---
tag: KBPJH3J56K
kind: concept
name: Hilbert C*-module
msc: [46L08]
sources:
  - url: https://en.wikipedia.org/wiki/Hilbert_C*-module
same_as:
  wikidata: Q5761182
  wikipedia: Hilbert C*-module
---

## Summary

A Hilbert C*-module over a C*-algebra $A$ is a module over $A$ with an inner product.
The values of this inner product lie in $A$ instead of in $\C$.
It generalizes a Hilbert space, which is the case $A = \C$.
Hilbert modules are the basic tool of Kasparov's KK-theory and of Morita equivalence of C*-algebras.

## Explanations

### L1 human

In a Hilbert space the inner product of two vectors is a complex number.
In a Hilbert C*-module the "numbers" are replaced by elements of a C*-algebra, for example by continuous functions on a space.
A typical picture is a continuous family of Hilbert spaces indexed by points of a space, with a pointwise inner product.
The simplest example is the algebra $A$ itself, with $\ip{a}{b} = a^* b$.