Dirichlet algebra
Let be a uniform algebra on
and
the algebra of all continuous functions on
(cf. also Algebra of functions). The algebra
is called a Dirichlet algebra if
is uniformly dense in
. Dirichlet algebras were introduced by A.M. Gleason [a4].
Let be a compact subset of the complex plane. Let
consist of those functions which are analytic on the interior of
and let
be the uniform closure in
of the functions analytic on a neighbourhood of
. T. Gamelin and J. Garnett [a3] determined exactly when
or
is a Dirichlet algebra on
. The disc algebra
is the algebra of all functions which are analytic in the open unit disc
and continuous in the closed unit disc
. The algebra
is a typical example of a Dirichlet algebra on the unit circle
. For
, the measure
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is the representing measure for the origin, that is,
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for . The origin gives a complex homomorphism for
. For
, the Hardy space
is defined as the closure of
in
(cf. also Hardy spaces). Let
be a Dirichlet algebra on
and
a non-zero complex homomorphism of
. If
is a representing measure on
for
, then
is unique. For
, the abstract Hardy space
is defined as the closure of
in
. A lot of theorems for the Hardy space
are valid for the abstract Hardy space
.
Let be a probability measure space (cf. also Probability measure; Measure space), let
be a subalgebra of
containing the constants and let
be multiplicative on
. The algebra
is called a weak
Dirichlet algebra if
is weak
dense in
. A Dirichlet algebra is a weak
Dirichlet algebra when
is a representing measure on it. Weak
Dirichlet algebras were introduced by T. Srinivasan and J. Wang [a9] as the smallest axiomatic setting on which each one of a lot of important theorems for the Hardy space
are equivalent to the fact that
is weak
dense in
.
K. Hoffman and H. Rossi [a6] gave an example such that even if is dense in
,
is not a weak
Dirichlet algebra. Subsequently, it was shown [a6] that if
is dense in
, then
is a weak
Dirichlet algebra. W. Arveson [a1] introduced non-commutative weak
Dirichlet algebras, which are also called subdiagonal algebras.
Examples of (weak
) Dirichlet algebras.
Let be a compact subset of the complex plane and suppose the algebra
consists of the functions in
that can be approximated uniformly on
by polynomials in
. Then
is a Dirichlet algebra on the outer boundary of
[a2].
Let be the real line
endowed with the discrete topology and suppose the algebra
consists of the functions in
whose Fourier coefficients are zero on the semi-group
, where
is the compact dual group of
. Then
is a Dirichlet algebra on
[a5].
Let be a fixed compact Hausdorff space upon which the real line
(with the usual topology) acts as a locally compact transformation group. The pair
is called a flow. The translate of an
by a
is written as
. A
is called analytic if for each
the function
of
is a boundary function which is bounded and analytic in the upper half-plane. If
is an invariant ergodic probability measure on
, then
is a weak
Dirichlet algebra in
[a7]. See also Hypo-Dirichlet algebra.
References
[a1] | W. Arveson, "Analyticity in operator algebras" Amer. J. Math. , 89 (1967) pp. 578–642 |
[a2] | H. Barbey, H. König, "Abstract analytic function theory and Hardy algebras" , Lecture Notes Math. : 593 , Springer (1977) |
[a3] | T. Gamelin, J. Garnett, "Pointwise bounded approximation and Dirichlet algebras" J. Funct. Anal. , 8 (1971) pp. 360–404 |
[a4] | A. Gleason, "Function algebras" , Sem. Analytic Functions , II , Inst. Adv. Study Princeton (1957) |
[a5] | H. Helson, "Analyticity on compact Abelian groups" , Algebras in Analysis; Proc. Instructional Conf. and NATO Adv. Study Inst., Birmigham, 1973 , Acad. Press (1975) pp. 1–62 |
[a6] | K. Hoffman, H. Rossi, "Function theory from a multiplicative linear functional" Trans. Amer. Math. Soc. , 102 (1962) pp. 507–544 |
[a7] | P. Muhly, "Function algebras and flows" Acta Sci. Math. , 35 (1973) pp. 111–121 |
[a8] | T. Nakazi, "Hardy spaces and Jensen measures" Trans. Amer. Math. Soc. , 274 (1982) pp. 375–378 |
[a9] | T. Srinivasan, J. Wang, "Weak![]() |
[a10] | J. Wermer, "Dirichlet algebras" Duke Math. J. , 27 (1960) pp. 373–381 |
Dirichlet algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dirichlet_algebra&oldid=50275