In
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...
and related areas of
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the group algebra is any of various constructions to assign to a
locally compact group
In mathematics, a locally compact group is a topological group ''G'' for which the underlying topology is locally compact and Hausdorff. Locally compact groups are important because many examples of groups that arise throughout mathematics are lo ...
an
operator algebra
In functional analysis, a branch of mathematics, an operator algebra is an algebra of continuous linear operators on a topological vector space, with the multiplication given by the composition of mappings.
The results obtained in the study of ...
(or more generally a
Banach algebra
In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach spa ...
), such that representations of the algebra are related to representations of the group. As such, they are similar to the
group ring
In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is the set of elements of the give ...
associated to a discrete group.
The algebra ''Cc''(''G'') of continuous functions with compact support
If ''G'' is a
locally compact Hausdorff group, ''G'' carries an essentially unique left-invariant countably additive
Borel measure
In mathematics, specifically in measure theory, a Borel measure on a topological space is a measure that is defined on all open sets (and thus on all Borel sets). Some authors require additional restrictions on the measure, as described below.
F ...
''μ'' called a
Haar measure In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups.
This measure was introduced by Alfréd Haar in 1933, though ...
. Using the Haar measure, one can define a
convolution
In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions ( and ) that produces a third function (f*g) that expresses how the shape of one is ...
operation on the space ''C
c''(''G'') of complex-valued continuous functions on ''G'' with
compact support
In mathematics, the support of a real-valued function f is the subset of the function domain containing the elements which are not mapped to zero. If the domain of f is a topological space, then the support of f is instead defined as the smallest ...
; ''C
c''(''G'') can then be given any of various
norm
Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envir ...
s and the
completion will be a group algebra.
To define the convolution operation, let ''f'' and ''g'' be two functions in ''C
c''(''G''). For ''t'' in ''G'', define
:
The fact that
is continuous is immediate from the
dominated convergence theorem
In measure theory, Lebesgue's dominated convergence theorem provides sufficient conditions under which almost everywhere convergence of a sequence of functions implies convergence in the ''L''1 norm. Its power and utility are two of the primary ...
. Also
:
where the dot stands for the product in ''G''. ''C
c''(''G'') also has a natural
involution
Involution may refer to:
* Involute, a construction in the differential geometry of curves
* '' Agricultural Involution: The Processes of Ecological Change in Indonesia'', a 1963 study of intensification of production through increased labour inpu ...
defined by:
:
where Δ is the
modular function
In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group, and also satisfying a growth condition. The theory of ...
on ''G''. With this involution, it is a
*-algebra.
Theorem. With the norm:
:
''Cc''(''G'') becomes an involutive normed algebra In mathematics, a normed algebra ''A'' is an algebra over a field which has a sub-multiplicative norm:
: \forall x,y\in A\qquad \, xy\, \le\, x\, \, y\, .
Some authors require it to have a multiplicative identity 1 such that ║1║ = 1.
See also ...
with an approximate identity
In mathematics, particularly in functional analysis and ring theory, an approximate identity is a net in a Banach algebra or ring (generally without an identity) that acts as a substitute for an identity element.
Definition
A right approximate ...
.
The approximate identity can be indexed on a neighborhood basis of the identity consisting of compact sets. Indeed, if ''V'' is a compact neighborhood of the identity, let ''f
V'' be a non-negative continuous function supported in ''V'' such that
:
Then
''V'' is an approximate identity. A group algebra has an identity, as opposed to just an approximate identity, if and only if the topology on the group is the
discrete topology
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are '' isolated'' from each other in a certain sense. The discrete topology is the finest to ...
.
Note that for discrete groups, ''C
c''(''G'') is the same thing as the complex group ring C
'G''
The importance of the group algebra is that it captures the
unitary representation In mathematics, a unitary representation of a group ''G'' is a linear representation π of ''G'' on a complex Hilbert space ''V'' such that π(''g'') is a unitary operator for every ''g'' ∈ ''G''. The general theory is well-developed in case ''G'' ...
theory of ''G'' as shown in the following
Theorem. Let ''G'' be a locally compact group. If ''U'' is a strongly continuous unitary representation of ''G'' on a Hilbert space ''H'', then
:
is a non-degenerate bounded *-representation of the normed algebra ''Cc''(''G''). The map
:
is a bijection between the set of strongly continuous unitary representations of ''G'' and non-degenerate bounded *-representations of ''Cc''(''G''). This bijection respects unitary equivalence and strong containment
Strong may refer to:
Education
* The Strong, an educational institution in Rochester, New York, United States
* Strong Hall (Lawrence, Kansas), an administrative hall of the University of Kansas
* Strong School, New Haven, Connecticut, United St ...
. In particular, ''U'' is irreducible if and only if ''U'' is irreducible.
Non-degeneracy of a representation of ''C
c''(''G'') on a Hilbert space ''H''
means that
:
is dense in ''H''
.
The convolution algebra ''L''1(''G'')
It is a standard theorem of
measure theory
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures ( length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simil ...
that the completion of ''C
c''(''G'') in the ''L''
1(''G'') norm is isomorphic to the space
''L''1(''G'') of equivalence classes of functions which are integrable with respect to the
Haar measure In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups.
This measure was introduced by Alfréd Haar in 1933, though ...
, where, as usual, two functions are regarded as equivalent if and only if they differ only on a set of Haar measure zero.
Theorem. ''L''1(''G'') is a Banach *-algebra
In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach spa ...
with the convolution product and involution defined above and with the ''L''1 norm. ''L''1(''G'') also has a bounded approximate identity.
The group C*-algebra ''C*''(''G'')
Let C
'G''be the
group ring
In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is the set of elements of the give ...
of a
discrete group
In mathematics, a topological group ''G'' is called a discrete group if there is no limit point in it (i.e., for each element in ''G'', there is a neighborhood which only contains that element). Equivalently, the group ''G'' is discrete if and on ...
''G''.
For a locally compact group ''G'', the group
C*-algebra
In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra ''A'' of continuous ...
''C*''(''G'') of ''G'' is defined to be the C*-enveloping algebra of ''L''
1(''G''), i.e. the completion of ''C
c''(''G'') with respect to the largest C*-norm:
:
where ranges over all non-degenerate *-representations of ''C
c''(''G'') on Hilbert spaces. When ''G'' is discrete, it follows from the triangle inequality that, for any such , one has:
:
hence the norm is well-defined.
It follows from the definition that, when G is a discrete group, ''C*''(''G'') has the following
universal property
In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently fro ...
: any *-homomorphism from C
'G''to some B(''H'') (the C*-algebra of
bounded operator
In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y.
If X and Y are normed vector s ...
s on some
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
''H'') factors through the
inclusion map
In mathematics, if A is a subset of B, then the inclusion map (also inclusion function, insertion, or canonical injection) is the function \iota that sends each element x of A to x, treated as an element of B:
\iota : A\rightarrow B, \qquad \iot ...
:
:
The reduced group C*-algebra ''Cr*''(''G'')
The reduced group C*-algebra ''C
r*''(''G'') is the completion of ''C
c''(''G'') with respect to the norm
:
where
:
is the ''L''
2 norm. Since the completion of ''C
c''(''G'') with regard to the ''L''
2 norm is a Hilbert space, the ''C
r*'' norm is the norm of the bounded operator acting on ''L''
2(''G'') by convolution with ''f'' and thus a C*-norm.
Equivalently, ''C
r*''(''G'') is the C*-algebra generated by the image of the left regular representation on ''ℓ''
2(''G'').
In general, ''C
r*''(''G'') is a quotient of ''C*''(''G''). The reduced group C*-algebra is isomorphic to the non-reduced group C*-algebra defined above if and only if ''G'' is
amenable.
von Neumann algebras associated to groups
The group von Neumann algebra ''W*''(''G'') of ''G'' is the enveloping von Neumann algebra of ''C*''(''G'').
For a discrete group ''G'', we can consider the
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
ℓ
2(''G'') for which ''G'' is an
orthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, ...
. Since ''G'' operates on ℓ
2(''G'') by permuting the basis vectors, we can identify the complex group ring C
'G''with a subalgebra of the algebra of
bounded operator
In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y.
If X and Y are normed vector s ...
s on ℓ
2(''G''). The weak closure of this subalgebra, ''NG'', is a
von Neumann algebra
In mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. It is a special type of C*-algebra.
Von Neumann algeb ...
.
The center of ''NG'' can be described in terms of those elements of ''G'' whose
conjugacy class
In mathematics, especially group theory, two elements a and b of a group are conjugate if there is an element g in the group such that b = gag^. This is an equivalence relation whose equivalence classes are called conjugacy classes. In other wor ...
is finite. In particular, if the identity element of ''G'' is the only group element with that property (that is, ''G'' has the
infinite conjugacy class property In mathematics, a group is said to have the infinite conjugacy class property, or to be an ICC group, if the conjugacy class of every group element but the identity is infinite.
The von Neumann group algebra of a group is a factor if and only if th ...
), the center of ''NG'' consists only of complex multiples of the identity.
''NG'' is isomorphic to the
hyperfinite type II1 factor if and only if ''G'' is
countable
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; ...
,
amenable, and has the infinite conjugacy class property.
See also
*
Graph algebra
In mathematics, especially in the fields of universal algebra and graph theory, a graph algebra is a way of giving a directed graph an algebraic structure. It was introduced by McNulty and Shallon, and has seen many uses in the field of universa ...
*
Incidence algebra In order theory, a field of mathematics, an incidence algebra is an associative algebra, defined for every locally finite partially ordered set
and commutative ring with unity. Subalgebras called reduced incidence algebras give a natural constr ...
*
Hecke algebra of a locally compact group
In mathematics, a Hecke algebra of a locally compact group is an algebra of bi-invariant measures under convolution.
Definition
Let (''G'',''K'') be a pair consisting of a unimodular locally compact topological group ''G'' and a closed subgroup ...
*
Path algebra
In graph theory, a quiver is a directed graph where loops and multiple arrows between two vertices are allowed, i.e. a multidigraph. They are commonly used in representation theory: a representation of a quiver assigns a vector space  ...
*
Groupoid algebra In mathematics, the concept of groupoid algebra generalizes the notion of group ring, group algebra.
Definition
Given a groupoid (G, \cdot) (in the sense of a category theory, category with all arrows invertible) and a field (mathematics), field ...
*
Stereotype algebra In mathematics, a topological algebra A is an algebra and at the same time a topological space, where the algebraic and the topological structures are coherent in a specified sense.
Definition
A topological algebra A over a topological field K is a ...
*
Stereotype group algebra
In functional analysis and related areas of mathematics, the group algebra is any of various constructions to assign to a locally compact group an operator algebra (or more generally a Banach algebra), such that representations of the algebra are ...
*
Hopf algebra Hopf is a German surname. Notable people with the surname include:
*Eberhard Hopf (1902–1983), Austrian mathematician
*Hans Hopf (1916–1993), German tenor
*Heinz Hopf (1894–1971), German mathematician
*Heinz Hopf (actor) (1934–2001), Swedis ...
Notes
References
*
*
*
*
*
* {{PlanetMath attribution, id=3628, title=Group $C^*$-algebra
Algebras
C*-algebras
von Neumann algebras
Unitary representation theory
Harmonic analysis
Lie groups