In operator theory, a branch of mathematics, every Banach algebra 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 str ...

. Consider the identity component :''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 i ...

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 examp ...

: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.

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 form ...

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 Multiplication, multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a rat ...

, not that it is an invertible function) if it does not map any element of T to zero. The group ''G''₀ consists of elements homotopic, 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 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 oper ...

''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 continuous ...

of ''L''(''H'') with respect to the compact operators. Suppose π is the quotient map. By

Atkinson's theorem In operator theory, Atkinson's theorem (named for Frederick Valentine Atkinson) gives a characterization of Fredholm operators. The theorem Let ''H'' be a Hilbert space and ''L''(''H'') the set of bounded operators on ''H''. The following is the ...

, an invertible elements in ''K'' is of the form π(''T'') where ''T'' is a Fredholm operators. 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 mathematics, Fredholm operators are certain 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 operator ''T'' : ''X ...

. 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