Banach Algebra - Examples

Examples

The prototypical example of a Banach algebra is, the space of (complex-valued) continuous functions on a locally compact (Hausdorff) space that vanish at infinity. is unital if and only if X is compact. The complex conjugation being an involution, is in fact a C*-algebra. More generally, every C*-algebra is a Banach algebra.

• The set of real (or complex) numbers is a Banach algebra with norm given by the absolute value.
• The set of all real or complex n-by-n matrices becomes a unital Banach algebra if we equip it with a sub-multiplicative matrix norm.
• Take the Banach space Rn (or Cn) with norm ||x|| = max |x_i| and define multiplication componentwise: (x₁,...,x_n)(y₁,...,y_n) = (x₁y₁,...,x_ny_n).
• The quaternions form a 4-dimensional real Banach algebra, with the norm being given by the absolute value of quaternions.
• The algebra of all bounded real- or complex-valued functions defined on some set (with pointwise multiplication and the supremum norm) is a unital Banach algebra.
• The algebra of all bounded continuous real- or complex-valued functions on some locally compact space (again with pointwise operations and supremum norm) is a Banach algebra.
• The algebra of all continuous linear operators on a Banach space E (with functional composition as multiplication and the operator norm as norm) is a unital Banach algebra. The set of all compact operators on E is a closed ideal in this algebra.
• If G is a locally compact Hausdorff topological group and μ its Haar measure, then the Banach space L1(G) of all μ-integrable functions on G becomes a Banach algebra under the convolution xy(g) = ∫ x(h) y(h−1g) dμ(h) for x, y in L1(G).
• Uniform algebra: A Banach algebra that is a subalgebra of C(X) with the supremum norm and that contains the constants and separates the points of X (which must be a compact Hausdorff space).
• Natural Banach function algebra: A uniform algebra whose all characters are evaluations at points of X.
• C*-algebra: A Banach algebra that is a closed *-subalgebra of the algebra of bounded operators on some Hilbert space.
• Measure algebra: A Banach algebra consisting of all Radon measures on some locally compact group, where the product of two measures is given by convolution.