \colon E \times E \to A$ such that for all $x, y, z \in E$ , $a \in A$ and $\lambda \in \C$ :

$\ip{x}{y + \lambda z} = \ip{x}{y} + \lambda \ip{x}{z}$ .
$\ip{x}{y a} = \ip{x}{y} a$ .
$\ip{y}{x} = \ip{x}{y}^*$ .
$\ip{x}{x}$ is a positive element of $A$ , and $\ip{x}{x} = 0$ only for $x = 0$ .

Then $\norm{x} = \norm{\ip{x}{x}}^{1/2}$ is a norm on $E$ . A Hilbert C-module* over $A$ is such a module $E$ that is complete for this norm.

Remarks

The inner product is linear in the second argument and conjugate-linear in the first, as is usual in this theory. A Hilbert $\C$ -module is the same as a complex Hilbert space with this convention.

Example: $E = A$ with $\ip{a}{b} = a^* b$ . Clauses 1 to 3 follow from the algebra rules and $(a^* b)^* = b^* a$ ; the element $a^* a$ is positive, and $a^* a = 0$ forces $a = 0$ . The norm is $\norm{a^* a}^{1/2} = \norm{a}$ by the C* identity, so $E$ is complete.

Another example is $A^n$ with $\ip{x}{y} = \sum_{i=1}^n x_i^* y_i$ .

Non-example: $E = \C$ over $A = \C$ with $\ip{x}{y} = -\overline{x} y$ . Clauses 1 to 3 hold, but $\ip{1}{1} = -1$ is not positive, so clause 4 fails.

Reading path

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Hilbert space concept
Positive element concept
Hilbert C*-module definition

Tag	`9K11FXYX5S`
URI	`m2://9K11FXYX5S`
Kind	definition
Writing	W0 stub
MSC	46L08
Defines	Hilbert C*-module
Uses	Positive element, Hilbert space
Changed	2 November 2026

Hilbert C*-module

Definition

Remarks

Reading path

Further reading

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