1. Computing Norms in Group C ∗ -Algebras
- Author
-
Charles A. Akemann and Phillip A. Ostrand
- Subjects
Discrete mathematics ,Discrete group ,General Mathematics ,Hilbert space ,Regular representation ,Group algebra ,Square (algebra) ,Combinatorics ,symbols.namesake ,Subgroup ,symbols ,Linear combination ,Operator norm ,Mathematics - Abstract
Every discrete group G generates a C*-algebra 9Q1(G) of operators on the Hilbert space L2(G) of square summable complex valued functions on G. 9Q1(G), in which elements are written as formal (generally infinite) complex linear combinations ExE -GaXx, is the closure in operator norm of the group algebra L(G) of finite linear combinations, which acts on L2(G) by left multiplication (the left regular representation of G). Our main result is a formula for the explicit calculation of the norm of certain operators in L(G). We show that, if xl, . . ., x, satisfy a certain "freedom condition" (in particular if they freely generate a free subgroup of G), then for arbitrary complex scalars al,,... ,an5 a~ ~ ~ ~ ~~~Z Z21 i1 _Z JZ>A7
- Published
- 1976