Positive-definite function

A complex-valued function $\phi$ on a group $G$ satisfying

$$\sum _ {i,j= 1 } ^ { m } \alpha _ {i} \overline \alpha \; _ {j} \phi ( x _ {j} ^ {-1} x _ {i} ) \geq 0$$

for all choices $x _ {1} \dots x _ {m} \in G$, $\alpha _ {1} \dots \alpha _ {m} \in \mathbf C$. The set of positive-definite functions on $G$ forms a cone in the space $M( G)$ of all bounded functions on $G$ which is closed with respect to the operations of multiplication and complex conjugation.

The reason for distinguishing this class of functions is that positive-definite functions define positive functionals (cf. Positive functional) on the group algebra $\mathbf C G$ and unitary representations of the group $G$( cf. Unitary representation). More precisely, let $\phi : G \rightarrow \mathbf C$ be any function and let $l _ \phi : \mathbf C G\rightarrow \mathbf C$ be the functional given by

$$l _ \phi \left ( \sum _ {g \in G } \alpha _ {g} g \right ) = \ \sum _ {g \in G } \phi ( g) \alpha _ {g} ;$$

then for $l _ \phi$ to be positive it is necessary and sufficient that $\phi$ be a positive-definite function. Further, $l _ \phi$ defines a $*$- representation of the algebra $\mathbf C G$ on a Hilbert space $H _ \phi$, and therefore a unitary representation $\pi _ \phi$ of the group $G$, where $\phi ( g) = ( \pi _ \phi ( g) \xi , \xi )$ for some $\xi \in H _ \phi$. Conversely, for any representation $\pi$ and any vector $\xi \in H _ \phi$, the function $g \rightarrow ( \pi ( g) \xi , \xi )$ is a positive-definite function.

If $G$ is a topological group, the representation $\pi _ \phi$ is weakly continuous if and only if the positive-definite function is continuous. If $G$ is locally compact, continuous positive-definite functions are in one-to-one correspondence with the positive functionals on $L _ {1} ( G)$.

For commutative locally compact groups, the class of continuous positive-definite functions coincides with the class of Fourier transforms of finite positive regular Borel measures on the dual group. There is an analogue of this assertion for compact groups: A continuous function $\phi$ on a compact group $G$ is a positive-definite function if and only if its Fourier transform $\widehat \phi ( b)$ takes positive (operator) values on each element of the dual object, i.e.

$$\int\limits _ { G } \phi ( g) ( \sigma ( g) \xi , \xi ) dg \geq 0$$

for any representation $\sigma$ and any vector $\xi \in H _ \sigma$, where $H _ \sigma$ is the space of $\sigma$.

References

[1]	E. Hewitt, K.A. Ross, "Abstract harmonic analysis" , 2 , Springer (1979)
[2]	M.A. Naimark, "Normed rings" , Reidel (1984) (Translated from Russian)

Comments

The representations of $\mathbf C G$ associated to positive functionals $l$ mentioned above are cyclic representations. A cyclic representation of a $C ^ {*}$- algebra ${\mathcal A}$ is a representation $\rho : {\mathcal A} \rightarrow B( H)$, the $C ^ {*}$- algebra of bounded operators on the Hilbert space $H$, such that there is a vector $\xi \in H$ such that the closure of $\{ {A \xi } : {A \in {\mathcal A} } \}$ is all of $H$. These are the basic components of any representation. Indeed, if $\rho$ is non-degenerate, i.e. $\{ {\xi \in H } : {\rho ( A) ( \xi ) = 0 \textrm{ for all } A \in {\mathcal A} } \} = 0$, then $\rho$ is a direct sum of cyclic representations. Cf. also Cyclic module for an analogous concept in ring and module theory.

The cyclic representation associated to a positive functional $l$ on ${\mathcal A}$ is a suitably completed quotient of the regular representation. More precisely, the construction is as follows. Define an inner product on ${\mathcal A}$ by

$$\langle A, B \rangle = l ( A ^ {*} B ) ,$$

and define a left ideal of ${\mathcal A}$ by

$${\mathcal I} = \{ {A \in {\mathcal A} } : {l( A ^ {*} A ) = 0 } \} .$$

The inner product just defined descends to define an inner product on the quotient space ${\mathcal A} / {\mathcal I}$. Now complete this space to obtain a Hilbert space $H _ {l}$, and define the representation $\pi _ {l}$ by:

$$\pi _ {l} ( A) ([ B ]) \simeq [ AB],$$

where $[ B]$ denotes the class of $B \in {\mathcal A}$ in ${\mathcal A} / {\mathcal I} \subset H _ {l}$. The operator $\pi _ {l} ( A)$ extends to a bounded operator on $H _ {l}$.

If ${\mathcal A}$ contains an identity, then the class of that identity is a cyclic vector for $\pi _ {l}$. If ${\mathcal A}$ does not contain an identity, such is first adjoined to obtain a $C ^ {*}$- algebra ${\mathcal A} tilde$ and the construction is repeated for ${\mathcal A} tilde$. To prove that then the class of 1 is cyclic for ${\mathcal A}$( not just ${\mathcal A} tilde$) one uses an approximate identity for ${\mathcal I}$, i.e. a net (directed set) $\{ E _ \alpha \}$ of positive elements $E _ \alpha \in {\mathcal I}$ such that $\| E _ \alpha \| \leq 1$, $\alpha \leq \beta$ implies $E _ \alpha \leq E _ \beta$ and $\lim\limits _ \alpha \| AE _ \alpha - A \| = 0$ for all $A \in {\mathcal I}$. Such approximate identities always exist. See e.g. [1], vol. 1, p. 321 and [a5], Sects. 2.2.3, 2.3.1 and 2.3.3 for more details on all this.

A positive functional on a $C ^ {*}$- algebra of norm 1 is often called a state, especially in the theoretical physics literature.

References

[a1]	S. Bochner, "Lectures on Fourier integrals" , Princeton Univ. Press (1959) (Translated from German)
[a2]	W. Rudin, "Fourier analysis on groups" , Wiley (1962)
[a3]	H. Reiter, "Classical harmonic analysis and locally compact groups" , Clarendon Press (1968)
[a4]	J. Dixmier, " algebras" , North-Holland (1977) (Translated from French)
[a5]	O. Bratteli, D.W. Robinson, "Operator algebras and quantum statistical mechanics" , 1 , Springer (1979)