Definitions

algebra

algebra

[al-juh-bruh]
algebra, branch of mathematics concerned with operations on sets of numbers or other elements that are often represented by symbols. Algebra is a generalization of arithmetic and gains much of its power from dealing symbolically with elements and operations (such as addition and multiplication) and relationships (such as equality) connecting the elements. Thus, a+a=2a and a+b=b+a no matter what numbers a and b represent.

Principles of Classical Algebra

In elementary algebra letters are used to stand for numbers. For example, in the equation ax2+bx+c=0, the letters a, b, and c stand for various known constant numbers called coefficients and the letter x is an unknown variable number whose value depends on the values of a, b, and c and may be determined by solving the equation. Much of classical algebra is concerned with finding solutions to equations or systems of equations, i.e., finding the roots, or values of the unknowns, that upon substitution into the original equation will make it a numerical identity. For example, x=-2 is a root of x2-2x-8=0 because (-2)2-2(-2)-8=4+4-8=0; substitution will verify that x=4 is also a root of this equation.

The equations of elementary algebra usually involve polynomial functions of one or more variables (see function). The equation in the preceding example involves a polynomial of second degree in the single variable x (see quadratic). One method of finding the zeros of the polynomial function f(x), i.e., the roots of the equation f(x)=0, is to factor the polynomial, if possible. The polynomial x2-2x-8 has factors (x+2) and (x-4), since (x+2)(x-4)=x2-2x-8, so that setting either of these factors equal to zero will make the polynomial zero. In general, if (x-r) is a factor of a polynomial f(x), then r is a zero of the polynomial and a root of the equation f(x)=0. To determine if (x-r) is a factor, divide it into f(x); according to the Factor Theorem, if the remainder f(r)—found by substituting r for x in the original polynomial—is zero, then (x-r) is a factor of f(x). Although a polynomial has real coefficients, its roots may not be real numbers; e.g., x2-9 separates into (x+3)(x-3), which yields two zeros, x=-3 and x=+3, but the zeros of x2+9 are imaginary numbers.

The Fundamental Theorem of Algebra states that every polynomial f(x)=anxn+an-1xn-1+ … +a1x+a0, with an≠0 and n≥1, has at least one complex root, from which it follows that the equation f(x)=0 has exactly n roots, which may be real or complex and may not all be distinct. For example, the equation x4+4x3+5x2+4x+4=0 has four roots, but two are identical and the other two are complex; the factors of the polynomial are (x+2)(x+2)(x+i)(x-i), as can be verified by multiplication.

Principles of Modern Algebra

Modern algebra is yet a further generalization of arithmetic than is classical algebra. It deals with operations that are not necessarily those of arithmetic and that apply to elements that are not necessarily numbers. The elements are members of a set and are classed as a group, a ring, or a field according to the axioms that are satisfied under the particular operations defined for the elements. Among the important concepts of modern algebra are those of a matrix and of a vector space.

Bibliography

See M. Artin, Algebra (1991).

B*-algebras are mathematical structures studied in functional analysis.

General Banach *-algebras

A Banach *-algebra A is a Banach algebra over the field of complex numbers, together with a map * : AA called involution which has the following properties:

  1. (x + y)* = x* + y* for all x, y in A.
  2. (lambda x)^* = bar{lambda}x^* for every λ in C and every x in A; here, bar{lambda} denotes the complex conjugate of λ.
  3. (xy)* = y* x* for all x, y in A.
  4. (x*)* = x for all x in A.

In most natural examples, one also has that the involution is isometric, i.e.

  • ||x*|| = ||x||,

B* algebras

A B*-algebra is a Banach *-algebra in which the involution satisfies the following further property:

  • ||x x*|| = ||x||2 for all x in A.

By a theorem of Gelfand and Naimark, given a B* algebra A there exists a Hilbert space H and an isometric *-homomorphism from A into the algebra B(H) of all bounded linear operators on H. Thus every B* algebra is isometrically *-isomorphic to a C*-algebra. Because of this, the term B* algebra is rarely used in current terminology, and has been replaced by the (overloading of) the term 'C* algebra'.

See also

References

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