Abstract index group
Encyclopedia
In operator theory
, every Banach algebra
can be associated with a group called its abstract index group.
. Consider the identity component
or in other words the connected component
containing the identity 1 of A; G0 is a normal subgroup
of G. The quotient group
is the abstract index group of A. Because G0, being the component of an open set, is both open and closed in G, the index group is a discrete group
.
Let T denote the unit circle in the complex plane. The algebra C(T) of continuous functions on T is a Banach algebra, with the topology of uniform convergence. An element of C(T) is invertible if its image does not contain 0. The group G0 consists of elements homotopic, in G, to the identity in G, the constant function 1. Thus the index group ΛC(T) is the set of homotopy classes, indexed by the winding number
of its members. It is a countable discrete group. One can choose the functions fn(z) = zn as representatives of distinct homotopy classes. Thus ΛC(T) is isomorphic to the fundamental group
of T.
The Calkin algebra
K is the quotient C*-algebra of L(H) with respect to the compact operators
. Suppose π is the quotient map. By Atkinson's theorem
, an invertible elements in K is of the form π(T) where T is a Fredholm operator
s. The index group ΛK is again a countable discrete group. In fact, ΛK is isomorphic to the additive group of integers Z, via the Fredholm index. In other words, for Fredholm operators, the two notions of index coincide.
Operator theory
In mathematics, operator theory is the branch of functional analysis that focuses on bounded linear operators, but which includes closed operators and nonlinear operators.Operator theory also includes the study of algebras of operators....
, every Banach algebra
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 which at the same time is also a Banach space...
can be associated with a group called its abstract index group.
Definition
Let A be a Banach algebra and G the group of invertible elements in A. The set G is open and a topological groupTopological group
In mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology. A topological group is a mathematical object with both an algebraic structure and a...
. Consider the identity component
Identity component
In mathematics, the identity component of a topological group G is the connected component G0 of G that contains the identity element of the group...
- G0,
or in other words the connected component
Connected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...
containing the identity 1 of A; G0 is a normal subgroup
Normal subgroup
In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group. Normal subgroups can be used to construct quotient groups from a given group....
of G. The quotient group
Quotient group
In mathematics, specifically group theory, a quotient group is a group obtained by identifying together elements of a larger group using an equivalence relation...
- ΛA = G/G0
is the abstract index group of A. Because G0, being the component of an open set, is both open and closed in G, the index group is a discrete group
Discrete group
In mathematics, a discrete group is a group G equipped with the discrete topology. With this topology G becomes a topological group. A discrete subgroup of a topological group G is a subgroup H whose relative topology is the discrete one...
.
Examples
Let L(H) be the Banach algebra of bounded operators on a Hilbert space. The set of invertible elements in L(H) is path connected. Therefore ΛL(H) is the trivial group.Let T denote the unit circle in the complex plane. The algebra C(T) of continuous functions on T is a Banach algebra, with the topology of uniform convergence. An element of C(T) is invertible if its image does not contain 0. The group G0 consists of elements homotopic, in G, to the identity in G, the constant function 1. Thus the index group ΛC(T) is the set of homotopy classes, indexed by the winding number
Winding number
In mathematics, the winding number of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point...
of its members. It is a countable discrete group. One can choose the functions fn(z) = zn as representatives of distinct homotopy classes. Thus ΛC(T) is isomorphic to the fundamental group
Fundamental group
In mathematics, more specifically algebraic topology, the fundamental group is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other...
of T.
The Calkin algebra
Calkin algebra
In functional analysis, the Calkin algebra, named after John Wilson Calkin, is the quotient of B, the ring of bounded linear operators on a separable infinite-dimensional Hilbert space H, by the ideal K of compact operators....
K is the quotient C*-algebra of L(H) with respect to the compact operators
Compact operator on Hilbert space
In functional analysis, compact operators on Hilbert spaces are a direct extension of matrices: in the Hilbert spaces, they are precisely the closure of finite-rank operators in the uniform operator topology. As such, results from matrix theory can sometimes be extended to compact operators using...
. Suppose π is the quotient map. By Atkinson's theorem
Atkinson's theorem
In operator theory, Atkinson's theorem gives a characterization of Fredholm operators.- The theorem :Let H be a Hilbert space and L the set of bounded operators on H...
, an invertible elements in K is of the form π(T) where T is a Fredholm operator
Fredholm operator
In mathematics, a Fredholm operator is an operator that arises in the Fredholm theory of integral equations. It is named in honour of Erik Ivar Fredholm....
s. The index group ΛK is again a countable discrete group. In fact, ΛK is isomorphic to the additive group of integers Z, via the Fredholm index. In other words, for Fredholm operators, the two notions of index coincide.