Functions on Locally Compact Groups
نویسنده
چکیده
1. Background. The subject of this address is a branch of mathematics which may be regarded as a combination of the classical theory of representations of finite groups by matrices and that part of analysis centering around the theory of Fourier series and integrals. The connection between these two apparently diverse subjects arises simply enough from the fact that the real numbers and the real numbers modulo 27T form groups under addition. In one form of the theory of representations of a finite group G a central role is played by the so-called group ring or group algebra. This is usually defined as the set of all formal linear combinations of group elements C1S1+C2S2+ • • • +cnsn, where each S;£G and each d is a complex number. Two such expressions are added in the obvious manner and are multiplied by writing down the formal product and simplifying by means of the distributive law and the given multiplication of group elements. I t may also be defined (and this is the definition we shall use) as the vector space of all complex-valued functions on G with multiplication defined by the formula
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