operator theory In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear oper ...

, a branch of mathematics, every

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 ...

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 group In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two ...

. Consider the

identity component In mathematics, specifically group theory, the identity component of a group ''G'' refers to several closely related notions of the largest connected subgroup of ''G'' containing the identity element. In point set topology, the identity comp ...

:''G''₀, or in other words the connected component containing the identity 1 of ''A''; ''G''₀ is a

normal subgroup In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G ...

of ''G''. The

quotient group A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored" out). For exam ...

:Λ_''A'' = ''G''/''G''₀ is the abstract index group of ''A''. Because ''G''₀, being the component of an open set, is both open and closed in ''G'', the index group is 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 o ...

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, Λ_''L''(''H'') is the trivial group. Let T denote the unit circle in the complex plane. The algebra ''C''(T) of continuous functions from T to the

complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...

s is a Banach algebra, with the topology of uniform convergence. A function in ''C''(T) is invertible (meaning that it has a

pointwise In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value f(x) of some function f. An important class of pointwise concepts are the ''pointwise operations'', that is, operations defined ...

multiplicative inverse In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a fraction ''a''/''b ...

, not that it is an

invertible function In mathematics, the inverse function of a function (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by f^ . For a function f\colon X\ ...

) if it does not map any element of T to zero. The group ''G''₀ consists of elements

homotopic In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a defor ...

, in ''G'', to the identity in ''G'', the constant function 1. One can choose the functions ''f_n''(''z'') = ''zⁿ'' as representatives in G of distinct homotopy classes of maps T→T. Thus the index group Λ_''C''(T) is the set of homotopy classes, indexed by the

winding number In mathematics, the winding number or winding index 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, i.e., the curve's number of tu ...

of its members. Thus Λ_''C''(T) is isomorphic to the

fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...

of T. It is a countable discrete group. The

Calkin algebra In functional analysis, the Calkin algebra, named after John Williams Calkin, is the quotient of ''B''(''H''), the ring of bounded linear operators on a separable infinite-dimensional Hilbert space ''H'', by the ideal ''K''(''H'') of compact ope ...

''K'' is the quotient

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 continu ...

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 In mathematics, Fredholm operators are certain Operator (mathematics), operators that arise in the Fredholm theory of integral equations. They are named in honour of Erik Ivar Fredholm. By definition, a Fredholm operator is a bounded linear operat ...

s. The index group Λ_''K'' is again a countable discrete group. In fact, Λ_''K'' 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.

References

* Zhu, Kehe (1993). ''An Introduction to Operator Algebras'', CRC Press, Boca Raton, LA, {{Functional analysis Operator theory Banach algebras Discrete groups