Let be a group and let be the *C*-algebra *of the group . This is a C*-algebra whose elements are complex-valued functions on the group . We define operations on in the ordinary way save for multiplication

,

and the adjoint . Note that the above multiplication is the same as defining and extending via linearity. Thence is abelian if and only if is.

To give the structure of a quantum group we define the following linear maps:

, .

,

, .

The functional defined by is the Haar state on . It is very easy to write down the :

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To do probability theory consider states on and form the product state:

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Whenever is a state of such that implies that , then the distribution of the random variables converges to .

At the moment we will use the one-norm to measure the distance to stationary:

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A quick calculation shows that:

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When, for example, when are transpositions in , then we have

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