Positive-definite function
A complex-valued function on a group
satisfying
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for all choices ,
. The set of positive-definite functions on
forms a cone in the space
of all bounded functions on
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 and unitary representations of the group
(cf. Unitary representation). More precisely, let
be any function and let
be the functional given by
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then for to be positive it is necessary and sufficient that
be a positive-definite function. Further,
defines a
-representation of the algebra
on a Hilbert space
, and therefore a unitary representation
of the group
, where
for some
. Conversely, for any representation
and any vector
, the function
is a positive-definite function.
If is a topological group, the representation
is weakly continuous if and only if the positive-definite function is continuous. If
is locally compact, continuous positive-definite functions are in one-to-one correspondence with the positive functionals on
.
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 on a compact group
is a positive-definite function if and only if its Fourier transform
takes positive (operator) values on each element of the dual object, i.e.
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for any representation and any vector
, where
is the space of
.
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 associated to positive functionals
mentioned above are cyclic representations. A cyclic representation of a
-algebra
is a representation
, the
-algebra of bounded operators on the Hilbert space
, such that there is a vector
such that the closure of
is all of
. These are the basic components of any representation. Indeed, if
is non-degenerate, i.e.
, then
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 on
is a suitably completed quotient of the regular representation. More precisely, the construction is as follows. Define an inner product on
by
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and define a left ideal of by
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The inner product just defined descends to define an inner product on the quotient space . Now complete this space to obtain a Hilbert space
, and define the representation
by:
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where denotes the class of
in
. The operator
extends to a bounded operator on
.
If contains an identity, then the class of that identity is a cyclic vector for
. If
does not contain an identity, such is first adjoined to obtain a
-algebra
and the construction is repeated for
. To prove that then the class of 1 is cyclic for
(not just
) one uses an approximate identity for
, i.e. a net (directed set)
of positive elements
such that
,
implies
and
for all
. 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 -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, "![]() |
[a5] | O. Bratteli, D.W. Robinson, "Operator algebras and quantum statistical mechanics" , 1 , Springer (1979) |
Positive-definite function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Positive-definite_function&oldid=19269